Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

Abstract

The main aim of this paper is to propose a new and simple algorithm namely homotopy analysis transform method (HATM), to obtain approximate analytical solutions of integral equations. Integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter ℏ to guarantee the convergence of the obtained series solution.

Solving Fractional Integro Differential Equations by Homotopy Analysis Transform Method

“…where q ∈ [0, 1] is an embedding parameter and ϕ (x, t; q) is the real function of x, t, and q. By means of generalizing the traditional homotopy methods, Liao [18][19][20][21][22][23][24] constructed the zero order deformation equation:…”

“…The main aim of this article is to present analytical and approximate solution of fractional integro-differential equations by using new mathematical tool like homotopy analysis transform method. The proposed method is coupling of the homotopy analysis method HAM and Laplace transform method [21][22][23][24][25][26]. We have studied some of linear and nonlinear fractional integro-differential equations with the help of homotopy analysis transform method.…”

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

“…(3.12) is linear and thus can be easily solved, especially by means of symbolic computation software such as Mathematica, Maple, Matlab. Yabushita et al [29] and Mohamed S. Mohamed et al [30,31,32] applied the homotopy analysis method to nonlinear ODE's and suggested the so called optimization method to find out the optimal convergence control parameters by minimum of the square residual error integrated in the whole region having physical meaning. Their approach is based on the square residual error.…”

“…Also the inverse of local fractional derivative to of f (x) order α in interval [a, b] is defined by [30,31]…”

Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

“…Several robust methods have been used to solve the FDEs, the fractional differential equations and dynamic systems containing fractional derivatives. Some of the most important methods are Adomian's decomposition method [10][11][12], the exp-function method [13], He's variational iteration method [14,15], the fractional subequation method [16], the first integral method [17], the homotopy analysis method [18], the (G ′ /G)-expansion method [19], the homotopy perturbation method [20,21], the spectral methods [22], and the transform methods [23]. In [24], the authors presented two methods, which are the exp(−ϕðξÞ)-expansion method and the Kudryashov method.…”