Parameterized Differential Transformation Method for Solving Initial and Boundary Value Problems

Abstract

Recently there has been developed various modifications of some numerical methods of
approximating the solution of high order ordinary and partial differential equations. The differential
transformation method (DTM) is one of the numerical methods that allows one to find an approximate
solution in the case of linear and nonlinear initial and/or boundary value problems for various type of
differential equations. This numerical method was first proposed by Zhou, in solving of linear and
nonlinear boundary value problems in electrical circuit analysis. The main advantage of DTM is that it
can be applied directly to solve nonlinear ordinary and partial differential equations without requiring
perturbation or linearization. This study develops a new extension/generalization of DTM called α
parameterized differential transform method (α-PDTM). The α -PDTM differs from classical DTM in the
calculation of differential transform coefficients. In this work, we applied the parameterized differential
transformation method to solve the following simple but illustrative differential equation.
�
�''(x) + u(x) +3ex = 0,
together with boundary conditions
�
�(0) = 3,
�
� ∈[0,1]
�
�'(0) = 0.
We also plotted the approximate solution to demonstrate the robustness and efficiency of our own
method. The results obtained showed that the proposed new method can become an alternative way to
solve boundary value problems of various types. Note that in some concrete values of the auxiliary
parameter our own method reduces to the classic DTM.