Abstract and Applied Analysis

Volume 2013 (2013), Article ID 705313, 8 pages

http://dx.doi.org/10.1155/2013/705313

Research Article

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

Murat Osmanoglu and Mustafa Bayram

Department of Mathematical Engineering, Chemical and Metallurgical Faculty, Yildiz Technical University, Esenler 34210, Istanbul, Turkey

Received 14 January 2013; Accepted 28 February 2013

Academic Editor: Adem Kiliçman

Copyright © 2013 Murat Osmanoglu and Mustafa Bayram. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We have considered linear partial differential algebraic equations (LPDAEs) of the form , which has at least one singular matrix of . We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.

1. 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. [2–4] 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].

There are numerous LPDAEs applications in scientific areas given, for instance, in the field of Navier-Stokes equations, in chemical engineering, in magnetohydrodynamics, and in the theory of elastic multibody systems [4, 8–12].

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 one-dimensional 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 [16–19]. 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 , , and . In (1) at least one of the matrices should be singular. If or , then (1) becomes ordinary differential equation or differential algebraic equation, so we assume that none of the matrices or is the zero matrix.

2. Indexes of Partial Differential Algebraic Equation

Let us consider (1), with initial values and boundary conditions given as follows:

where , is the set of indices of components of for which boundary conditions can be prescribed arbitrarily, and , is the set of indices of components of for which initial conditions can be prescribed arbitrarily. The initial boundary value problem (IBVP) (1) has only one solution where a function is a solution of the problem, if it is sufficiently smooth, uniquely determined by its initial values (IVs) and boundary values (BVs), and if it solves the LPDAE point wise.

Definition of the indexes can be given using the following assumptions.

(i)Each component of the vectors , and satisfy the following condition:

where and are independent of and . (ii), called as the matrix pencil, is regular. (iii) is regular for all , where is an eigenvalue of the operator together with prescribed BCs. (iv)The vector and the initial vector are sufficiently smooth.

If we use Laplace transform, from assumption (ii), (1) can be transformed into

if is a singular matrix, then (4) is a DAE depending on the parameter . To characterize , we introduce as the set of indices of components of for which boundary conditions can be prescribed arbitrarily.

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 such that

where and is a nilpotent Jordan chain matrix with is the unit matrix of order . The Riesz index (or nilpotency) of is denoted by (i.e., ).

Here, we will assume that there is a real number such that the index set is independent of the Laplace parameter , provided .

Definition 1. Let be a number with , such that for all with

(1)the matrix pencil is regular, (2) is independent of , i.e., ,(3)the nilpotency of is .

Then is called the “differential spatial index” of the LPDAE. If , then the differential spatial index of LPDAE is defined to be zero.

If we use Fourier transform, (1) can be transformed into

with and

for , which results from partial integration of the term .

If is a singular matrix, then (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 . is again a nilpotent Jordan chain matrix with Riesz index , where .

To characterize , we introduce as the set of indices of components of for which initial conditions can be prescribed arbitrarily. Therefore, we always assume in the context of a Fourier analysis of that is independent of , i.e., .

Definition 2. Assume for that

(1)the matrix pencil is regular,(2) is independent of , i.e., ,(3)the nilpotency of is .

Then the PDAE (1) is said to have uniform differential time index .

The differential spatial and time indexes are used to decide which initial and boundary values can be taken to solve the problem.

3. Two-Dimensional Differential Transform Method

The two-dimensional differential transform of function is defined as

where it is noted that upper case symbol is used to denote the two-dimensional differential transform of a function represented by a corresponding lower case symbol . The differential inverse transform of 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.

Theorem 3. Differential transform of the function is

see [17].

Theorem 4. Differential transform of the function is

see [17].

Theorem 5. Differential transform of the function is

see [17].

Theorem 6. Differential transform of the function is

see [17].

Theorem 7. Differential transform of the function is

see [17].

Theorem 8. Differential transform of the function is

see [17].

Theorem 9. Differential transform of the function is

see [17], where

Theorem 10. Differential transform of the function is

Proof. From Definition 1, we can write

where

hence,

Theorem 11. Differential transform of the function is

Proof. From Definition 2, we can write

Hence, we can write

Using Definition 2, we obtain

Theorem 12. Differential transform of the function is

Proof. Let be differential transform of the function . From Theorem 7, we can write that differential transform of the function is

from Theorem 4, we can write

If we substitute (30) into (29), we find

4. 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

If nonsingular matrices , , , and are chosen such as

matrices and are found as

From (38), we have and . Then the PDAE (32) has differential spatial index 1 and differential time index 1. So, it is enough to take and to solve the problem.

Taking differential transformation of (32), we obtain

The Taylor series of functions and about are

The values in (39) and (40) are coefficients of polynomials (41) and (42). If we use Theorem 3 for boundary values, we obtain

In order to write and in (40), we have

If we take in (43) and (44), we obtain

From (45) and (46), we find

In this manner, from (40), (44), and (45), the coefficients of the are obtained as follows:

Using the initial values for the second component, we obtain the following coefficients:

The coefficients of the can be found using (47), (48), (49), and taking and in (39) as follows:

If we write the above values in (39) and (40), then we have

Numerical and exact solution of the given problem has been compared in Tables 1 and 2, and simulations of solutions have been depicted in Figures 1, 2, 3, and 4, respectively.

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

0.1
−0.0895789043
−0.0895789043
0

0.2
−0.0810543445
−0.0810543422
0.0000000023

0.3
−0.0733410038
−0.0733409887
0.0000000151

0.4
−0.0663616845
−0.0663616355
0.0000000490

0.5
−0.0600465353
−0.0600464409
0.0000000944

0.6
−0.0543323519
−0.0543322800
0.0000000719

0.7
−0.0491619450
−0.0491621943
0.0000002493

0.8
−0.0444835674
−0.0444849422
0.0000013748

0.9
−0.0402503963
−0.0402546487
0.0000042524

1.0
−0.0364200646
−0.0364305555
0.0000104909

Table 2: The numerical and exact solution of the test problem (32), where is exact solution and is numerical solution, for .

0.1
−0.9949046649
−0.9949046653
0.0000000004

0.2
−0.9799685711
−0.9799685778
0.0000000067

0.3
−0.9552409555
−0.9552409875
0.0000000320

0.4
−0.9209688879
−0.9209689778
0.0000000899

0.5
−0.8774948036
−0.8774949653
0.0000001637

0.6
−0.8252530813
−0.8252532000
0.0000001187

0.7
−0.7647657031
−0.7647652653
0.0000004378

0.8
−0.6966370386
−0.6966345778
0.0000024608

0.9
−0.6215478073
−0.6215398875
0.0000079198

1.0
−0.5402482757
−0.5402277778
0.0000204979

Figure 1: The graphic of the function in the test problem (32).

Figure 2: The graphic of the function in the test problem (32).

Figure 3: The graphic of the function in the test problem (32).

Figure 4: The graphic of the function in the test problem (32).

5. 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.

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