We present a brief report on the different methods for finding exact solutions of nonlinear evolution equations. Explicit exact traveling wave solutions are the most amenable besides implicit and parametric ones. It is shown that most of methods that exist in the literature are equivalent to the “generalized mapping method” that unifies them. By using this method a class of formal exact solutions for reaction diffusion equations with finite memory transport is obtained. Attention is focused to the finite-memory-transport-Fisher and Nagumo equations.

Towards a Unified Method for Exact Solutions of Evolution Equations. An Application to Reaction Diffusion Equations with Finite Memory Transport

Differential transform method for solving linear and non-linear systems of partial differential equations

In this article, we apply the new iterative method proposed by Daftardar-Gejji and Jafari (J Math Anal Appl 316, (2006), 753–763) for solving various linear and nonlinear evolution equations. The results obtained are compared with the results by existing methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Solving evolution equations using a new iterative method

An Approximate Solution of Blasius Equation by using HPM Method

In this paper, a general framework of the reduced differential transform method is presented for solving the generalized Korteweg–de Vries equations. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology with some known techniques shows that the present approach is effective and powerful. In addition, three test problems of mathematical physics are discussed to illustrate the effectiveness and the performance of the reduced differential transform method.

https://www.mdpi.com/2297-8747/15/3/382/pdf

Reduced Differential Transform Method for Generalized KdV Equations

/pdf/explicit-solution-of-nonlinear-zk-bbm-wave-equation-using-3xqord8574.pdf

Explicit Solution of Nonlinear ZK-BBM Wave Equation Using Exp-Function Method

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

The objective of this paper is to present an analytical investigation to analyze the
 vibration of parametrically excited oscillator with strong cubic negative nonlinearity based on
 Mathieu-Duffing equation. The analytic investigation was conducted by using He's homotopy-perturbation method (HPM). In order to obtain the analytical solution of Mathieu-Duffing equation, homotopy-perturbation method has been utilized. The Runge-Kutta's (RK) algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate
 solution, has been shown graphically and compared with that of numerical solution. Afterward, the
 effects of variation of the parameters on the accuracy of the homotopy-perturbation method were studied.

/pdf/comparative-vibration-analysis-of-a-parametrically-nonlinear-99i2aeu3hl.pdf

I. Khatami

Papers

Explicit Solution of Nonlinear ZK-BBM Wave Equation Using Exp-Function Method

Assessment of two analytical approaches in some nonlinear problems arising in engineering sciences

Comparative Vibration Analysis of a Parametrically Nonlinear Excited Oscillator Using HPM and Numerical Method

Application of Homotopy Analysis Method and Variational Iteration Method for Shock Wave Equation

On the Non-Linear Deformation of Elastic Beams in an Analytic Solution