A New Algorithm Based on Differential Transform Method for Solving Partial Differential Equation System with Initial and Boundary Conditions

In this paper, a new Fourier-differential transform method (FDTM) based on differential transformation method (DTM) is proposed. The method can effectively and quickly solve linear and nonlinear partial differential equations with initial boundary value (IBVP). According to boundary condition, the initial condition is expanded into a Fourier series. After that, the IBVP is transformed to an iterative relation in K-domain. The series solution or exact solution can be obtained. The rationality and practicability of the algorithm FDTM are verified by comparisons of the results obtained by FDTM and the existing analytical solutions.


Introduction
The differential transform method (DTM) is a powerful approximate analytic method for solving linear and nonlinear differential equations. Thus it provides widely applicable technique to construct an analytical solution of differential equations in a polynomial form. Since the basic idea of the DTM was introduced by Pukhov [1], the method has been studied intensely and has substantially grown.
A large amount of literature about the DTM, its applications and its extensions is available, e.g. [2]- [28].

One-Dimensional Differential Transform
The definitions and operations of one-dimensional differential transform method (DTM) are introduced in [3]- [13]. The basic definitions of the DTM are given as follows: The differential transform of the k th differentiable function ( ) Some of the fundamental mathematical operations performed by one-dimensional differential transform can be obtained in Table 1.

Two-Dimensional Differential Transform
Consider a function of two variables ( ) , : , u x t is analytic and continuously differentiable with respect to variables in the domain of interest, the two-dimensional transform at ( ) 0, 0 are given as follows [18] [20] [22] [24] ( ) ( ) The spectrum function ( ) , U i j is the transformed function, which is also called the T-function. The function ( ) , u x t is the original function. The differential inverse transform of ( ) , U i j is defined as follows.
Substituting Equation (6) into Equation (5), it can be obtained as Unlike the traditional high-order Taylor series method which requires complicated symbolic computations, with this method, the given equation and related conditions are transformed into a recurrence relation, through which one can easily obtain the coefficients of a Taylor series solution.
The main advantage of the DTM is that it provides an explicit and numerical solution with minimal calculations. Another important advantage is that this method can be applied directly to nonlinear problems without linearization, discretization or perturbation. Some of the fundamental mathematical operations performed by two-dimensional differential transform can be obtained in Table  2. Table 1. One-dimensional differential transform.

Original function
Transformed function

The Fourier-Differential Transform Method (FDTM)
Many engineering phenomenon are mathematically modeled by categorizing them in the initial boundary value problems (IBVPs). In this study, an efficient algorithm based upon the differential transform method (DTM) is considered to solve the system of partial differential equations with initial and zero boundary conditions. In order to illustrate the algorithm, a more general form of equations with initial conditions and zero Dirichlet boundary conditions or zero Neumann boundary conditions or zero mixed boundary conditions where, L is liner differential operators, N is nonliner differential operators, the vector of main unknown functions, x is spatial variable and t is time.
On the basis of the DTM, we shall introduce an effective algorithm to obtain the approximate solution of Equation (8) under the initial conditions Equation (9) and the boundary conditions Equations (12)- (17). The main steps of the algorithm are: Step 1, Appling the differential transformation in Equation (8), and an iterative formula is obtained.
where, , 0,1, 2, , Step 2, Duce According to the characteristics of boundary conditions, the initial function is respectively expanded into corresponding Fourier series [29] [30]. The expanded form of the initial conditions see Table 3.
Step 3, Imposing the truncated series solution obtained in Step 2 on the initial and boundary conditions, a linear or nonlinear algebraic equations system can be obtained by where, , 0,1, 2, , 1 i j n = −  .
Step 4, Solve system in Step3 to determine

Expmple
In order to verify the effectiveness of the above method, three typical initial boundary value problems are solved. It should be noted that the following examples only serve as illustrations and the more complex problems can be tackled by the proposed technique, e.g. higher-ordered ODEs, PDEs (system) or other functional equations involving nonlinear terms with zero boundary conditions. Example 1. The linear heat equation was first considered at with the initial data ( ) and subject to the boundary conditions where, 0 α > is constant. Based on the FDTM, we have ones For the initial condition, it can be obtained as From the definition of inverse differential transform, we get Note that Equation (2) is often used to describe the law of heat conduction. Because its heat conduction coefficient is constant, it belongs to the category of linear partial differential equations. For linear partial differential equations, the separation variable method is often used to solve, but this paper gives different solutions method. The solution obtained by the FDTM converge to the solution by separation of variables method [31]. This also confirms the rationality of this method.
Example 2. Partial differential equations with constant coefficients are considered in this example.
and boundary conditions where, 1 2 3 , , C C C are costant coefficient and 1 2 Remark: The system described above has been widely used in geotechnical engineering [32].
Based on the FDTM, we have ones where 1 λ and 2 λ are the eigenvalues of the matrix It shoud be noted that the series expression of a sine function is ( ) ( ) with the initial condition i.e.