Comparison between homotopy analysis method and optimal homotopy asymptotic method for nth-order integro-differential equation

Abstract

Communicated by I. StratisThis paper presents general framework for solving the nth-order integro-differential equation using homotopy analysis method (HAM) and optimal homotopy asymptotic method (OHAM). OHAM is parameter free and can provide better accuracy over the HAM at the same order of approximation. Furthermore, in OHAM the convergence region can be easily adjusted and controlled. Comparison, via two examples, between our solution using HAM and OHAM and the exact solution shows that the HAM and the OHAM are effective and accurate in solving the nth-order integro-differential equation.

On the polynomial solution of the first Painlevé equation

“…OHAM reduces the size of the computational domain and it has been successfully applied to a number of nonlinear differential equations in science and engineering, e.g., to study steady flow of a fourth-grade fluid through a porous medium [20], oscillators with discontinuities and fractional-power restoring force [21], periodic solutions for the motion of a particle on a rotating parabola [22], thin film flow of a fourth-grade fluid [23], nonlinear heat transfer equations [24], and nonlinear problems in elasticity [25]. In particular, using OHAM Islam et al [26] investigated Couette and Poiseuille flows of a third-grade fluid with heat transfer analysis, Idrees et al [27] analyzed the Korteweg-de Vries (KDV) equation, Mohsen et al [28] studied viscous flow in a semi-porous channel with uniform magnetic field, and Ghoreishi et al [29] provided a comparative study for nth-order integral-differential equations. In the next section we present the mathematical formalities that will be used in the rest of the paper.…”

Modified iteration method for solving fractional gas dynamics equation

“…Nonlinear equations are much more difficult to solve than linear ones, especially analytically. There are some methods to find out solutions of fractional differential equations, such as Adomian decomposition method [5][6][7][8][9][10], variational iteration method [11][12][13][14][15], homotopy analysis method [16][17][18][19][20], homotopy perturbation method [21][22][23][24][25], and other approximate methods [26][27][28][29][30]. The new iteration method, which firstly proposed by Gejji and Jafari [31], is simple to understand and an effective method for nonlinear equations, and it is also easy to implement using computer packages.…”

An analytical method for solving the two-phase inverse Stefan problem

“…This method has found an application for solving many problems formulated with the aid of ordinary and partial differential equations [24][25][26][27], including the heat conduction problems [28][29][30][31], fractional differential equations [32,33] (for some other applications of the fractional calculus see for example [34][35][36]), integral equations [37][38][39], integro-differential equations [40,41] and others. A particular case of the homotopy analysis method is the homotopy perturbation method [16,17,42].…”