Approximate solutions of coupled Ramani equation by using RDTM with compared DTM and exact solutions

Abstract

In this paper, we present the new approximate solutions of famous coupled Ramani Equation. In order to obtain the solution, we use the semi-analytic methods differential transform method (DTM) and reduced form of DTM called reduced differential transform method (RDTM). We compare our proposed methods with exact solution and also compare with together. Numerical results show clearly that DTM and RDTM are very effective and also provide very accurate solutions. Additionally, it should be noted that RDTM is applied very easily, fast and more convergent than DTM for these kind of problems.

Approximate Analytic Solutions of Two-Dimensional Nonlinear Klein–Gordon Equation by Using the Reduced Differential Transform Method

“…We then compute the set of values of u k (x, y) n k�0 . Substituting equations ( 27), (31), and (32) into equation (26)…”

“…It is an iterative procedure based on the use of the Taylor series solution of differential equations. RDTM has been successfully applied to solve various nonlinear partial differential equations [18][19][20][21][22][23][24][25][26][27][28].…”

“…irdly, this method can reduce the size of the calculations and can provide an analytic approximation, in many cases exact solutions, in rapidly convergent power series form with elegantly computed terms (see [29] and the references therein). Moreover, the reduced differential transform method (RDTM) has an alternative approach of solving problems to overcome the demerit of discretization, linearization, or perturbations of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration [29,30]. e structure of the remaining parts of this paper is organized as follows.…”

Analytical Solution of Two-Dimensional Sine-Gordon Equation

“…Thirdly, this method can reduce the size of the calculations and can provide an analytic approximation, in many cases exact solutions, in rapidly convergent power series form with elegantly computed terms ( [37] and see the references therein). Moreover, the reduced differential transform method (RDTM) has an alternative approach of solving problems to overcome the demerit of discretization, linearization, or perturbations of well-known numerical and analytical methods such as Adomian decomposition, differential transform, homotopy perturbation, and variational iteration [37][38][39].…”

A Comparison between the Reduced Differential Transform Method and Perturbation-Iteration Algorithm for Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

“…Since it does not require any parameter, discretization, linearization or small perturbations, thus it reduces the size of computations and can be easily used. The RDTM was used for solving the generalized Korteweg-de Vries equation [14], the fractional Benney-Lin equation [15], the Wu-Zhang equation [16], the equal width wave equation and the inviscid Burgers equation [17], the Sine-Gordon equation [18], the Burgers and Huxley equations [19], the time-fractional telegraph equation [20], the generalized Drinfeld-Sokolov equations and Kaup-Kupershmidt equation [21], the Zakharov-Kuznetsov equations [22], the heat-like equations [23], the coupled Ramani equations [25], two integral members of nonlinear Kadomtsev-Petviashvili hierarchy equations [26], and the second order hyperbolic telegraph equation [27]. Few studies have been applied RDTM to solve the Navier-Stocks equations, which is one of the reasons for choosing it as a method for solving the lid-driven cavity flow.…”