Chebyshev wavelets method for solving Bratu’s problem

Boundary Value Problems Chebyshev wavelets method for solving Bratu's problem Changqing Yang 0 Jianhua Hou 0 0 Department of Science, Huaihai Institute of Technology , Lianyungang, Jiangsu 222005 , China 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  = with the boundary conditions u() = u() = . For λ >  is a constant, the exact solution of equation () is given by [] u(x) = – ln cosh(.θ (x – .)) cosh(.θ ) It was evaluated in [–] that the critical value λc is given by λc = .. In addition, an initial value problem of Bratu’s problem 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 [–]. A substantial amount of research work has been done for the study of Bratu’s problem. Boyd [, ] employed Chebyshev polynomial expansions and the Gegenbauer as base functions. Syam and Hamdan [] presented the Laplace decomposition method for solving Bratu’s problem. Also, Aksoy and Pakdemirli [] developed a perturbation solution to Bratu-type equations. Wazwaz [] 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 [–]. 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 [, , ]. 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 []. 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 () 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). When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets []: Chebyshev wavelets ψn,m = ψ (k, n, m, x) have four arguments, n = , , . . . , k–, k can assume any positive integer, m is the degree of Chebyshev polynomials of first kind and x denotes the time. nk–– ≤ x < kn– ; and m = , , , . . . , M – , n = , , . . . , k–. Here Tm(x) are the well-known Chebyshev polynomials of order m, which are orthogonal with respect to the weight function ω(x) = √ /  – x and satisfy the following recursive formula: T(x) = , T(x) = x, Tm+(x) = xTm(x) – Tm–(x). We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight function ωn(x) = ω(kx – n + ). The derivative of Chebyshev polynomials is a linear combination of lower-order Chebyshev polynomials, in fact [], km=– Tk(x), km=– Tk(x) + mT(x), m odd. 2.2 Function approximation A function u(x) defined over [, ) may be expanded as where cnm = (u(x), ψnm(x)), in which (·, ·) denotes the inner product with the weight function ωn(x). If u(x) in () is truncated, then () can be written as ∞ ∞ n= m= n= m= u(x) ≈ C = [c, c, . . . , ck– ]T , ci = [ci, ci, . . . , ci,M–], 3 Chebyshev wavelets operational matrix of derivative km=– Tk(kx – n + ), ψi (x) = ψi(x)M, i = , , . . . , k–, ⎜⎜  ⎝  ⎜⎜  ⎝  √  √  √ · · · (M – )√⎞      · · ·  ⎟⎟      · · · (M – ) ⎟⎟ ... ... ... ... ... ·.·. ·. (M... – ) ⎟⎟⎟⎟⎟⎠        √  √  √ · · ·  ⎞      · · ·  ⎟⎟ ... ... ... ... ... ··.··. ··. ((MM... –– ))⎟⎟⎟⎟⎠⎟⎟⎟        for even M, for odd M. In fact we have shown that D = diag MT , MT , . . . , MT . (...truncated)