Open Access

Well-posedness of fractional parabolic equations

Boundary Value Problems20132013:31

DOI: 10.1186/1687-2770-2013-31

Received: 2 October 2012

Accepted: 25 January 2013

Published: 18 February 2013

Abstract

In the present paper, we consider the abstract Cauchy problem for the fractional differential equation

d u ( t ) d t + D t 1 2 u ( t ) + A ( t ) u ( t ) = f ( t ) , 0 < t < 1 , u ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ1_HTML.gif
(1)

in an arbitrary Banach space E with the strongly positive operators A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif. The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2m th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of this problem is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2m th order multidimensional fractional parabolic equation are obtained.

MSC:65M12, 65N12.

Keywords

fractional parabolic equation Basset problem well-posedness coercive stability

1 Introduction

It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, etc. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [143] and the references given therein).

The role played by coercive stability inequalities (well-posedness) in the study of boundary value problems for parabolic partial differential equations is well known (see, e.g., [4451]). In the present paper, the initial value problem
d u ( t ) d t + D t 1 2 u ( t ) + A ( t ) u ( t ) = f ( t ) , 0 < t < 1 , u ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ2_HTML.gif
(2)

for the fractional differential equation in an arbitrary Banach space E with the linear (unbounded) operators A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif is considered. Here u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif and f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq3_HTML.gif are the unknown and the given functions, respectively, defined on [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq4_HTML.gif with values in E. The derivative u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq5_HTML.gif is understood as the limit in the norm of E of the corresponding ratio of differences. A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif is a given closed linear operator in E with the domain D ( A ( t ) ) = D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq6_HTML.gif, independent of t and dense in E. Finally, u ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq7_HTML.gif.

Here D t 1 2 = D 0 + 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq8_HTML.gif is the standard Riemann-Liouville derivative of order 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq9_HTML.gif. This fractional differential equation corresponds to the Basset problem [9]. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Recently, fractional Basset equations with independent in t operator coefficients A ( t ) = A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq10_HTML.gif have been studied extensively (see, e.g., [5256] and the references given therein).

In the present paper, the well-posedness of problem (2) with dependent in t operator coefficients A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif in spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of problems for 2m th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of initial value problem (2)
{ τ 1 ( u k u k 1 ) + A k u k + 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! u m u m 1 τ 1 2 = f k , f k = f ( t k ) , A k = A ( t k ) , t k = k τ , 1 k N , N τ = 1 , u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ3_HTML.gif
(3)

is presented. Here Γ ( k m + 1 2 ) = 0 t k m 1 2 e t d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq11_HTML.gif.

The paper is organized as follows. The well-posedness of problem (2) in spaces of smooth functions is established in Section 2. In Section 3 the coercive stability estimates for the solution of problems for 2m th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions are obtained. The well-posedness of (3) in difference analogues of spaces of smooth functions is established and the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2m th order multidimensional fractional parabolic equation are obtained in Section 4.

2 The well-posedness of problem (2)

A function u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif is called a solution of problem (2) if the following conditions are satisfied:
  1. (i)

    u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif is continuously differentiable on the segment [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq12_HTML.gif. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.

     
  2. (ii)

    The element u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif belongs to D ( A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq13_HTML.gif for all t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq14_HTML.gif and the function A ( t ) u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq15_HTML.gif is continuous on the segment [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq12_HTML.gif.

     
  3. (iii)

    u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif satisfies the equation and the initial condition (2).

     
A solution of problem (2) defined in this manner will from now on be referred to as a solution of problem (2) in the space C ( E ) = C ( [ 0 , 1 ] , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq16_HTML.gif of all continuous functions φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq17_HTML.gif defined on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq12_HTML.gif with values in E equipped with the norm
φ C ( E ) = max 0 t 1 φ ( t ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ4_HTML.gif
(4)

In this paper, positive constants, which can differ in time, are indicated with an M. On the other hand, M ( α , β , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq18_HTML.gif is used to focus on the fact that the constant depends only on α , β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq19_HTML.gif .

The well-posedness in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq20_HTML.gif of boundary value problem (2) means that the coercive inequality
u C ( E ) + A ( ) u C ( E ) M f C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ5_HTML.gif
(5)

is true for its solution u ( t ) C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq21_HTML.gif.

Suppose that for each t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq14_HTML.gif the operator A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq22_HTML.gif generates an analytic semigroup exp { s A ( t ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq23_HTML.gif ( s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq24_HTML.gif) with an exponentially decreasing norm, when s + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq25_HTML.gif, i.e., the following estimates
exp ( s A ( t ) ) E E , s A ( t ) exp ( s A ( t ) ) E E M e δ s ( s > 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ6_HTML.gif
(6)

hold for some M [ 1 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq26_HTML.gif, δ ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq27_HTML.gif. From this inequality it follows the operator A 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq28_HTML.gif exists and is bounded, and hence A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif is closed in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq20_HTML.gif.

Suppose that the operator A ( t ) A 1 ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq29_HTML.gif is Hölder continuous in t in the uniform operator topology for each fixed s, that is,
[ A ( t ) A ( τ ) ] A 1 ( s ) E E M | t τ | ε , 0 < ε 1 , 0 t , s , τ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ7_HTML.gif
(7)
An operator-valued function v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq30_HTML.gif, defined and strongly continuous jointly in t, s for 0 s < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq31_HTML.gif, is called a fundamental solution of (2) if
  1. (1)

    the operator v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq30_HTML.gif is strongly continuous in t and s for 0 s < t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq32_HTML.gif,

     
  2. (2)

    the following identity holds:

     
v ( t , s ) = v ( t , τ ) v ( τ , s ) , v ( t , t ) = I for  0 s τ t 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equa_HTML.gif
  1. (3)

    the operator v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq30_HTML.gif maps the region D into itself. The operator u ( t , s ) = A ( t ) v ( t , s ) A 1 ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq33_HTML.gif is bounded and strongly continuous in t and s for 0 s < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq31_HTML.gif,

     
  2. (4)
    on the region D the operator v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq30_HTML.gif is strongly differentiable relative to t and s, while
    v ( t , s ) t = A ( t ) v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ8_HTML.gif
    (8)
     
and
v ( t , s ) s = v ( t , s ) A ( s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ9_HTML.gif
(9)
Now, let us obtain the representation for the solution of problem (2). The initial value problem
d u d t + A ( t ) u ( t ) = F ( t ) , 0 < t < 1 , u ( 0 ) = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ10_HTML.gif
(10)
has a unique solution [54] and the following formula holds:
u ( t ) = v ( t , 0 ) u 0 + 0 t v ( t , s ) F ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ11_HTML.gif
(11)
Using u ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq7_HTML.gif and the formula F ( s ) = f ( s ) D s 1 2 u ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq34_HTML.gif, we get
u ( t ) = 0 t v ( t , s ) D s 1 2 u ( s ) d s + 0 t v ( t , s ) f ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ12_HTML.gif
(12)

Now, we will give a series of interesting lemmas and estimates concerning the fundamental solution v ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq30_HTML.gif of (2) which will be useful in the sequel.

Lemma 2.1 For any 0 s < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq31_HTML.gif and u D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq35_HTML.gif, the following identities hold:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ13_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ14_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ15_HTML.gif
(15)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ16_HTML.gif
(16)
Lemma 2.2 For any 0 s < t t + r 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq36_HTML.gif, 0 α 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq37_HTML.gif and 0 ε 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq38_HTML.gif, the following estimates hold:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ20_HTML.gif
(20)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ21_HTML.gif
(21)
Theorem 2.1 Let A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif be a strongly positive operator in a Banach space E and f ( t ) C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq39_HTML.gif. Then for the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq20_HTML.gif of initial value problem (2), the following stability inequality holds:
D t 1 2 u C ( E ) + u + A ( ) u C ( E ) M f C ( E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ22_HTML.gif
(22)
Proof Using formula (12), we get
u ( t ) = D t 1 2 u ( t ) + f ( t ) + 0 t A ( t ) v ( t , s ) D s 1 2 u ( s ) d s 0 t A ( t ) v ( t , s ) f ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ23_HTML.gif
(23)
Applying formula (23) and the formula
D t 1 2 u ( t ) = 0 t u ( p ) d p π ( t p ) 1 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ24_HTML.gif
(24)
we obtain
D t 1 2 u ( t ) = 0 t 1 π ( t s ) 1 2 ( D s 1 2 u ( s ) + f ( s ) ) d s + 0 t s t 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p D s 1 2 u ( s ) d s 0 t s t 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p f ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ25_HTML.gif
(25)
Let us first obtain the estimate
s t 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p E E M t s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ26_HTML.gif
(26)
for any 0 s < t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq31_HTML.gif. We have that
s t 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p = t + s 2 t 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ27_HTML.gif
(27)
+ s t + s 2 1 π ( t p ) 1 2 A ( p ) v ( p , s ) d p = J 1 + J 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ28_HTML.gif
(28)
Applying estimate (21), we get
J 1 E E M t + s 2 t 1 π ( t p ) 1 2 1 p s d p 2 M t s t + s 2 t 1 π ( t p ) 1 2 d p = M 1 t s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ29_HTML.gif
(29)
Now, we will estimate J 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq40_HTML.gif. We have that
J 2 = 1 π t s I v ( t + s 2 , s ) 2 π t s + s t + s 2 1 2 π ( t p ) 3 2 v ( p , s ) d p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ30_HTML.gif
(30)
Applying estimate (17), we get
J 2 E E 1 t s + M 2 t s + M s t + s 2 1 2 π ( t p ) 3 2 d p M 2 t s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ31_HTML.gif
(31)

Estimate (26) follows from estimates (29) and (31).

Now, let us first estimate z ( t ) = D t 1 2 u ( t ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq41_HTML.gif. Applying the triangle inequality and estimate (26), we get
z ( t ) M 0 t 1 t s z ( s ) d s + M 0 t 1 t s f ( s ) E d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ32_HTML.gif
(32)
for any t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq14_HTML.gif. Applying the above inequality and the integral inequality, we obtain
max 0 t 1 z ( t ) M max 0 t 1 f ( t ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ33_HTML.gif
(33)
Using the triangle inequality and equation (2), we get
max 0 t 1 u t + A ( t ) u ( t ) E [ max 0 t 1 f ( t ) E + max 0 t 1 D t 1 2 u ( t ) E ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ34_HTML.gif
(34)
M 1 max 0 t 1 f ( t ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ35_HTML.gif
(35)

Estimate (22) follows from estimates (33) and (35). Theorem 2.1 is proved. □

With the help of A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif, we introduce the fractional spaces E α ( E , A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq42_HTML.gif, 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif, consisting of all v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq44_HTML.gif for which the following norms are finite:
v E α = sup z > 0 z 1 α A ( t ) exp { z A ( t ) } v E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ36_HTML.gif
(36)

From (6) and (7) it follows that

Theorem 2.2 E α ( E , A ( t ) ) = E α ( E , A ( 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq45_HTML.gif for all 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif and 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq46_HTML.gif.

Problem (2) is not well posed in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq20_HTML.gif for arbitrary E. It turns out that a Banach space E can be restricted to a Banach space E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq47_HTML.gif in such a manner that the restricted problem (2) in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq47_HTML.gif will be well posed in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq48_HTML.gif. The role of E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq49_HTML.gifwill be played here by the fractional spaces E α = E α ( A ( t ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq50_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif).

Theorem 2.3 Suppose f ( t ) C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq51_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif). Suppose that assumptions (6) and (7) hold and 0 < α ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq52_HTML.gif. Then for the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif in C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq53_HTML.gif of problem (2), the coercive inequality
u C ( E α ) + A ( ) u C ( E α ) M α 1 ( 1 α ) 1 f C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ37_HTML.gif
(37)

holds.

Proof

By Theorem 2.1,
D t 1 2 u C ( E α ) M f C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ38_HTML.gif
(38)
for the solution of initial value problem (2). The proof of the estimate
A ( ) u C ( E α ) M α 1 ( 1 α ) 1 f C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ39_HTML.gif
(39)
for the solution of initial value problem (2) is based on formula (12), estimate (38) and the following estimates [54]:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ40_HTML.gif
(40)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ41_HTML.gif
(41)
Using equation (2) and the triangle inequality, we get
max 0 t 1 u ( t ) E α [ max 0 t 1 f ( t ) E α + max 0 t 1 A ( t ) u ( t ) E α + max 0 t 1 D t 1 2 u ( t ) E α ] M 1 α 1 ( 1 α ) 1 max 0 t 1 f ( t ) E α . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ42_HTML.gif
(42)

Estimate (37) follows from estimates (39) and (42). Theorem 2.3 is proved. □

Let us give, without proof, the following result.

Theorem 2.4 Suppose that assumption (6) holds. Suppose that the operator A ( t ) A 1 ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq29_HTML.gif is Hölder continuous in t in the uniform operator topology for each fixed s, that is,
[ A ( t ) A ( τ ) ] A 1 ( s ) E α E α M | t τ | ε , 0 < ε 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ43_HTML.gif
(43)
where M and ε are positive constants independent of t, s and τ for 0 t , s , τ T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq54_HTML.gif. Suppose f ( t ) C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq55_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif). Then for the solution u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq2_HTML.gif in C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq53_HTML.gif of problem (2), the coercive inequality
u C ( E α ) + A ( ) u C ( E α ) M α 1 ( 1 α ) 1 f C ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ44_HTML.gif
(44)

holds.

3 Applications

Now, we consider the applications of Theorems 2.1, 2.3 and 2.4.

First, the Cauchy problem on the range { 0 t 1 , x R n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq56_HTML.gif for the 2m-order multidimensional fractional parabolic equation is considered:
{ v ( t , x ) t + D t 1 2 v ( t , x ) + | r | = 2 m a r ( t , x ) | r | v ( t , x ) x 1 r 1 x n r n + σ v ( t , x ) = f ( t , x ) , 0 < t < 1 , v ( 0 , x ) = 0 , x R n , | r | = r 1 + + r n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ45_HTML.gif
(45)

where a r ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq57_HTML.gif and f ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq58_HTML.gif are given as sufficiently smooth functions. Here, σ is a sufficiently large positive constant.

Let us consider a differential operator with constant coefficients of the form
B = | r | = 2 m b r r 1 + + r n x 1 r 1 x n r n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ46_HTML.gif
(46)
acting on functions defined on the entire space R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq59_HTML.gif. Here r R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq60_HTML.gif is a vector with nonnegative integer components, | r | = r 1 + + r n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq61_HTML.gif. If φ ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq62_HTML.gif ( y = ( y 1 , , y n ) R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq63_HTML.gif) is an infinitely differentiable function that decays at infinity together with all its derivatives, then by means of the Fourier transformation, one establishes the equality
F ( B φ ) ( ξ ) = B ( ξ ) F ( φ ) ( ξ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ47_HTML.gif
(47)
Here the Fourier transform operator is defined by the following rule:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ48_HTML.gif
(48)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ49_HTML.gif
(49)
The function B ( ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq64_HTML.gif is called the symbol of the operator B and is given by
B ( ξ ) = | r | = 2 m b r ( i ξ 1 ) r 1 ( i ξ n ) r n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ50_HTML.gif
(50)
We will assume that the symbol
B t , x ( ξ ) = | r | = 2 m a r ( t , x ) ( i ξ 1 ) r 1 ( i ξ n ) r n , ξ = ( ξ 1 , , ξ n ) R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ51_HTML.gif
(51)
of the differential operator of the form
B t , x = | r | = 2 m a r ( t , x ) | r | x 1 r 1 x n r n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ52_HTML.gif
(52)
acting on functions defined on the space R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq59_HTML.gif, satisfies the inequalities
0 < M 1 | ξ | 2 m ( 1 ) m B t , x ( ξ ) M 2 | ξ | 2 m < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ53_HTML.gif
(53)

for ξ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq65_HTML.gif. Problem (45) has a unique smooth solution. This allows us to reduce problem (45) to the abstract Cauchy problem (2) in a Banach space E = C μ ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq66_HTML.gif of all continuous bounded functions defined on R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq59_HTML.gif satisfying the Hölder condition with the indicator μ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq67_HTML.gif with a strongly positive operator A t , x = B t , x + δ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq68_HTML.gif defined by (52) (see [57, 58]).

Theorem 3.1 For the solution of boundary problem (45), the following estimates are satisfied:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ54_HTML.gif
(54)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ55_HTML.gif
(55)

The proof of Theorem 3.1 is based on the abstract Theorems 2.1, 2.3, 2.4 and the coercivity inequality for an elliptic operator A t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq69_HTML.gif in C μ ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq70_HTML.gif and on the following theorem on the structure of the fractional spaces E α ( C μ ( R n ) , A t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq71_HTML.gif.

Theorem 3.2 E α ( C μ ( R n ) , A t , x ) = C 2 m α + μ ( R n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq72_HTML.gif for all 0 < 2 m α + μ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq73_HTML.gif and 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq46_HTML.gif [59].

Second, we consider the mixed boundary value problem for the fractional parabolic equation
{ v ( t , x ) t + D t 1 2 v ( t , x ) a ( t , x ) 2 v ( t , x ) x 2 + σ v ( t , x ) = f ( t , x ) , 0 < t < 1 , 0 < x < 1 , v ( 0 , x ) = 0 , 0 x 1 , u ( t , 0 ) = u ( t , 1 ) , u x ( t , 0 ) = u x ( t , 1 ) , 0 t 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ56_HTML.gif
(56)

where a ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq74_HTML.gif and f ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq58_HTML.gif are given sufficiently smooth functions and a ( t , x ) a > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq75_HTML.gif. Here, σ is a sufficiently large positive constant.

We introduce the Banach spaces C β [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq76_HTML.gif ( 0 < β < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq77_HTML.gif) of all continuous functions φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq78_HTML.gif satisfying the Hölder condition for which the following norms are finite:
φ C β [ 0 , 1 ] = φ C [ 0 , 1 ] + sup 0 x < x + τ 1 | φ ( x + τ ) φ ( x ) | τ β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ57_HTML.gif
(57)
where C [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq79_HTML.gif is the space of all continuous functions φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq78_HTML.gif defined on [0,1] with the usual norm
φ C [ 0 , 1 ] = max 0 x 1 | φ ( x ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ58_HTML.gif
(58)
It is known that the differential expression [60]
A t , x v = a ( t , x ) v ( t , x ) + σ v ( t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ59_HTML.gif
(59)

defines a positive operator A t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq69_HTML.gif acting in C β [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq76_HTML.gif with the domain C β + 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq80_HTML.gif and satisfying the conditions v ( t , 0 ) = v ( t , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq81_HTML.gif, v x ( t , 0 ) = v x ( t , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq82_HTML.gif. Therefore, we can replace the mixed problem (56) by the abstract boundary value problem (2). Using the results of Theorems 2.1, 2.3, 2.4, we can obtain the following theorem.

Theorem 3.3 For the solution of mixed problem (56), the following estimates are valid:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ60_HTML.gif
(60)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ61_HTML.gif
(61)

The proof of Theorem 3.3 is based on abstract Theorems 2.1, 2.3, 2.4 and on the following theorem on the structure of the fractional spaces E α ( C [ 0 , 1 ] , A t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq83_HTML.gif.

Theorem 3.4 E α ( C [ 0 , 1 ] , A t , x ) = C 2 α [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq84_HTML.gif for all 0 < α < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq85_HTML.gif, 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq46_HTML.gif [60].

4 The well-posedness of problem (3)

Let us first obtain the representation for the solution of problem (3). It is clear that the first order of accuracy difference scheme
τ 1 ( u k u k 1 ) + A k u k = F k , 1 k N , N τ = 1 , u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ62_HTML.gif
(62)
has a solution and the following formula holds:
u k = s = 1 k u τ ( k , s ) F s τ , 1 k N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ63_HTML.gif
(63)
where
u τ ( k , j ) = { R k R j + 1 , k > j , I , k = j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ64_HTML.gif
(64)
Here R k = ( I + τ A k ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq86_HTML.gif. Denote that
D τ 1 2 u k = 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! u m u m 1 τ 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ65_HTML.gif
(65)
Applying the formula F k = f k D τ 1 2 u k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq87_HTML.gif, we get
u k = s = 1 k u τ ( k , s ) D τ 1 2 u s τ + s = 1 k u τ ( k , s ) f s τ , 1 k N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ66_HTML.gif
(66)

So, formula (66) gives the representation for the solution of problem (3).

Let F τ ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq88_HTML.gif be the linear space of mesh functions φ τ = { φ k } 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq89_HTML.gif with values in the Banach space E. Next on F τ ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq88_HTML.gif we introduce the Banach space C τ ( E ) = C ( [ 0 , 1 ] τ , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq90_HTML.gif with the norm
φ τ C τ ( E ) = max 1 k N φ k E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ67_HTML.gif
(67)
Theorem 4.1 Let A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif be a strongly positive operator in a Banach space E. Then for the solution u τ = { u k } 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq91_HTML.gif in C τ ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq92_HTML.gif of initial value problem (3), the stability inequality
{ D τ 1 2 u k } 1 N C τ ( E ) + { τ 1 ( u k u k 1 ) + A k u k } 1 N C τ ( E ) M f τ C τ ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ68_HTML.gif
(68)

holds.

Proof Using formula (66), we get
τ 1 ( u k u k 1 ) = D τ 1 2 u k + s = 1 k A k u τ ( k , s ) D τ 1 2 u s τ + f k s = 1 k A k u τ ( k , s ) f s τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ69_HTML.gif
(69)
Applying formulas (69) and (65), we obtain
D τ 1 2 u k = 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! τ 1 2 [ D τ 1 2 u m + f m ] + 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! [ s = 1 m A k u τ ( k , s ) D τ 1 2 u s τ 3 2 s = 1 m A k u τ ( k , s ) f s τ 3 2 ] = 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! τ 1 2 [ D τ 1 2 u m + f m ] + 1 π s = 1 k m = s k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) D τ 1 2 u s τ 3 2 1 π s = 1 k m = s k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) f s τ 3 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ70_HTML.gif
(70)
Let us first obtain the estimate
1 π m = s k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) τ 1 2 E E M ( k s ) τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ71_HTML.gif
(71)
for any 1 s < k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq93_HTML.gif. We have that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ72_HTML.gif
(72)
Using estimates
A k u τ ( k , s ) E E M ( k s + 1 ) τ , u τ ( k , s ) E E M , 1 k N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ73_HTML.gif
(73)
and the following elementary inequality:
Γ ( k m + 1 2 ) ( k m ) ! 1 k m , 0 m < k , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ74_HTML.gif
(74)
we obtain
J 1 E E 1 π m = [ s + k 2 ] k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) E E τ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ75_HTML.gif
(75)
M 1 π m = [ s + k 2 ] k Γ ( k m + 1 2 ) ( k m ) ! 1 ( m s + 1 ) τ τ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ76_HTML.gif
(76)
2 M ( k s ) τ 1 π m = [ s + k 2 ] k τ ( k m ) τ M 1 ( k s ) τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ77_HTML.gif
(77)
Now, we will estimate J 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq40_HTML.gif. We have that
J 2 = 1 π Γ ( k s + 1 2 ) ( k s ) ! τ 1 2 1 π Γ ( k [ s + k 2 ] + 3 2 ) ( k [ s + k 2 ] + 1 ) ! u τ ( [ s + k 2 ] , s ) τ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ78_HTML.gif
(78)
+ 1 π m = s + 1 [ s + k 2 ] 1 [ Γ ( k m + 1 2 ) ( k m ) ! Γ ( k m + 3 2 ) ( k m + 1 ) ! ] u τ ( m 1 , s ) τ 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ79_HTML.gif
(79)
Applying estimates (73) and (74), we get
J 2 E E 1 π 1 ( k s ) τ + 1 π u τ ( [ s + k 2 ] , s ) E E 1 ( k [ s + k 2 ] + 1 ) τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ80_HTML.gif
(80)
+ 1 π m = s + 1 [ s + k 2 ] 1 | Γ ( k m + 1 2 ) ( k m ) ! Γ ( k m + 3 2 ) ( k m + 1 ) ! | u τ ( m 1 , s ) E E τ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ81_HTML.gif
(81)
1 π 1 ( k s ) τ + 1 π M 2 ( k s ) τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ82_HTML.gif
(82)
+ M 1 2 π m = s + 1 [ s + k 2 ] 1 τ ( k m + 1 ) τ ( k m ) τ M 2 ( k s ) τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ83_HTML.gif
(83)

Estimate (71) follows from estimates (75) and (80).

Now, let us first estimate z k = D τ 1 2 u k E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq94_HTML.gif. Applying the triangle inequality and estimate (71), we get
z k 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! τ 1 2 [ z m + f m E ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ84_HTML.gif
(84)
+ 1 π s = 1 k m = s k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) E E z s τ 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ85_HTML.gif
(85)
+ 1 π s = 1 k m = s k Γ ( k m + 1 2 ) ( k m ) ! A m u τ ( m , s ) E E f s E τ 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ86_HTML.gif
(86)
M 3 s = 1 k 1 1 ( k s ) τ τ [ z s + f s E ] + M 4 [ z k + f k E ] τ 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ87_HTML.gif
(87)
for any k = 1 , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq95_HTML.gif. Applying the above inequality and the difference analogue of the integral inequality, we obtain
{ D τ 1 2 u k } 1 N C τ ( E ) M { f k } 1 N C τ ( E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ88_HTML.gif
(88)
Using the triangle inequality and equation (3), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ89_HTML.gif
(89)

Estimate (68) follows from estimates (88) and (89). Theorem 4.1 is proved. □

With the help of A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq1_HTML.gif, we introduce the fractional spaces E α = E α ( E , A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq96_HTML.gif, 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif, consisting of all v E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq44_HTML.gif for which the following norms are finite:
v E α = sup λ > 0 λ α A ( t ) ( λ + A ( t ) ) 1 v E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ90_HTML.gif
(90)

From (73) it follows that

Theorem 4.2 E α ( E , A ( t ) ) = E α ( E , A ( 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq97_HTML.gif for all 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif and 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq46_HTML.gif.

Problem (3) is not well posed in C τ ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq92_HTML.gif for arbitrary E. It turns out that a Banach space E can be restricted to a Banach space E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq47_HTML.gif in such a manner that the restricted problem (3) in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq47_HTML.gif will be well posed in C ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq48_HTML.gif. The role of E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq47_HTML.gif will be played here by the fractional spaces E α = E α ( A ( t ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq98_HTML.gif ( 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq43_HTML.gif).

Theorem 4.3 Suppose that assumptions (6) and (7) hold and 0 < α ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq99_HTML.gif. Then for the solution u τ = { u k } 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq100_HTML.gif in C τ ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq101_HTML.gif of initial value problem (3), the coercive stability inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ91_HTML.gif
(91)

holds.

Proof

By Theorem 4.1,
{ D τ 1 2 u k } 1 N C τ ( E α ) M f τ C τ ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ92_HTML.gif
(92)
for the solution of initial value problem (3). The proof of the estimate
{ A k u k } 1 N C τ ( E α ) M α 1 ( 1 α ) 1 f τ C τ ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ93_HTML.gif
(93)
for the solution of initial value problem (3) is based on estimate (92) and the following estimates [51]:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ94_HTML.gif
(94)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ95_HTML.gif
(95)
Using the triangle inequality and equation (3), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ96_HTML.gif
(96)

Estimate (91) follows from estimates (93) and (96). Theorem 4.3 is proved. □

Let us give, without proof, the following result.

Theorem 4.4 Suppose that assumptions (6) and (43) hold. Then for the solution u τ = { u k } 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq91_HTML.gif in C τ ( E α ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq102_HTML.gif of initial value problem (3), the coercive stability inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ97_HTML.gif
(97)

holds.

Note that by passing to the limit for τ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq103_HTML.gif, one can recover Theorems 2.1-2.3 and 2.4.

5 Applications

Now, we consider the applications of Theorems 4.1, 4.3 and 4.4.

First, initial value problem (45) is considered. The discretization of problem (45) is carried out in two steps. In the first step, the grid space R h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq104_HTML.gif ( 0 < h h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq105_HTML.gif) is defined as the set of all points of the Euclidean space R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq59_HTML.gif whose coordinates are given by
x k = s k h , s k = 0 , ± 1 , ± 2 , , k = 1 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ98_HTML.gif
(98)
The difference operator A h t , x = B h t , x + σ I h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq106_HTML.gif is assigned to the differential operator A x = B x + σ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq107_HTML.gif, defined by (52). The operator
B h t , x = h 2 m 2 m | s | S b s t , x Δ 1 s 1 Δ 1 + s 2 Δ n s 2 n 1 Δ n + s 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ99_HTML.gif
(99)
acts on functions defined on the entire space R h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq104_HTML.gif. Here s R 2 n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq108_HTML.gif is a vector with nonnegative integer coordinates,
Δ k ± f h ( x ) = ± ( f h ( x ± e k h ) f h ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ100_HTML.gif
(100)

where e k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq109_HTML.gif is the unit vector of the axis x k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq110_HTML.gif.

An infinitely differentiable function φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq111_HTML.gif of the continuous argument x R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq112_HTML.gif that is continuous and bounded together with all its derivatives is said to be smooth. We say that the difference operator A h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq113_HTML.gif is a λ th order ( λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq114_HTML.gif) approximation of the differential operator A t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq69_HTML.gif if the inequality
sup x R h n | A h t , x φ ( x ) A t , x φ ( x ) | M ( φ ) h λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ101_HTML.gif
(101)

holds for any smooth function φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq115_HTML.gif. The coefficients b s t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq116_HTML.gif are chosen in such a way that the operator A h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq113_HTML.gif approximates in a specified way the operator A t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq69_HTML.gif. It will be assumed that the operator A h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq113_HTML.gif approximates the differential operator A t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq69_HTML.gif with any prescribed order [57, 58].

The function A t , x ( ξ h , h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq117_HTML.gif is obtained by replacing the operator Δ k ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq118_HTML.gif in the right-hand side of equality (99) with the expression ± ( exp { ± i ξ k h } 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq119_HTML.gif, respectively, and is called the symbol of the difference operator B h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq120_HTML.gif.

It will be assumed that for | ξ k h | π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq121_HTML.gif and fixed x, the symbol A t , x ( ξ h , h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq122_HTML.gif of the operator B h t , x = A h t , x σ I h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq123_HTML.gif satisfies the inequalities
( 1 ) m A t , x ( ξ h , h ) M | ξ | 2 m , | arg A t , x ( ξ h , h ) | ϕ < ϕ 0 π 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ102_HTML.gif
(102)
Suppose that the coefficient b s x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq124_HTML.gif of the operator B h t , x = A h t , x σ I h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq123_HTML.gif is bounded and satisfies the inequalities
| b s t , x + e k h b s t , x | M h ϵ , x R h n , ϵ ( 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ103_HTML.gif
(103)
With the help of A h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq113_HTML.gif, we arrive at the nonlocal boundary value problem
{ d v h ( t , x ) d t + D t 1 2 v h ( t , x ) + A h t , x v h ( t , x ) = f h ( t , x ) , 0 < t < 1 , x R h n , v h ( 0 , x ) = 0 , x R h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ104_HTML.gif
(104)

for an infinite system of ordinary differential equations.

In the second step, problem (104) is replaced by the difference scheme
{ u k h ( x ) u k 1 h ( x ) τ + 1 π m = 1 k Γ ( k m + 1 2 ) ( k m ) ! u m h u m 1 h τ 1 2 + A h k , x u k h = f k h ( x ) , f k h ( x ) = f h ( t k , x ) , t k = k τ , 1 k N 1 , N τ = 1 , x R h n , u 0 h ( x ) = 0 , x R h n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ105_HTML.gif
(105)
Based on the number of corollaries of the abstract theorems given in the above, to formulate the result, one needs to introduce the spaces C h = C ( R h n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq125_HTML.gif and C h β = C β ( R h n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq126_HTML.gif of all bounded grid functions u h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq127_HTML.gif defined on R h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq104_HTML.gif, equipped with the norms
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ106_HTML.gif
(106)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ107_HTML.gif
(107)
Theorem 5.1 Suppose that assumptions (102) and (103) for the operator A h k , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq128_HTML.gif hold. Then, the solutions of difference scheme (105) satisfy the following stability estimates:
max 1 k N D τ 1 2 u k h C h μ M 1 ( μ ) max 1 k N f k h C h μ , 0 μ 1 , { τ 1 ( u k h u k 1 h ) } 1 N C τ ( C h μ + 2 m α ) + { A k u k } 1 N C τ ( C h μ + 2 m α ) M ( α , μ ) max 1 k N f k h C h μ + 2 m α , 0 < 2 m α + μ < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ108_HTML.gif
(108)

The proof of Theorem 5.1 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq129_HTML.gif defined by (114) in C h μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq130_HTML.gif and on the following two theorems on the coercivity inequality for the solution of the elliptic difference equation in C h β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq131_HTML.gif and on the structure of the fractional space E α ( C h , A h x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq132_HTML.gif.

Theorem 5.2 Suppose that assumptions (102) and (103) for the operator A h k , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq128_HTML.gif hold. Then for the solutions of the elliptic difference equation
A h k , x u h ( x ) = ω h ( x ) , x R h n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ109_HTML.gif
(109)
the estimates [54]
2 m | s | S h 2 m Δ 1 s 1 Δ 1 + s 2 Δ n s 2 n 1 Δ n + s 2 n u h C h β M ( σ , β ) ω h C h β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ110_HTML.gif
(110)

are valid.

Theorem 5.3 Suppose that assumptions (102) and (103) for the operator A h k , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq128_HTML.gif hold. Then for any 0 < α < 1 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq133_HTML.gif, the norms in the spaces E α ( C h , A h x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq132_HTML.gif and C h 2 m α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq134_HTML.gif are equivalent uniformly in h [51].

Second, we consider mixed boundary value problem (56). The discretization of problem (56) is carried out in two steps. In the first step, let us define the grid space
[ 0 , 1 ] h = { x : x r = r h , 0 r K , K h = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ111_HTML.gif
(111)
We introduce the Banach space C h β = C β ( [ 0 , 1 ] h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq135_HTML.gif ( 0 < β < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq77_HTML.gif) of the grid functions φ h ( x ) = { φ r } 1 K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq136_HTML.gif defined on [ 0 , 1 ] h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq137_HTML.gif, equipped with the norm
φ h C h β = φ h C h + sup 1 k < k + τ K 1 | φ k + r φ k | τ β , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ112_HTML.gif
(112)
where C h = C ( [ 0 , 1 ] h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq138_HTML.gif is the space of the grid functions φ h ( x ) = { φ r } 1 K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq139_HTML.gif defined on [ 0 , 1 ] h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq137_HTML.gif, equipped with the norm
φ h C h = max 1 k K 1 | φ k | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ113_HTML.gif
(113)
To the differential operator A generated by problem (56), we assign the difference operator A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq129_HTML.gif by the formula
A h t , x φ h ( x ) = { ( a ( t , x ) φ x ) x , r + δ φ r } 1 K 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ114_HTML.gif
(114)
acting in the space of grid functions φ h ( x ) = { φ r } 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq140_HTML.gif satisfying the conditions φ 0 = φ K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq141_HTML.gif, φ 1 φ 0 = φ K φ K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq142_HTML.gif. With the help of A h x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq129_HTML.gif, we arrive at the initial boundary value problem
{ d v h ( t , x ) d t + D t 1 2 v h ( t , x ) + A h t , x v h ( t , x ) = f h ( t , x ) , 0 < t < 1 , x [ 0 , 1 ] h , v h ( 0 , x ) = 0 , x [ 0 , 1 ] h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ115_HTML.gif
(115)
for an infinite system of ordinary fractional differential equations. In the second step, we replace problem (115) by difference scheme (3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ116_HTML.gif
(116)
Theorem 5.4 Let τ and h be sufficiently small numbers. Then, the solutions of difference scheme (116) satisfy the following stability estimates:
max 1 k N D τ 1 2 u k h C h μ M 1 ( μ ) max 1 k N f k h C h μ , 0 μ 1 , { τ 1 ( u k h u k 1 h ) } 1 N C τ ( C h μ + 2 α ) + { A k u k } 1 N C τ ( C h μ + 2 α ) M ( α , μ ) max 1 k N f k h C h μ + 2 α , 0 < 2 α + μ < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_Equ117_HTML.gif
(117)

The proof of Theorem 5.4 is based on the abstract Theorems 4.1, 4.3, 4.4 and the strong positivity of the operator A h t , x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq113_HTML.gif defined by (114) in C h μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq130_HTML.gif and on the following theorem on the structure of the fractional space E α ( C h , A h t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq143_HTML.gif.

Theorem 5.5 For any 0 < α < 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq85_HTML.gif, the norms in the spaces E α ( C h , A h t , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq144_HTML.gif and C h 2 α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq145_HTML.gif are equivalent uniformly in h and t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-31/MediaObjects/13661_2012_Article_290_IEq146_HTML.gif [60].

Declarations

Acknowledgements

The author would like to thank Prof. P. E. Sobolevskii for his helpful suggestions to the improvement of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, Fatih University

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