Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Ordinary and Partial Differential Equations

Abstract

In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.

Optical soliton solutions to the $(2+1)$-dimensional Chaffee–Infante equation and the dimensionless form of the Zakharov equation

“…For the Chaffee-Infante equation, we have discussed two different cases, m = 1 and m = 2 in Eq. (13), and other values of m are ignored, since the algebraic complexity arises rapidly, however, no viable solution can be found in this proportion. Case 1: First suppose m = 1, then by comparing the coefficients of Y i (i = 2, 1, 0) on both sides of (14), we obtaiṅ…”

“…Closed form solitary wave solutions provide better internal information about those phenomena. Therefore, considerable efforts have been made by many mathematicians and physical scientists to obtain closed form wave solutions of such NLEEs and a number of powerful and efficient methods, such as the Bäcklund transformation method [1], the first integral method [2], the modified simple equation method [3,4], the Exp-function method [5], the (G /G)-expansion method [6], the sine-cosine method [7], the modified Kudryashov method [8], the homogeneous balance method [9], the F-expansion method [10], the variational iteration method [11], the tanh-function method [12], the Adomian decomposition method [13], the projective Riccati equation method [14], the homotopy analysis method [15], and the (G /G, 1/G)expansion method [16] have been developed.…”

Stable soliton solutions to the nonlinear low-pass electrical transmission lines and the Cahn-Allen equation

“…The exact solutions can also serve as the basis for the excellence and precision of computer algebra software in solving NLEEs. Therefore, a number of important methods for the explicit and detailed stable soliton solutions of nonlinear physical models have currently been developed with the aid of Matlab, Mathematica, etc., such as, the differential transformation [1] method, the Hirota's bilinear method [2] , [3] , [4] , [5] , the approach of modified simple equation [6] , [7] , the F-expansion method [8] , the Exp-function [9] method, the modified exponential-function method [10] , the -expansion [11] , [12] , [13] scheme, the improved -expansion method [14] , the rational -method [15] , the extended trial equation method [16] , the improved -expansion method [17] , the first integral method [18] , the generalized Kudryashov [19] approach, the homotopy analysis [20] technique, the mean finite difference Monte-Carlo [21] method, the sine-Gordon expansion method [22] , the -expansion method [23] , the modified Kudryashov [24] scheme, the Adomian decomposition method [25] , the generalized projective Riccati equation method [26] , the multi-symplectic Runge-Kutta method [27] , the -expansion method [28] , the modified extended tanh method [29] , [30] , the exponential rational function method [31] , the generalized rational function method [32] , the unified method [33] , [34] ,…”

Bernstein Operator Approach for Solving Linear Differential Equations

“…Only in four years, collocation method that is one of the numerical method has been developed for numerical solution of many types of equations like integral equations [17], integro-differential equations [11,37] and differential equations [25]. The Bernstein polynomials have also been modified the Adomian decomposition [31] and Laplace decomposition method [33] for solving linear and nonlinear differential equations. The spectral Petrov-Galerkin method [20] has been applied for numerical solution of fractional partial differential equations that are one of the most popular and remarkable equations in the science world.…”