Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

Abstract

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).

Analytical Solutions of the Nonlinear Time-Fractional Coupled Boussinesq-Burger Equations Using Laplace Residual Power Series Technique

“…This section reviews some definitions and theorems for the fractional operators and the LT [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] which are essential in constructing the L-RPS solutions for the nonlinear T-FCB-BEs as in Equations ( 1) and ( 2). Definition 1.…”

“…In the past twenty years, partial fractional differential equations (P-FDEs) have been motivated due to their various applications in several fields of science such as fluid and layer flows, multi-energy groups of neutron diffusion processes, neutral and multi pantograph systems, dynamic and hyperbolic systems, statistical mechanics model, material sciences and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. These important phenomena and applications are well described by P-FDEs.…”

“…The main advantage of using fractional derivatives with an arbitrary order is that they are flexible more than classical derivatives and also they are not-local. The two famous and important fractional deriavtives in applications are: The Riemann-Liouville FD (R-L-FD) and C-FD [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The relationship between the R-L and the C-FDs are very closed since the R-L-FD can be converted to the C-FD under some regularity assumptions of the function.…”

“…In most cases, exact solutions do not exist for many Partial differential equations (PDEs), therefore, several numerical methods are created and applied to get the approximate series solutions for such P-FDEs such as the homotopy analysis, asymptotic and perturbation methods [1][2][3]5], variational iteration and Adomian decomposition methods [2,4,8], LT and differential transform techniques, RPS method [9][10][11][12][13][14] and L-RPS method [15][16][17][18][19].…”

Fourth-order Compact Iterative Scheme for the Two-dimensional Time Fractional Sub-diffusion Equations

“…Fractional differential equations have gained more attention over the last few years because of their application in various fields of science and technology such as in physics, engineering, and biology [1][2][3][4]. Most of the fractional differential equations, however cannot be solved analytically; therefore researchers turn to numerical methods for alternative solvers.…”