Two-dimensional differential transform for partial differential equations

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Abstract

The differential transform is a numerical method for solving differential equations. In this paper, we present the definition and operation of the two-dimensional differential transform. A distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations. Partial differential equation of parabolic, hyperbolic, elliptic and nonlinear types can be solved by the differential transform. We demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation.

Introduction

Integral transforms, such as the Laplace and the Fourier transforms are commonly used to solve differential equations in engineering problems. The merit of the integral transform is its ability to transform differential equations to algebraic equations, which leads to a systematic and simple solution procedure. However, nonlinear problems increase in complexity when integral transform techniques are applied.

The differential transform is a numerical method for solving differential equations. The concept of the differential transform was first proposed in Zhou [7], and its main application therein is to solve both linear and nonlinear initial-value problems in electric circuit analysis. This method constructs an analytical solution in the form of a polynomial. It is different from the traditional high-order Taylor series method, which requires symbolic computation of the necessary derivatives of the data functions. The Taylor series method is computationally expensive for large orders. The differential transform is an iterative procedure for obtaining analytic Taylor series solutions of differential equations.

In this paper, the definition and operation of the two-dimensional differential transform is introduced. Different types of partial differential equations are solved using the differential transformation technique, which demonstrates its feasibility in solving partial differential equations.

Section snippets

One-dimensional differential transform

The basic definitions and operations of the one-dimensional differential transform are introduced in [1], [5], [7] as follows:

Definition 1

If x(t) is analytic in the time domain T then let

ϕt,k=dkxtdtk∀t∈T.

For t=ti where ϕt,kti,k, where k belongs to the set of non-negative integer, denoted as the K domain. Therefore, Eq. (1) can be rewritten asXikti,k=dkxtdtkt=ti∀k∈K,where X(k) is called the spectrum of x(t) at t=ti in the K domain.

If x(t) is analytic then x(t) can be represented asxt=k=0t−tikk!Xk.

Eq.

Numerical case studies

Four examples are used to demonstrate the procedure of solving partial differential equation by the two-dimensional differential transformation technique. The first two examples are well-posed problems, that is, the initial conditions and the boundary conditions in such problems are sufficiently differentiable. In these cases, differential transform can be applied to such problems without any modification. In the third example, the initial condition is an arbitrary continuous function, but

Conclusions

In this paper, the definition and operation of two-dimensional differential transform were introduced. Using the differential transform, differential equations in the time domain can be transformed to algebraic equations in the K (spectrum) domain, and the resulting algebraic equations are called iterative equations or K-equations. The overall spectra can be calculated through the initial and boundary conditions in association with the K-equations. According to the inverse operation of proposed

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