Homotopy analysis method to obtain numerical solutions of the Painlevé equations

Abstract

Communicated by M. EfendievIn this paper, the homotopy analysis method (HAM) is presented to obtain the numerical solutions for the two kinds of the Painlevé equations with a number of initial conditions. Then, a numerical evaluation and comparison with the results obtained via the HAM are included. It illustrates the validity and the great potential of the HAM in solving Painlevé equations. Although the HAM contains the auxiliary parameter, the convergence region of the series solution can be controlled in a simple way.

Approximate Solutions for Solving Fractional-order Painlevé Equations

“…By substituting the collocation points (11) into 12, we get the following system of matrix equations…”

“…To obtain a system of matrix equations for the fractional derivative, we insert the collocation points (11) into (14) to get…”

“…There has been significant interest in developing analytical, approximative as well as numerical schemes for the solution of the classical and fractional Painlevé differential equations (1)- (2). The most significat analytical schemes include Adomian's decomposition (ADM) and homotopy perturbation methods (HPM) [7] , variational iteration method (VIM) and HPM [12] , homotopy analysis method [8,11] , sinc collocation method and VIM [23] , optimal homotopy asymptotic method [16] , to name but a few. On the other hand, numerical techniques such as Chebyshev series [6,13] , computational intelligence technique based on neural networks and particle swarm optimization [21,22] , and reproducing kernel Hilbert space algorithms [3] have been developed in the past to solve the nonlinear equation (1)- (2).…”

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

“…Hesameddini and Latifizadeh [29] obtained solutions for the first and second types of the Painlevé equations with 8, 10 and 12 iterations. Wang [30] developed an approximate analytical solution of relativistic Toda lattice system and concluded that using HAM may be very effective for solving differential difference equations (DDEs).…”

Meromorphic solutions of some algebraic differential equations related Painlevé equation IV and its applications

“…The Painlevé equations, although discovered from strictly mathematical considerations, can be used to describe several physical phenomena 12 including nonlinear waves, statistical mechanics, general relativity, quantum field theory, and quantum gravity. 13 Many powerful methods for constructing the solutions of Painlevé equations have been developed, including the isomonodormy system, 14 the Padé, 15,16 the functional fitting, 17 the Sinc-Galerkin, 18 the homotopy analysis, 19 and the adomian decomposition methods. 20 Recently, Yuan et al presented a new method, named complex method, to get the meromorphic solutions of the Fisher-type equations.…”