Modification of the Differential Transform Method With Search Direction Along the Spatial Axis

Abstract

This paper presents a modification of the Differential Transform Method (ModDTM) formulated so that the search direction of the solution is along a spatial axis (x). It is shown that the method is applicable to a wide spectrum of 1+1 partial differential equations, integro partial differential equations, and integral equations. The solutions obtained are in the form of a Taylor series, the coefficients of which are determined by recursively operating a differential transform of the equation. These solutions are either exact and in closed form, or are good approximations. To illustrate the application of the method, as well as to show its versatility, examples are chosen from all the categories mentioned earlier. These include equations such as the nonlinear Fisher, combined KdV-mKdV, Hunter–Saxton (H-S), Fornberg–Whitham (FW), coupled systems such as the Whitham–Broer–Kaup (WBK) and the sine-Gordon (s-G) equations, integral equations, Volterra partial integro-differential equations (PIDE), and linear and nonlinear versions of the (complex) Schröedinger equation. The Tables and 3D-plots show the good rate of convergence of the obtained solutions with the exact ones. It is thus found that the new scheme is successful in producing satisfactory results as any other method in which the conventional approach of searching for solutions along the temporal axis is followed. Further, the procedures involved are simple, and hence may be operated with great ease.