A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method

and Applied Analysis 3 3. Two-Dimensional Differential Transform Method The two-dimensional differential transform of function w(x, y) is defined as W(k, h) = 1 k!h! [ ∂ k+h w (x, y) ∂x∂y ] x=0 y=0 , (9) where it is noted that upper case symbol W(k, h) is used to denote the two-dimensional differential transform of a function represented by a corresponding lower case symbol w(x, y). The differential inverse transform of W(k, h) is defined as


Introduction
The partial differential algebraic equation was first studied by Marszalek.He also studied the analysis of the partial differential algebraic equations [1].Lucht et al. [2][3][4] studied the numerical solution and indexes of the linear partial differential equations with constant coefficients.A study about characteristics analysis and differential index of the partial differential algebraic equations was given by Martinson and Barton [5,6].Debrabant and Strehmel investigated the convergence of Runge-Kutta method for linear partial differential algebraic equations [7].
On the other hand, the differential transform method was used by Zhou [13] to solve linear and nonlinear initial value problems in electric circuit analysis.Analysis of nonlinear circuits by using differential Taylor transform was given by Köksal and Herdem [14].Using onedimensional differential transform, Abdel-Halim Hassan [15] proposed a method to solve eigenvalue problems.The two-dimensional differential transform methods have been applied to the partial differential equations [16][17][18][19].The differential transform method extended to solve differential-difference equations by Arikoglu and Ozkol [20].Jang et al. have used differential transform method to solve initial value problems [21].The numerical solution of the differential-algebraic equation systems has been studied by using differential transform method [22,23].
In this paper, we have considered linear partial differential equations with constant coefficients of the form where  = [0, ∞), Ω = [−, ],  > 0, and , ,  ∈ R × .In (1) at least one of the matrices ,  ∈ R × should be singular.If  = 0 or  = 0, then (1) becomes ordinary differential equation or differential algebraic equation, so we assume that none of the matrices  or  is the zero matrix.

Indexes of Partial Differential Algebraic Equation
Let us consider (1), with initial values and boundary conditions given as follows: where  ∈ M  ⊆ {1, 2, . .
where  and  are independent of  and .(ii) (,  + ), Re() > , called as the matrix pencil, is regular.(iii) (,    + ) is regular for all , where   is an eigenvalue of the operator  2 / 2 together with prescribed BCs.(iv) The vector (, ) and the initial vector () are sufficiently smooth.
In order to define a spatial index, we need the Kronecker normal form of the DAE (4).Assumption (iii) guarantees that there are nonsingular matrices  , ,  , ∈ C × such that where  , ∈ C  , = 0, Here, we will assume that there is a real number  * ≥  such that the index set M ()   is independent of the Laplace parameter , provided Re() ≥  * .Definition 1.Let  * ∈ R + be a number with  * ≥ , such that for all  ∈ C with Re() ≥  * (1) the matrix pencil (,  + ) is regular, (2) M ()   is independent of , i.e., M ()  = M  , (3) the nilpotency of  , is ]  ≥ 1.
Then ] , = 2]  − 1 is called the "differential spatial index" of the LPDAE.If ]  = 0, then the differential spatial index of LPDAE is defined to be zero.
If we use Fourier transform, (1) can be transformed into with   () = ( 1 (), . . .,   ())  and for  ∉ M  , which results from partial integration of the term 6) is a DAE depending on the parameter   which can be solved uniquely with suitable ICs under the assumptions (iv) and (v).Analogous to the case of the Laplace transform, the above assumption (iv) implies that there exist regular matrices  , ,  , such that With  , ∈ R  1 × 1 . , ∈ R  2 × 2 is again a nilpotent Jordan chain matrix with Riesz index ] , , where  1 +  2 = .
To characterize M  , we introduce M ()  ⊆ {1, 2, . . ., } as the set of indices of components of û for which initial conditions can be prescribed arbitrarily.Therefore, we always assume in the context of a Fourier analysis of The differential spatial and time indexes are used to decide which initial and boundary values can be taken to solve the problem.

Two-Dimensional Differential Transform Method
The two-dimensional differential transform of function (, ) is defined as where it is noted that upper case symbol (, ℎ) is used to denote the two-dimensional differential transform of a function represented by a corresponding lower case symbol (, ).The differential inverse transform of (, ℎ) is defined as From ( 9) and ( 10), we obtain The concept of two-dimensional differential transform is derived from two-dimensional Taylor series expansion, but the method doesn't evaluate the derivatives symbolically.

Application
We have considered the following PDAE as a test problem: with initial values and boundary values The right hand side function  is and the exact solutions are matrices  ,  , and  ,  , are found as From (38), we have  , = 0 and  , = 0. Then the PDAE (32) has differential spatial index 1 and differential time index 1.So, it is enough to take M ()  = {1} and M ()  = {2} to solve the problem.

Conclusion
The computations associated with the example discussed above were performed by using Computer Algebra Techniques [24].We show the results in Tables 1 and 2 for the solution of (32) by numerical method.The numerical values on Tables 1 and 2 obtained above are in full agreement with the exact solutions of (32).This study has shown that the differential transform method often shows superior performance over series approximants, providing a promising tool for using in applied fields.

Table 1 :
The numerical and exact solution of the test problem(32), where  1 (, ) is the exact solution and  * 1 (, ) is the numerical solution, for  = 0.1.