Recent Developments in Integral Transforms, Special Functions, and Their Extensions to Distributions Theory
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A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method
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 RungeKutta method for linear partial differential algebraic equations [7].
There are numerous LPDAEs applications in scientific areas given, for instance, in the field of NavierStokes 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 onedimensional differential transform, AbdelHalim Hassan [15] proposed a method to solve eigenvalue problems. The twodimensional differential transform methods have been applied to the partial differential equations [16–19]. The differential transform method extended to solve differentialdifference 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 differentialalgebraic 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. TwoDimensional Differential Transform Method
The twodimensional differential transform of function is defined as where it is noted that upper case symbol is used to denote the twodimensional 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 twodimensional differential transform is derived from twodimensional 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.


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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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.