A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein‐Gordon Equation of Arbitrary (Fractional) Orders

Abstract

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

“…Theorem 5. Let satisfy the Lipschitz condition (13) and then the problem (11) has unique solution ( , ), whenever 0 < < 1.…”

“…For example, [11] applied HPM to solve a class of initialboundary value problems of fractional partial differential equations over finite domain. [12] used HPM for solving the Klein-Gordon partial differential equations of fractional order. Furthermore, many authors applied HPM for solving and investigating linear and nonlinear partial differential equations of fractional ordering; see [13,14].…”

“…[12] used HPM for solving the Klein-Gordon partial differential equations of fractional order. Furthermore, many authors applied HPM for solving and investigating linear and nonlinear partial differential equations of fractional ordering; see [13,14]. For more details about homotopy perturbation method and its applications, we refer to [15,16].…”

An approximate analytical solution of the fractional multi-dimensional Burgers equation by the homotopy perturbation method

“…The HPM has successfully been applied to solve many linear and nonlinear differential equations, see [2-5, 19, 29], etc. It is well known that the homotopy perturbation method (HPM) is an effective method which provides a simple solution without any assumption of linearization [12] and [20], Therefore, we use the HPM technique to obtain the approximate analytical solution for the fractional multi-dimensional Burgers equation.…”

A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein-Gordon equations

“…The numerical solutions of the nonlinear Klein-Gordon equation have received considerable attention in the literature and fall into two groups: the analytical methods and the discrete ones. The analytical methods express the exact solution in the form of the elementary functions and convergent function series, such as Adomian's decomposition method [4][5][6][7][8], homotopy perturbation method [9][10][11], variational iteration method [12][13][14][15][16][17], differential transform method [18][19][20], and other methods [21]. Unlike the analytical methods, the discrete ones approximate the exact solution on a finite set of distinct points, such as finite-difference methods [22][23][24][25][26], spectral method [27][28][29], wavelets method [30,31], and others [32][33][34][35].…”