New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

Abstract

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan's convergence parameter, are derived using the DuanRach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (ME ) or the error remainder functions (ER ( )) of each problem are calculated.

The novel family of transcendental Leal-functions with applications to science and engineering

“…Given the solution for (34) is important, several authors have proposed approximate solutions employing different analytical and semianalytical methodologies [77,78,79], among others. The planar onedimensional Bratu problem is given by (34) with exact solution [80,81] given by:…”

Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

“…ere are two essential types of solutions for NPDEs, which are analytical and exact solutions. Some examples of the schemes used to obtain analytical approximate solutions to NPDEs are the homotopy analysis method (HAM) [3], the Adomian decomposition method (ADM) [4,5], the modified Laplace variational iteration method (ML-VIM) [6], and the reduced differential transform method [7], while many effective methods have been proposed to obtain exact solutions of NPDEs including fractional order partial differential equations such as the generalized Kudryashov method [8][9][10], the amplitude ansatz method [11], the He's semiinverse method [12,13], the exp-function method [14,15], the auxiliary equation method [16,17], the extended trial equation method [18,19], and the extended direct algebraic method [20]. Moreover, the sine-cosine method [21,22], the tanh-coth method [23,24], the extended sech-tanh method [25], the sine-Gordon expansion method [26][27][28], and the (G ′ /G)-expansion method [29][30][31] have been recently utilized to find analytical exact solutions of NPDEs as well.…”

Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

“…Over the last few decades, many kinds of solutions of NLEEs, including exact solutions, analytical approximate solutions, and numerical solutions, have been successfully obtained using various and efficient methods. Examples of the methods for obtaining analytical approximate solutions of NLEEs are the Adomian decomposition method (ADM) [7,8], the revised variational iteration method (RVIM) [9], the reduced differential transform method [10], and the homotopy perturbation method (HPM) [11,12]. Useful methods for solving NLEEs numerically are those such as the finite element method [13], the finite volume method [14], and the finite-difference predictor-corrector method [15].…”