New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Abstract

In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques,unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations.

A new efficient technique for solving modified Chua’s circuit model with a new fractional operator

“…Deniz and Bildik presented the optimal perturbation iteration algorithms (OPIA) in 2016 [43]. It has been proven that these algorithms can be safely applied to many types of differential equations [44][45][46][47][48][49][50][51][52]. In this section, we give a short description of OPIA for systems of ordinary differential equation.…”

“…This technique has been applied to many types of ODEs and PDEs such as Bratu-type [43], Riccati differential equation [44], heat transfer equations [45], nonlinear systems [46], Lane-Emden types [47], generalized regularized long wave equations [48]. More studies can also be found in [49][50][51][52][53][54][55][56][57]. The rest of the current paper is planned as follows.…”

Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

“…Optimal perturbation iteration method (OPIM) is first developed by using the ideas of perturbation iteration [8,9] and optimal homotopy asymptotic methods [1][2][3]. It has been used for solving many different types of nonlinear differential equations [41][42][43][44][45][46]. In this section, we use OPIM and Laplace transform to obtain approximate solutions of the extended fractional damped Burgers' equation.…”

On the stability analysis of the time-fractional variable order Klein-Gordon equation and a numerical simulation

“…As it is well known, partial di¤erential equations are encountered frequently in many …elds of applied physics [16][17][18][19][20][21][22][23]. One of them is Klein -Gordon equation which models many problems in quantum mechanics, condensed matter physics, etc.…”