Chebyshev wavelets method for solving Bratu’s problem

  • Changqing Yang1Email author and

    Affiliated with

    • Jianhua Hou1

      Affiliated with

      Boundary Value Problems20132013:142

      DOI: 10.1186/1687-2770-2013-142

      Received: 14 May 2013

      Accepted: 16 May 2013

      Published: 2 June 2013

      Abstract

      A numerical method for one-dimensional Bratu’s problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. The matrix together with the collocation method are then utilized to transform the differential equation into a system of algebraic equations. Numerical examples are presented to verify the efficiency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.

      1 Introduction

      In this paper, we consider the boundary-value problem and initial value problem of Bratu’s problem. It is well known that Bratu’s boundary value problem in one-dimensional planar coordinates is of the form
      u + λ e u = 0 , 0 < x < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ1_HTML.gif
      (1)
      with the boundary conditions u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq1_HTML.gif. For λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq2_HTML.gif is a constant, the exact solution of equation (1) is given by [1]
      u ( x ) = 2 ln [ cosh ( 0.5 θ ( x 0.5 ) ) cosh ( 0.25 θ ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ2_HTML.gif
      (2)
      where θ satisfies
      θ = 2 λ sinh ( 0.25 θ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ3_HTML.gif
      (3)
      The problem has zero, one or two solutions when λ > λ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq3_HTML.gif, λ = λ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq4_HTML.gif and λ < λ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq5_HTML.gif, respectively, where the critical value λ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq6_HTML.gif satisfies the equation
      1 = 1 4 2 λ c cosh ( 1 4 θ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equa_HTML.gif

      It was evaluated in [13] that the critical value λ c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq6_HTML.gif is given by λ c = 3.513830719 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq7_HTML.gif.

      In addition, an initial value problem of Bratu’s problem
      u + λ e u = 0 , 0 < x < 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ4_HTML.gif
      (4)

      with the initial conditions u ( 0 ) = u ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq8_HTML.gif will be investigated.

      Bratu’s problem is also used in a large variety of applications such as the fuel ignition model of the thermal combustion theory, the model of thermal reaction process, the Chandrasekhar model of the expansion of the universe, questions in geometry and relativity about the Chandrasekhar model, chemical reaction theory, radiative heat transfer and nanotechnology [411].

      A substantial amount of research work has been done for the study of Bratu’s problem. Boyd [2, 12] employed Chebyshev polynomial expansions and the Gegenbauer as base functions. Syam and Hamdan [8] presented the Laplace decomposition method for solving Bratu’s problem. Also, Aksoy and Pakdemirli [13] developed a perturbation solution to Bratu-type equations. Wazwaz [10] presented the Adomian decomposition method for solving Bratu’s problem. In addition, the applications of spline method, wavelet method and Sinc-Galerkin method for solution of Bratu’s problem have been used by [1417].

      In recent years, the wavelet applications in dealing with dynamic system problems, especially in solving differential equations with two-point boundary value constraints have been discussed in many papers [4, 16, 18]. By transforming differential equations into algebraic equations, the solution may be found by determining the corresponding coefficients that satisfy the algebraic equations. Some efforts have been made to solve Bratu’s problem by using the wavelet collocation method [16].

      In the present article, we apply the Chebyshev wavelets method to find the approximate solution of Bratu’s problem. The method is based on expanding the solution by Chebyshev wavelets with unknown coefficients. The properties of Chebyshev wavelets together with the collocation method are utilized to evaluate the unknown coefficients and then an approximate solution to (1) is identified.

      2 Chebyshev wavelets and their properties

      2.1 Wavelets and Chebyshev wavelets

      In recent years, wavelets have been very successful in many science and engineering fields. They constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet ψ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq9_HTML.gif. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets [19]:
      ψ a , b ( x ) = | a | 1 / 2 ψ ( x b a ) , a , b R , a 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equb_HTML.gif
      Chebyshev wavelets ψ n , m = ψ ( k , n , m , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq10_HTML.gif have four arguments, n = 1 , 2 , , 2 k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq11_HTML.gif, k can assume any positive integer, m is the degree of Chebyshev polynomials of first kind and x denotes the time.
      ψ n , m ( x ) = { α m 2 ( k 1 ) / 2 π T m ( 2 k x 2 n + 1 ) , n 1 2 k 1 x < n 2 k 1 ; 0 , otherwise , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ5_HTML.gif
      (5)
      where
      α m = { 2 , m = 0 ; 2 , m = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equc_HTML.gif
      and m = 0 , 1 , 2 , , M 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq12_HTML.gif, n = 1 , 2 , , 2 k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq11_HTML.gif. Here T m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq13_HTML.gif are the well-known Chebyshev polynomials of order m, which are orthogonal with respect to the weight function ω ( x ) = 1 / 1 x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq14_HTML.gif and satisfy the following recursive formula:
      T 0 ( x ) = 1 , T 1 ( x ) = x , T m + 1 ( x ) = 2 x T m ( x ) T m 1 ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equd_HTML.gif

      We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight function ω n ( x ) = ω ( 2 k x 2 n + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq15_HTML.gif.

      The derivative of Chebyshev polynomials is a linear combination of lower-order Chebyshev polynomials, in fact [20],
      { T m ( x ) = 2 m k = 1 m 1 T k ( x ) , m  even ; T m ( x ) = 2 m k = 1 m 1 T k ( x ) + m T 0 ( x ) , m  odd . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ6_HTML.gif
      (6)

      2.2 Function approximation

      A function u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif defined over [ 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq17_HTML.gif may be expanded as
      u ( x ) = n = 1 m = 0 c n m ψ n m ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ7_HTML.gif
      (7)
      where c n m = ( u ( x ) , ψ n m ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq18_HTML.gif, in which ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq19_HTML.gif denotes the inner product with the weight function ω n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq20_HTML.gif. If u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif in (7) is truncated, then (7) can be written as
      u ( x ) n = 1 2 k 1 m = 0 M 1 c n m ψ n m ( x ) = C T Ψ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ8_HTML.gif
      (8)
      where C and Ψ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq21_HTML.gif are 2 k 1 M × 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq22_HTML.gif matrices given by
      C = [ c 1 , c 2 , , c 2 k 1 ] T , Ψ ( x ) = [ ψ 1 , ψ 2 , , ψ 2 k 1 ] T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Eque_HTML.gif
      and
      c i = [ c i 0 , c i 1 , , c i , M 1 ] , ψ i ( x ) = [ ψ i 0 ( x ) , ψ i 1 ( x ) , , ψ i , M 1 ( x ) ] , i = 1 , 2 , 3 , , 2 k 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equf_HTML.gif

      3 Chebyshev wavelets operational matrix of derivative

      In this section we first derive the operational matrix D of derivative which plays a great role in dealing with Bratu’s problem.

      In the interval [ ( n 1 ) / 2 k 1 , n / 2 k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq23_HTML.gif,
      ψ n , m ( x ) = α m 2 ( k 1 ) / 2 π T m ( 2 k x 2 n + 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equg_HTML.gif
      Applying (6) the derivative of ψ n , m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq24_HTML.gif is
      ψ n , m ( x ) = { α m 2 ( k 1 ) / 2 π 2 k 2 m k = 1 m 1 T k ( 2 k x 2 n + 1 ) , m  even ; α m 2 ( k 1 ) / 2 π 2 k [ 2 m k = 1 m 1 T k ( 2 k x 2 n + 1 ) + m T 0 ( 2 k x 2 n + 1 ) ] , m  odd . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equh_HTML.gif
      The function ψ i ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq25_HTML.gif is zero outside the interval [ ( i 1 ) / 2 k 1 , i / 2 k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq26_HTML.gif, so
      ψ i ( x ) = ψ i ( x ) M , i = 1 , 2 , , 2 k 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ9_HTML.gif
      (9)
      where
      M = 2 k ( 0 2 0 3 2 0 5 2 ( M 1 ) 2 0 0 4 0 8 0 0 0 0 0 6 0 10 2 ( M 1 ) 0 0 0 0 0 0 2 ( M 1 ) 0 0 0 0 0 0 0 0 ) M × M for even  M , M = 2 k ( 0 2 0 3 2 0 5 2 0 0 0 4 0 8 0 0 0 0 0 6 0 10 2 ( M 1 ) 0 0 0 0 0 0 2 ( M 1 ) 0 0 0 0 0 0 0 0 ) M × M for odd  M . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equi_HTML.gif
      In fact we have shown that
      Ψ ( x ) = D Ψ ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ10_HTML.gif
      (10)
      where
      D = diag ( M T , M T , , M T ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equj_HTML.gif
      From (10), it can be generalized for any n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq27_HTML.gif as
      d n Ψ ( x ) d x n = D n Ψ ( x ) , n = 1 , 2 , 3 , . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ11_HTML.gif
      (11)

      4 Solution of Bratu’s problem

      Consider Bratu’s problem given in (1). In order to use Chebyshev wavelets, we first approximate u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif as
      u ( x ) = C T Ψ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equk_HTML.gif
      Applying (11) we can get
      u ( x ) = C T D 2 Ψ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equl_HTML.gif
      Thus we have
      C T D 2 Ψ ( x ) + λ e C T Ψ ( x ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ12_HTML.gif
      (12)
      We now collocate (12) at 2 k 1 M 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq28_HTML.gif points at x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq29_HTML.gif as
      C T D 2 Ψ ( x i ) + λ e C T Ψ ( x i ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ13_HTML.gif
      (13)
      Suitable collocation points are
      x i = 1 2 [ 1 + cos ( ( i 1 ) π 2 k 1 M 1 ) ] , i = 2 , 3 , , 2 k 1 M 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equm_HTML.gif
      Thus with the boundary conditions u ( 0 ) = u ( 1 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq1_HTML.gif, we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ14_HTML.gif
      (14)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ15_HTML.gif
      (15)

      Equations (13), (14) and (15) generate 2 k 1 M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq30_HTML.gif set of nonlinear equations. The approximate solution of the vector C is obtained by solving the nonlinear system using the Gauss-Newton method.

      5 Error analysis

      Theorem 5.1 A function u ( x ) L ω 2 ( [ 0 , 1 ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq31_HTML.gif, with bounded second derivative, say | u ( x ) | N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq32_HTML.gif, can be expanded as an infinite sum of Chebyshev wavelets, and the series converges uniformly to u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif, that is [21],
      u ( x ) = n = 1 m = 0 c n m ψ n m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equn_HTML.gif
      Since the truncated Chebyshev wavelets series is an approximate solution of Bratu’s problem, so one has an error function E ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq33_HTML.gif for u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif as follows:
      E ( x ) = | u ( x ) C T Ψ ( x ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equo_HTML.gif

      The error bound of the approximate solution by using Chebyshev wavelets series is given by the following theorem.

      Theorem 5.2 Suppose that u ( x ) C m [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq34_HTML.gif and C T Ψ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq35_HTML.gif is the approximate solution using the Chebyshev wavelets method. Then the error bound would be obtained as follows:
      E ( x ) 2 m ! 4 m 2 m ( k 1 ) max x [ 0 , 1 ] | u m ( x ) | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equp_HTML.gif
      Proof Applying the definition of norm in the inner product space, we have
      E ( x ) 2 = 0 1 [ u ( x ) C T Ψ ( x ) ] 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equq_HTML.gif
      Because the interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq36_HTML.gif is divided into 2 k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq37_HTML.gif subintervals I n = [ ( n 1 ) / 2 k 1 , n / 2 k 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq38_HTML.gif, n = 1 , 2 , , 2 k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq11_HTML.gif, then we can obtain
      E ( x ) 2 = 0 1 [ u ( x ) C T Ψ ( x ) ] 2 d x = k = 1 2 k 1 n 1 2 k 1 n 2 k 1 [ u ( x ) C T Ψ ( x ) ] 2 d t k = 1 2 k 1 n 1 2 k 1 n 2 k 1 [ u ( x ) P m ( x ) ] 2 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equr_HTML.gif
      where P m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq39_HTML.gif is the interpolating polynomial of degree m which agrees with u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif at the Chebyshev nodes on I n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq40_HTML.gif with the following error bound for interpolating [22, 23]:
      | u ( x ) P m ( x ) | 2 m ! 4 m 2 m ( k 1 ) max x I n | u m ( x ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equs_HTML.gif
      Therefore, using the above equation, we would get
      E ( x ) 2 k = 1 2 k 1 n 1 2 k 1 n 2 k 1 [ u ( x ) P m ( x ) ] 2 d t k = 1 2 k 1 n 1 2 k 1 n 2 k 1 [ 2 m ! 4 m 2 m ( k 1 ) max x I n | u m ( x ) | ] 2 d t k = 1 2 k 1 n 1 2 k 1 n 2 k 1 [ 2 m ! 4 m 2 m ( k 1 ) max x [ 0 , 1 ] | u m ( x ) | ] 2 d t = 0 1 [ 2 m ! 4 m 2 m ( k 1 ) max x [ 0 , 1 ] | u m ( x ) | ] 2 d t = 2 m ! 4 m 2 m ( k 1 ) max x [ 0 , 1 ] | u m ( x ) | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equt_HTML.gif

       □

      6 Numerical examples

      To illustrate the ability and reliability of the method for Bratu’s problem, some examples are provided. The results reveal that the method is very effective and simple.

      Example 6.1 Consider the first case for Bratu’s equation as follows, when λ = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq41_HTML.gif [14, 15]:
      u + 2 e u = 0 , 0 < x < 1 , u ( 0 ) = u ( 1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ16_HTML.gif
      (16)
      We solve the equation by using the Chebyshev wavelets method with k = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq42_HTML.gif, M = 4 , 6 , 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq43_HTML.gif. The numerical results obtained are presented in Table 1. Table 1 shows the comparison between the absolute error of exact and approximate solutions for various values of M (with k = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq42_HTML.gif). Moreover, higher accuracy can be achieved by taking higher order approximations.
      Table 1

      Computed absolute errors for Example 6.1

      x

      k = 2, M = 4

      k = 2, M = 6

      k = 2, M = 8

      0.1

      1.33462 × 10−3

      1.18825 × 10−5

      5.01033 × 10−7

      0.2

      2.57841 × 10−3

      2.62388 × 10−5

      1.16521 × 10−6

      0.3

      5.88781 × 10−3

      2.26840 × 10−5

      2.33696 × 10−6

      0.4

      3.36183 × 10−3

      5.20260 × 10−5

      5.48713 × 10−6

      0.5

      5.98507 × 10−3

      4.44306 × 10−5

      1.33512 × 10−6

      0.6

      3.59903 × 10−3

      2.53383 × 10−5

      2.40800 × 10−6

      0.7

      5.14675 × 10−3

      4.17100 × 10−5

      3.73327 × 10−6

      0.8

      1.43561 × 10−3

      3.25335 × 10−5

      7.18825 × 10−6

      0.9

      1.25962 × 10−3

      1.71598 × 10−5

      1.46832 × 10−6

      Example 6.2 Consider the initial value problem [10, 1416, 24]
      u 2 e u = 0 , 0 < x < 1 , u ( 0 ) = 0 , u ( 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ17_HTML.gif
      (17)
      The exact solution is u ( x ) = 2 ln ( cos ( x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq44_HTML.gif. Here we solve it using Chebyshev wavelets, with k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq45_HTML.gif, M = 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq46_HTML.gif. First we assume that the unknown function u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_IEq16_HTML.gif is given by
      u ( x ) = C T Ψ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equu_HTML.gif
      Applying (12) we get
      C T D 2 Ψ ( x i ) 2 e C T Ψ ( x i ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ18_HTML.gif
      (18)
      Using the initial condition, we obtain
      C T Ψ ( 0 ) = 0 , C T D Ψ ( 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Equ19_HTML.gif
      (19)
      Equations (18) and (19) generate a system of nonlinear equations. These equations can be solved for unknown coefficients of the vector C. A comparison between the exact and the approximate solutions is demonstrated in Figure 1. From Figure 1, it can be found that the obtained approximate solutions are very close to the exact solution. In addition, Table 2 shows the exact and approximate solutions using the method presented in Section 3 and compares the results with the method presented in [16]. Also, by comparing the results of the table, we see that the results of the proposed method are more accurate.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-142/MediaObjects/13661_2013_Article_391_Fig1_HTML.jpg
      Figure 1

      Comparison of solutions for Example 6.2.

      Table 2

      Comparison of the results of the Chebyshev and Legendre wavelets method for Example 6.2

      x

      Chebyshev wavelets

      Legendre wavelets

      Exact solutions

      0.1

      1.0016711 × 10−2

      1.0016801 × 10−2

      1.0016711 × 10−2

      0.2

      4.0269541 × 10−2

      4.0269696 × 10−2

      4.0269546 × 10−2

      0.3

      9.1383326 × 10−2

      9.1382697 × 10−2

      9.1383311 × 10−2

      0.4

      1.6445871 × 10−1

      1.6444915 × 10−1

      1.6445803 × 10−1

      0.5

      2.6116111 × 10−1

      2.6111176 × 10−1

      2.6116848 × 10−1

      0.6

      3.8339360 × 10−1

      3.8367456 × 10−1

      3.8393033 × 10−1

      0.7

      5.3617551 × 10−1

      5.3524690 × 10−1

      5.3617151 × 10−1

      0.8

      7.2271751 × 10−1

      7.1991951 × 10−1

      7.2278149 × 10−1

      0.9

      9.5086960 × 10−1

      9.4297240 × 10−1

      9.5088488 × 10−1

      7 Conclusions

      The aim of present work is to develop an efficient and accurate method for solving Bratu’s problems. The Chebyshev wavelet operational matrix of derivative together with the collocation method are used to reduce the problem to the solution of nonlinear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

      Declarations

      Acknowledgements

      Project is supported by the Huaihai Institute of Technology (No. Z2001151).

      Authors’ Affiliations

      (1)
      Department of Science, Huaihai Institute of Technology

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      © Yang and Hou; licensee Springer. 2013

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