An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method

Abstract

We introduce Chebyshev wavelet analysis method to solve the nonlinear Troesch and Bratu problems. Chebyshev wavelets expansions together with operational matrix of derivative are employed to reduce the computation of nonlinear problems to a system of algebraic equations. Several examples are given to validate the efficiency and accuracy of the proposed technique. We compare the results with those ones reported in the literature in order to demonstrate that the method converges rapidly and approximates the exact solution very accurately by using only a small number of Chebyshev wavelet basis functions. Convergence analysis is also included.

Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

“…In another study, restarted Adomian's decomposition method is used the approximate the analytical solution by Vahidi and Hasanzade [14]. Several numerical techniques, such as the finite difference method [15], weighted residual method [12], the shooting method [10], decomposition technique [16], Legendre wavelets [17], wavelet analysis method [18], B-spline method [19], Jacobi-Gauss collocation method [20], Laplace transform decomposition method [21], He's variational iteration [22,23], have been implemented to solve the Bratu model numerically. On the other hand, Duffing model [24] arises in several scientific fields such as classical oscillator in chaotic systems, non-uniformity caused by an infinite domain, nonlinear mechanical oscillators, magnetic-pliancy mechanical systems, nonlinear vibration of beams and plates, prediction of diseases.…”

Laguerre wavelet method for solving Troesch equation

“…In literature, several numerical methods have been employed to solve this nonlinear problem. We can list these methods as: Finite difference method [1], Chebyshev wavelet method [2], Chebysev collocation method [3], A finite-element approach based on cubic B-spline collocation [4], An accurate asymptotic approximation [5], Adomian decomposition method and the reproducing kernel method [6], Christov rational functions [7], Decomposition method [8], Differential transform method [9], High-Order Difference Schemes [10], Homotopy perturbation method [11], Hybrid heuristic computing [12], Jacobi-Gauss collocation method [13], Laplace transform and a modified decomposition technique [14], Modified Homotopy perturbation method [15], Newton-Raphson-Kantorovich approximation method [16], Optimal Homotopy asymptotic method [17], Perturbation Method and Laplace-Padé Approximation [18], Scott and the Kagiwada-Kalaba algorithms [19], Modified nonlinear Shooting method [20], Sinc-Collocation Method [21], sinc-Galerkin method [22], Variational iteration method [23,24].…”