Open Access

r-Modified Crank-Nicholson difference scheme for fractional parabolic PDE

Boundary Value Problems20142014:76

DOI: 10.1186/1687-2770-2014-76

Received: 2 December 2013

Accepted: 18 March 2014

Published: 31 March 2014

Abstract

The second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation are presented using by r-modified Crank-Nicholson difference scheme. Stability estimate for the solution of this difference scheme is obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations. Numerical results for this scheme and the Crank-Nicholson scheme are compared in test examples.

1 Introduction

At present, there is a huge number of theoretical and applied works devoted to the study of fractional differential equations. Solutions of various problems for fractional differential equations can be found, for example, in the monographs of Podlubny [1], Kilbas, Srivastava, and Trujillo [2], Diethelm [3], and in [411]. These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations.

Many problems in fluid flow, dynamical and diffusion processes, control theory, mechanics, and other areas of physics can be reduced fractional partial differential equations.

In [12] the simple connection of fractional derivatives with fractional powers of first order differential operator was presented. This approach is important to obtain the formula for the fractional difference derivative. Presently, many mathematicians apply this approach and operator tools to investigate various problems for fractional partial differential equations which appear in applied problems (see, e.g., [1320] and the references therein).

In previous paper [17] authors investigated stability estimates for Crank-Nicholson schemes for the Dirichlet problem for the fractional parabolic equation
{ u ( t , x ) t + D t 1 / 2 u ( t , x ) p = 1 m ( a p ( x ) u x p ) x p + σ u ( t , x ) = f ( t , x ) , x = ( x 1 , , x m ) Ω , 0 < t < T , u ( t , x ) = 0 , x S , u ( 0 , x ) = 0 , x Ω ¯ .
(1.1)
Here D t 1 / 2 = D 0 + 1 / 2 is the standard Riemann-Liouville’s derivative of order 1 / 2 and Ω is the open cube in the m-dimensional Euclidean space
R m : { x Ω : x = ( x 1 , , x m ) ; 0 < x p < 1 , 1 p m }

with boundary S, Ω ¯ = Ω S , a p ( x ) ( x Ω ) and f ( t , x ) ( t ( 0 , T ) , x Ω ) are given smooth functions and a p ( x ) a > 0 , σ > 0 .

In [18] the authors investigated stability estimates for Crank-Nicholson schemes for the Neumann problem for the fractional parabolic equation
{ u ( t , x ) t + D t 1 / 2 u ( t , x ) p = 1 m ( a p ( x ) u x p ) x p + σ u ( t , x ) = f ( t , x ) , x = ( x 1 , , x m ) Ω , 0 < t < T , u ( t , x ) = 0 , x S , u ( t , x ) n = 0 , x Ω ¯ .
(1.2)

The role played by stability inequalities (well posedness) in the study of boundary-value problems for parabolic partial differential equations is well known (see, e.g., [2126]).

In the present paper, we consider an r-modified Crank-Nicholson difference scheme of the above mentioned two problems (1.1), (1.2). This r-modified scheme is of the second order of accuracy in t and in space variables difference schemes for the approximate solution of problems. The stability estimate for the solution of this difference scheme is established. We use a procedure of a modified Gauss elimination method for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations.

2 Stability of difference scheme

Let us define the grid space
{ Ω ¯ h = { x = x p = ( h 1 p 1 , , h m p m ) , p = ( p 1 , , p m ) , Ω ¯ h = 0 p j M j , h j M j = 1 , j = 1 , , m } , Ω h = Ω ¯ h Ω , S h = Ω ¯ h S .
We introduce the Hilbert space L 2 h = L 2 ( Ω ¯ h ) of the grid function φ h ( x ) = { φ ( h 1 j 1 , , h m j m ) } defined on Ω ¯ , equipped with the norm
φ h L 2 ( Ω ¯ h ) = ( x Ω ¯ h | φ h ( x ) | 2 h 1 h m ) 1 / 2 .
To the differential operator A x generated by problem (1.1) or (1.2), respectively, we assign the difference operator A h x by the formula
A h x u h = p = 1 m ( a p ( x ) u x ¯ p h ) x p , j p + σ u h
(2.1)
acting in the space of grid functions u h ( x ) , satisfying the conditions u h ( x ) = 0 or u ( t , x ) n = 0 ( x S h ). It is known that A h x is a self-adjoint positive definite operator in L 2 ( Ω ¯ h ) . Here,
φ x p , j p = 1 h p ( φ ( h 1 j 1 , , h j ( j j + 1 ) , , h m j m ) φ ( h 1 j 1 , , h j j j , , h m j m ) ) , φ x ¯ p , j p = 1 h p ( φ ( h 1 j 1 , , h j j j , , h m j m ) φ ( h 1 j 1 , , h j ( j j 1 ) , , h m j m ) ) .
With the help of A h x , 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 x v h ( t , x ) = f h ( t , x ) , 0 < t < T , x Ω h , v h ( 0 , x ) = 0 , x Ω ¯
(2.2)

for a finite system of ordinary fractional differential equations.

We denote
d = 2 π τ , λ ( q ) = q + 1 / 2 q 1 / 2 , μ ( q ) = 1 3 ( ( q + 1 / 2 ) 3 / 2 ( q 1 / 2 ) 3 / 2 ) .
Applying the second order of the approximation formula
D t k τ 2 1 / 2 u k = d i = 0 k α k , i u i
(2.3)
for
D t k τ 2 1 / 2 u ( t k τ / 2 ) = 1 Γ ( 1 / 2 ) 0 t k τ / 2 ( t k τ / 2 s ) 1 / 2 u ( s ) d s
(see [16]) and the Crank-Nicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to t and to x. Here,
α 1 , 0 = 2 / 3 , α 1 , 1 = 2 / 3 , k = 1 , α 2 , 0 = 2 6 / 5 , α 2 , 1 = 6 / 5 , α 2 , 2 = 6 / 5 , k = 2 , α k , 0 = ( k 2 ) λ ( k 2 ) + μ ( k 2 ) , α k , 1 = ( k 3 ) λ ( k 3 ) + μ ( k 3 ) + ( 3 2 k ) λ ( k 2 ) 2 μ ( k 2 ) , α k , i = ( k i 2 ) λ ( k i 2 ) + μ ( k i 2 ) α k , i = + ( 2 i 2 k + 1 ) λ ( k i 1 ) 2 μ ( k i 1 ) α k , i = + ( k i + 1 ) λ ( k i ) + μ ( k i ) , 2 i k 3 , α k , k 2 = 3 λ ( 1 ) 2 μ ( 1 ) + 3 λ ( 2 ) + μ ( 2 ) 1 6 2 , α k , k 1 = 2 λ ( 1 ) + μ ( 1 ) 2 3 , 3 k N .
Now, we introduce the second order accuracy r-modified Crank-Nicholson difference scheme in the following form:
{ u k h ( x ) u k 1 h ( x ) τ + D t k 1 / 2 u k h ( x ) + A h x u k h ( x ) = f k h ( x ) , x Ω ¯ h , 1 k r , u k h ( x ) u k 1 h ( x ) τ + D t k 1 / 2 u k h ( x ) + 1 2 A h x [ u k h ( x ) + u k 1 h ( x ) ] = f k h ( x ) , x Ω ¯ h , r + 1 k N , f k h ( x ) = f ( t k τ 2 , x ) , N τ = T , t k = k τ , 1 k N , u 0 h ( x ) = 0 , x Ω ¯ h
(2.4)

for the approximate solution of problem (2.2).

Theorem 2.1 Let τ and | h | be sufficiently small positive numbers. Then the solutions of the difference scheme (2.4) satisfy the following stability estimate:
max 1 k N u k h L 2 h C max 1 k N f k h L 2 h ,
(2.5)

where C does not depend on τ, h, and f k h , 1 k N .

Proof Consider the difference scheme (2.4). We have
u k h ( x ) = { s = 1 k R 1 k s + 1 τ [ f s h ( x ) D t s 1 / 2 u s h ( x ) ] , 1 k r , s = 1 k r B k r R 1 r s + 1 τ [ f s h ( x ) D t s 1 / 2 u s h ( x ) ] + l = 1 k r B k r l R τ [ f r + l h ( x ) D t r + l 1 / 2 u r + l h ( x ) ] , r + 1 k N ,
(2.6)
where
R 1 1 = ( I + τ A h x ) , R 1 = ( I + τ 2 A h x ) , B = R ( I τ 2 A h x ) .
We obtain
max 1 k N D t k 1 / 2 u k h L 2 h M max 1 k N f k h L 2 h .
(2.7)
Let us write z k = D t k 1 / 2 u k h L 2 h . Using (2.6), we have
D t k 1 / 2 u k h ( x ) = { d i = 0 k α k , i [ s = 1 i R 1 i s + 1 τ ( f s h ( x ) D t s 1 / 2 u s h ( x ) ) ] , 1 k r , d i = 0 k α k , i [ s = 1 i r B i r R 1 r s + 1 τ ( f s h ( x ) D t s 1 / 2 u s h ( x ) ) + l = 1 i r B i r l R τ ( f r + l h ( x ) D t r + l 1 / 2 u r + l h ( x ) ) ] , r + 1 k N .
Now, let us estimate z k = D t k 1 / 2 u k h L 2 h , 1 k N . From the triangle inequality, it follows that
z 1 α 1 , 1 R 1 L 2 h L 2 h ( f 1 h ( x ) L 2 h + D τ 1 / 2 u 1 L 2 h ) τ M 1 τ ( f 1 h ( x ) L 2 h + z 1 ) .
(2.8)
Applying the triangle inequality and the estimates [24]
R 1 k L 2 h L 2 h M k τ , B i r R 1 r s + 1 L 2 h L 2 h M k τ , B i r l R L 2 h L 2 h M , 1 k N ,
(2.9)
we have
z k i = 0 k s = 1 i α k , i R 1 i s + 1 L 2 h L 2 h ( f s h ( x ) L 2 h + D t s 1 / 2 u s h ( x ) L 2 h ) τ z k M 2 s = 1 k 1 [ 1 ( k s ) τ τ ( f s h ( x ) L 2 h + z s ) ] z k + M 3 ( f k h ( x ) L 2 h + z k ) τ , 2 k r ,
(2.10)
z k i = 0 k s = 1 i r α k , i B i r R 1 r s + 1 L 2 h L 2 h ( f s h ( x ) L 2 h + D t s 1 / 2 u s h ( x ) L 2 h ) τ z k + i = 0 k l = 1 i r α k , i B i r l R L 2 h L 2 h ( f r + l h ( x ) L 2 h + D t r + l 1 / 2 u r + l h ( x ) L 2 h ) τ z k M 2 s = 1 k 1 [ 1 ( k s ) τ τ ( f s h ( x ) L 2 h + z s ) ] z k + M 3 ( f k h ( x ) L 2 h + z k ) τ z k + M 2 ′′ l = 1 k 1 [ 1 ( k s ) τ τ ( f r + l h ( x ) L 2 h + z r + l ) ] z k + M 3 ′′ ( f k h ( x ) L 2 h + z k ) τ , r + 1 k N .
(2.11)
Hence, applying the difference analog of the integral inequality and inequalities (2.8), (2.10), and (2.11), we get
{ z k } 1 N L 2 h = { D τ 1 2 u k } 1 N L 2 h M f τ L 2 h .
(2.12)

The proof of estimate (2.5) for the solution of (2.4) follows from (2.6), (2.9), and (2.12). Note that M , M 0 are independent from τ, h, and f k h , 1 k N . Theorem 2.1 is proved. □

3 Numerical analysis

We consider two examples for numerical results.

Example 3.1 We consider the following initial boundary value problem with Dirichlet condition for the one-dimensional fractional parabolic partial differential equation:
{ u ( t , x ) t + D t 1 / 2 u ( t , x ) x ( ( 1 + x ) u ( t , x ) x ) = f ( t , x ) , f ( t , x ) = [ 3 t 2 + 3 π t 4 + ( 1 + x ) π 2 t 3 / 2 ] sin π x f ( t , x ) = π t 3 / 2 cos π x , 0 < t < 1 , 0 < x < 1 , u ( t , 0 ) = u ( t , 1 ) = 0 , 0 t 1 , u ( 0 , x ) = 0 , 0 x 1 .
(3.1)
It is clear that the exact solution of problem (3.1) is
u ( t , x ) = t 3 / 2 sin π x .
Applying the r-modified Crank-Nicholson difference scheme (2.4), we get
{ u n k u n k 1 τ + D t k τ 2 1 / 2 u n k [ ( 1 + x n ) u n + 1 k 2 u n k + u n 1 k h 2 + u n + 1 k u n 1 k 2 h ] = φ n k , 1 k r , u n k u n k 1 τ + D t k τ 2 1 / 2 u n k 1 2 [ ( 1 + x n ) u n + 1 k 2 u n k + u n 1 k h 2 + u n + 1 k u n 1 k 2 h + ( 1 + x n ) u n + 1 k 1 2 u n k 1 + u n 1 k 1 h 2 + u n + 1 k 1 u n 1 k 1 2 h ] = φ n k , r + 1 k N , φ n k = f ( t k τ 2 , x n ) , t k = k τ , x n = n h , 1 k N , 1 n M 1 , u 0 k = u M k = 0 , 0 k N , u n 0 = 0 , 0 n M ,
where D t k τ 2 1 / 2 u n k is defined by (2.3) for any n, 1 n M 1 . We can rewrite it in the system of equations with matrix coefficients
{ A U n + 1 + B U n + C U n 1 = D φ n , 1 n M 1 , U 0 = 0 ˜ , U M = 0 ˜ .
(3.2)
Here and in the sequel 0 ˜ is the ( N + 1 ) × 1 zero matrix,
A = [ 0 0 0 0 0 0 0 0 0 0 0 a n 0 0 0 0 0 0 0 0 0 0 a n 0 0 0 0 0 0 0 0 0 0 a n 0 0 0 0 0 0 0 0 0 0 a n 0 0 0 0 0 0 0 0 0 z n z n 0 0 0 0 0 0 0 0 0 z n z n 0 0 0 0 0 0 0 0 0 0 z n z n 0 0 0 0 0 0 0 0 0 z n z n ] ( N + 1 ) × ( N + 1 ) ,
(3.3)
B = B 1 + B 2 ,
(3.4)
B 1 = [ b 11 0 0 0 0 b 21 b 22 0 0 0 b 31 b 32 b 33 0 0 b N , 1 b N , 2 b N , 3 b N , N 0 b N + 1 , 1 b N + 1 , 2 b N + 1 , 3 b N + 1 , N b N + 1 , N + 1 ] ( N + 1 ) × ( N + 1 ) , B 2 = [ 1 0 0 0 0 0 0 0 0 0 1 / τ η n 0 0 0 0 0 0 0 0 0 1 / τ η n 0 0 0 0 0 0 0 0 0 0 η n 0 0 0 0 0 0 0 0 0 1 / τ η n 0 0 0 0 0 0 0 0 0 v n w n 0 0 0 0 0 0 0 0 0 v n w n 0 0 0 0 0 0 0 0 0 0 v n w n 0 0 0 0 0 0 0 0 0 v n w n ] ( N + 1 ) × ( N + 1 ) , C = [ 0 0 0 0 0 0 0 0 0 0 0 c n 0 0 0 0 0 0 0 0 0 0 c n 0 0 0 0 0 0 0 0 0 0 c n 0 0 0 0 0 0 0 0 0 0 c n 0 0 0 0 0 0 0 0 0 y n y n 0 0 0 0 0 0 0 0 0 y n y n 0 0 0 0 0 0 0 0 0 0 y n y n 0 0 0 0 0 0 0 0 0 y n y n ] ( N + 1 ) × ( N + 1 ) , φ n = [ φ n 0 φ n 1 φ n 2 φ n N 1 φ n N ] ( N + 1 ) × 1 , U q = [ u q 0 u q 1 u q 2 u q N 1 u q N ] ( N + 1 ) × 1 , q = n ± 1 , n , a n = 1 + x n h 2 1 2 h , c n = 1 + x n h 2 + 1 2 h , z n = 1 2 ( 1 + x n h 2 + 1 2 h ) , y n = 1 2 ( 1 + x n h 2 1 2 h ) , η n = 1 τ + 2 ( 1 + x n ) h 2 , v n = 1 τ + 1 + x n h 2 , w n = 1 τ + 1 + x n h 2 , b 11 = 1 , b 21 = 2 2 3 π τ , b 22 = 2 2 3 π τ , b 31 = 4 6 5 π τ , b 32 = 2 6 5 π τ , b 33 = 2 6 5 π τ , b 41 = d [ 1 λ ( 1 ) + μ ( 1 ) ] , b 42 = d [ 3 λ ( 1 ) 2 μ ( 1 ) ] d / 6 2 , b 43 = d [ 2 λ ( 1 ) + μ ( 1 ) ] 4 d / 6 2 , b 44 = 5 d / 6 2 , b 51 = d [ 2 λ ( 2 ) + μ ( 2 ) ] , b 52 = d [ 5 λ ( 2 ) 2 μ ( 2 ) + 1 λ ( 1 ) + μ ( 1 ) ] , b 53 = d [ 3 λ ( 2 ) + μ ( 2 ) 3 λ ( 1 ) 2 μ ( 1 ) ] d / 6 2 , b 54 = d [ 2 λ ( 1 ) + μ ( 1 ) ] 4 d / 6 2 , b 55 = 5 d / 6 2 , b i j = { d [ ( i 3 ) λ ( i 3 ) + μ ( i 3 ) ] , j = 1 , d [ ( 5 2 i ) λ ( i 3 ) 2 μ ( i 3 ) + ( i 4 ) λ ( i 4 ) + μ ( i 4 ) ] , j = 2 , d [ ( i j + 1 ) λ ( i j ) + μ ( i j ) + ( 2 j 2 i + 1 ) λ ( i j 1 ) 2 μ ( i j 1 ) + ( i j 2 ) λ ( i j 2 ) + μ ( i j 2 ) ] , 3 j i 3 , d [ 3 λ ( 2 ) + μ ( 2 ) 3 λ ( 1 ) 2 μ ( 1 ) ] d / 6 2 , j = i 2 , d [ 2 λ ( 1 ) + μ ( 1 ) ] 4 d / 6 2 , j = i 1 , 5 d / 6 2 , j = i , 0 , i < j r + 1
(3.5)
for i = 6 , 7 , , N + 1 , and
φ n k = [ 3 k τ 2 + 3 π k τ 4 + π 2 ( k τ ) 3 / 2 ( 1 + n h ) ] sin ( π n h ) π ( k τ ) 3 / 2 cos ( π n h ) .
(3.6)
For solving (3.2) we use a modified Gauss elimination method [27]. Hence, we seek a solution of the matrix equation in the following form:
U j = α j + 1 U j + 1 + β j + 1 , j = M 1 , , 2 , 1 ,
(3.7)
where α j ( j = 1 , 2 , , M ) are ( N + 1 ) × ( N + 1 ) square matrices and β j ( j = 1 , 2 , , M ) are ( N + 1 ) × 1 column matrices defined by
α j + 1 = ( B + C α j ) 1 A ,
(3.8)
β j + 1 = ( B + C α j ) 1 ( D φ j C β j ) , j = 1 , 2 , , M 1 ,
(3.9)

where j = 1 , 2 , , M 1 , α 1 is the ( N + 1 ) × ( N + 1 ) zero matrix and β 1 is the ( N + 1 ) × 1 zero matrix and U M = 0 .

Example 3.2 We consider the following initial boundary value problem with Neumann condition for the one-dimensional fractional parabolic partial differential equation:
{ u ( t , x ) t + D t 1 / 2 u ( t , x ) x ( ( 1 + x ) u ( t , x ) x ) + u ( t , x ) = f ( t , x ) , f ( t , x ) = ( 3 + t + 16 t 5 π + π 2 t ( 1 + x ) ) t 2 cos π x + π t 3 sin π x , 0 < t < 1 , 0 < x < 1 , u x ( t , 0 ) = u x ( t , 1 ) = 0 , 0 t 1 , u ( 0 , x ) = 0 , 0 x 1 .
(3.10)
It is clear that the exact solution of problem (3.10) is
u ( t , x ) = t 3 cos π x .
Applying the r-modified Crank-Nicholson difference scheme (2.4), we get
{ u n k u n k 1 τ + D t k τ 2 1 / 2 u n k [ ( 1 + x n ) u n + 1 k 2 u n k + u n 1 k h 2 + u n + 1 k u n 1 k 2 h + u n k ] = φ n k , 1 k r , u n k u n k 1 τ + D t k τ 2 1 / 2 u n k 1 2 [ ( 1 + x n ) u n + 1 k 2 u n k + u n 1 k h 2 + u n + 1 k u n 1 k 2 h + u n k + ( 1 + x n ) u n + 1 k 1 2 u n k 1 + u n 1 k 1 h 2 + u n + 1 k 1 u n 1 k 1 2 h + u n k 1 ] = φ n k , r + 1 k N , φ n k = f ( t k τ 2 , x n ) , t k = k τ , x n = n h , 1 k N , 1 n M 1 , u 0 0 = 0 , k = 0 , h 4 τ u 0 k 1 + [ 1 h + h 2 D t 1 / 2 + h 2 ] u 0 k + h 4 τ u 0 k + 1 = 1 h u 1 k + h 2 φ 0 k , 1 k N 1 , h 4 τ u 0 N 2 + h τ u 0 N 1 + [ 1 h + 3 h 4 τ + h 2 D t 1 / 2 + h 2 ] u 0 N = 1 h u 1 N + h 2 φ 0 N , k = N , 3 u M k 4 u M 1 k + u M 2 k = 0 , 0 k N , u n 0 = 0 , 0 n M ,
where D t k τ 2 1 / 2 u n k is defined by (2.3). We can rewrite it in the form of a system of equations with matrix coefficients
{ A U n + 1 + B U n + C U n 1 = D φ n , 1 n M 1 , E U 0 = F U 1 + R φ 0 , 3 U M 4 U M 1 + U M 2 = 0 ˜ .
Here, A, B, C are defined by (3.3), (3.4), and (3.5):
F = [ 0 0 0 0 0 0 1 / h 0 0 0 0 0 1 / h 0 0 0 0 0 1 / h 0 0 0 0 0 1 / h ] ( N + 1 ) × ( N + 1 ) , R = [ 0 0 0 0 0 0 h / 2 0 0 0 0 0 h / 2 0 0 0 0 0 h / 2 0 0 0 0 0 h / 2 ] ( N + 1 ) × ( N + 1 ) , E = [ e 11 0 0 0 0 e 21 e 22 0 0 0 e 31 e 32 e 33 0 0 e N 1 e N 2 e N 3 e N N 0 e N + 1 , 1 e N + 1 , 2 e N + 1 , 3 e N + 1 , N e N + 1 , N + 1 ] ( N + 1 ) × ( N + 1 ) , e 11 = 1 , e 21 = h 4 τ 4 h 3 π τ , e 22 = 1 h + h 2 + 4 h 3 π τ , e 23 = h 4 τ , e 31 = 2 2 h 15 π τ , e 32 = 16 2 h 15 π τ h 4 τ , e 33 = 1 h + h 2 + 14 2 h 15 π τ , e 34 = h 4 τ , e 41 = d h 2 [ ( 1 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) ] , e 42 = d h 2 [ 4 λ ( 1 ) 2 μ ( 1 ) + 1 / 2 λ ( 0 ) + μ ( 0 ) ] , e 43 = h 4 τ + d h 2 [ ( 2 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) 2 2 ( 1 / 3 ) ] , e 44 = 1 h + h 2 + d h 2 [ ( 1 + 1 / 2 ) λ ( 0 ) + μ ( 0 ) ] , e 45 = h 4 τ , e 51 = d h 2 [ ( 2 + 1 / 2 ) λ ( 2 ) + μ ( 2 ) ] , e 52 = d h 2 [ 2 3 λ ( 2 ) 2 μ ( 2 ) + ( 1 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) ] , e 53 = d h 2 [ ( 2 + 1 + 1 / 2 ) λ ( 2 ) + μ ( 2 ) 2 2 λ ( 1 ) 2 μ ( 1 ) + 1 / 2 λ ( 0 ) + μ ( 0 ) ] , e 54 = h 4 τ + d h 2 [ ( 1 + 1 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) 2 λ ( 0 ) 2 μ ( 0 ) ] , e 55 = 1 h + h 2 + d h 2 [ ( 1 + 1 / 2 ) λ ( 0 ) + μ ( 0 ) ] , e 56 = h 4 τ , e i j = { d h 2 [ ( i 3 + 1 / 2 ) λ ( i 3 ) + μ ( i 3 ) ] , j = 1 , d h 2 [ 2 ( i 2 ) λ ( i 3 ) 2 μ ( i 3 ) + ( i 4 + 1 / 2 ) λ ( i 4 ) + μ ( i 4 ) ] , j = 2 , d h 2 [ ( i j + 1 + 1 / 2 ) λ ( i j ) + μ ( i j ) 2 ( i j ) λ ( i j 1 ) 2 μ ( i j 1 ) + ( i j 2 + 1 / 2 ) λ ( i j 2 ) + μ ( i j 2 ) ] , 3 j i 2 , h 4 τ + d h 2 [ ( 2 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) 2 λ ( 0 ) 2 μ ( 0 ) ] , j = i 1 , 1 h + h 2 + d h 2 [ ( 1 + 1 / 2 ) λ ( 0 ) + μ ( 0 ) ] , j = i , h 4 τ , j = i + 1 , h 4 τ + d h 2 [ ( i N + 2 + 1 / 2 ) λ ( i N + 1 ) + μ ( i N + 1 ) 2 ( i N + 1 ) λ ( i N ) 2 μ ( i N ) + ( i N 1 + 1 / 2 ) λ ( i N 1 ) + μ ( i N 1 ) ] , j = N 1 , h τ + d h 2 [ ( 2 + 1 / 2 ) λ ( 1 ) + μ ( 1 ) 2 λ ( 0 ) 2 μ ( 0 ) ] , j = N , 1 h + h 2 + 3 h 4 τ + d h 2 [ ( 1 + 1 / 2 ) λ ( 0 ) + μ ( 0 ) ] , j = N + 1 0 , j > i + 1
for i = 6 , 7 , , N + 1 , and
φ 0 k = ( 3 + k τ + 16 k τ 5 π + π 2 k τ ) ( k τ ) 2 , φ n k = [ 3 + k τ + 16 k τ 5 π + π 2 ( k τ ) ( 1 + n h ) ] ( k τ ) 2 cos ( π n h ) + π ( k τ ) 3 sin ( π n h ) .
(3.11)
For solving this matrix equation we will use the same method as for Example 3.1. Namely, we use (3.7), (3.8), (3.9), and
u M = [ 3 I 4 α M + α M 1 α M ] 1 [ ( 4 I α M 1 ) β M β M 1 ] , α 1 = E 1 F , β 1 = E 1 R φ 0 .
Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of the modification parameter r, and discretization parameters N and M. Besides u n k represents the numerical solutions of these difference schemes at ( t k , x n ) . The error is computed by the following formula:
E M N = max 1 k N , 1 n M 1 | u ( t k , x n ) u n k | .
Table 1 and Table 2 are constructed for N = M = 40  and  80 , respectively. As can be seen from Table 1, the r-modified Crank-Nicholson difference scheme is more accurate than the Crank-Nicholson and Rothe difference schemes. Table 2 shows that the r-modified Crank-Nicholson difference scheme has the same order error as the Crank-Nicholson difference scheme.
Table 1

Error analysis for Dirichlet problem

Method

N = M = 40

N = M = 80

Rothe

0.019436847

0.009569477

Crank-Nicholson

0.000525352

0.000149122

One-modified Crank-Nicholson

0.000525046

0.000149050

Two-modified Crank-Nicholson

0.000503791

0.000144560

Three-modified Crank-Nicholson

0.001309365

0.000260588

Table 2

Error analysis for Neumann problem

Method

N = M = 40

N = M = 80

Rothe

0.038312769

0.018981758

Crank-Nicholson

0.009051376

0.002281280

One-modified Crank-Nicholson

0.011770581

0.002979532

Two-modified Crank-Nicholson

0.011770790

0.002979547

Three-modified Crank-Nicholson

0.011770831

0.002979551

4 Conclusion

In this study, the second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation is investigated. We have obtained a stability estimate for the solution of this difference scheme. The theoretical statements for the solution of this difference scheme for one-dimensional parabolic equations are supported by numerical examples obtained by computer.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Fatih University
(2)
Department of Mathematics, ITTU
(3)
Department of Mathematical Engineering, Gumushane University

References

  1. Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
  2. Kilbas AA, Srivastava HM, Trujillo JJ 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
  3. Diethelm K: The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
  4. Diethelm K, Ford NJ: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 2004, 154(3):621-640. 10.1016/S0096-3003(03)00739-2MathSciNetView ArticleGoogle Scholar
  5. El-Sayed AMA, El-Mesiry AEM, El-Saka HAA: Numerical solution for multi-term fractional (arbitrary) orders differential equations. Comput. Appl. Math. 2004, 23(1):33-54.MathSciNetGoogle Scholar
  6. De la Sen M: Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays. Abstr. Appl. Anal. 2011., 2011: Article ID 161246 10.1155/2011/161246Google Scholar
  7. Yakar A, Koksal ME: Existence results for solutions of nonlinear fractional differential equations. Abstr. Appl. Anal. 2012., 2012: Article ID 267108 10.1155/2012/267108Google Scholar
  8. Yuan C:Two positive solutions for ( n 1 , 1 ) -type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17(2):930-942. 10.1016/j.cnsns.2011.06.008MathSciNetView ArticleGoogle Scholar
  9. De la Sen M, Agarwal RP, Ibeas A, Alonso-Quesada S: On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination. Adv. Differ. Equ. 2011., 2011: Article ID 748608 10.1155/2011/748608Google Scholar
  10. Yuan C:Multiple positive solutions for semipositone ( n , p ) -type boundary value problems of nonlinear fractional differential equations. Anal. Appl. 2011, 9(1):97-112. 10.1142/S0219530511001753MathSciNetView ArticleGoogle Scholar
  11. Agarwal RP, de Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 2010, 11: 3532-3554. 10.1016/j.nonrwa.2010.01.002MathSciNetView ArticleGoogle Scholar
  12. Ashyralyev A: A note on fractional derivatives and fractional powers of operators. J. Math. Anal. Appl. 2009, 357(1):232-236. 10.1016/j.jmaa.2009.04.012MathSciNetView ArticleGoogle Scholar
  13. Ashyralyev A, Dal F, Pinar Z: A note on fractional hyperbolic differential and difference equations. Appl. Math. Comput. 2011, 217(9):4654-4664. 10.1016/j.amc.2010.11.017MathSciNetView ArticleGoogle Scholar
  14. Berdyshev AS, Cabada A, Karimov ET: On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator. Nonlinear Anal. 2011, 75(6):3268-3273. 10.1016/j.na.2011.12.033MathSciNetView ArticleGoogle Scholar
  15. Amanov D, Ashyralyev A: Initial-boundary value problem for fractional partial differential equations of higher order. Abstr. Appl. Anal. 2012., 2012: Article ID 973102 10.1155/2012/973102Google Scholar
  16. Cakir Z: Stability of difference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions. Abstr. Appl. Anal. 2012., 2012: Article ID 463746 10.1155/2012/463746Google Scholar
  17. Ashyralyev A, Cakir Z: On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 696179 10.1155/2012/696179Google Scholar
  18. Ashyralyev A, Cakir Z: FDM for fractional parabolic equations with the Neumann condition. Adv. Differ. Equ. 2013., 2013: Article ID 120 10.1186/1687-1847-2013-120Google Scholar
  19. Ashyralyev A, Hicdurmaz B: A note on the fractional Schrodinger differential equations. Kybernetes 2011, 40(5-6):736-750. 10.1108/03684921111142287MathSciNetGoogle Scholar
  20. Ashyralyev A: Well-posedness of the Basset problem in spaces of smooth functions. Appl. Math. Lett. 2011, 24(7):1176-1180. 10.1016/j.aml.2011.02.002MathSciNetView ArticleGoogle Scholar
  21. Clement P, Guerre-Delabrire S: On the regularity of abstract Cauchy problems and boundary value problems. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 1999, 9(4):245-266.Google Scholar
  22. Agarwal RP, Bohner M, Shakhmurov VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl. 2005, 1: 9-42. 10.1155/BVP.2005.9MathSciNetGoogle Scholar
  23. Lunardi A Operator Theory: Advances and Applications. In Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel; 1995.Google Scholar
  24. Ashyralyev A, Sobolevskii PE: Well-Posedness of Parabolic Difference Equations. Birkhäuser, Basel; 1994.View ArticleGoogle Scholar
  25. Rassias JM, Karimov ET: Boundary-value problems with non-local condition for degenerate parabolic equations. Contemp. Anal. Appl. Math. 2013, 1(1):42-48.Google Scholar
  26. Selitskii AM: The space of initial data for the second boundary-value problem for parabolic differential-difference equation. Contemp. Anal. Appl. Math. 2013, 1(1):34-41.Google Scholar
  27. Samarskii AA, Nikolaev ES Iterative Methods 2. In Numerical Methods for Grid Equations. Birkhäuser, Basel; 1989.Google Scholar

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