On Solving System of Linear Differential-Algebraic Equations Using Reduction Algorithm

In this paper, we present a new reduction algorithm for solving system of linear differential-algebraic equations with power series coefficients. In the proposed algorithm, we transform the given system of differential-algebraic equations into another simple equivalent system using the elementary algebraic techniques. is algorithm would help to implement the manual calculations in commercial packages such as Mathematica, Maple, MATLAB, Singular, and Scilab. Maple implementation of the proposed algorithm is discussed, and sample computations are presented to illustrate the proposed algorithm.

In this paper, we are concerned with a linear system of differential-algebraic equations of the following form: where x is a complex variable, A(x) and B(x) are m × n matrices with analytic functions entries, f(x) is an m-dimensional column matrix with analytic functions entries, u(x) is an n-dimensional unknown column matrix which is going to be determined, and z � (d/dx) is a differential operator. In this paper, we focus on creating a new reduction algorithm using elementary algebraic techniques as well as the implementation of the proposed algorithm in Maple.
Using this algorithm, we can transform a given system of DAEs into another equivalent system, where we can solve the reduced system easily. e rest of the paper is organized as follows: in Section 2, we present a new reduction algorithm to solve the given system of DAEs with certain examples to illustrate the proposed reduction algorithm and Section 3 discusses the Maple implementation of the proposed algorithm with sample computations.

Reduction Algorithm.
e following lemma is one of the essential steps to create a new reduction algorithm. It shows that any matrix of formal power series centered at origin can be transformed into a block matrix.
Lemma 1 (see [23,26,29] where A ij denote the i-th row and j-th column block of matrix A, A 11 ∈ K[[x]] r×r is a block matrix, and r is the rank of matrix A.
Suppose that L ∈ K[[x]][z] m×n is a given matrix differential operator. Using Lemma 1, we can construct two unimodular matrix differential operators S 1 and T 1 , by finding the basis of left null space and right null space of the matrix differential operator L, such that where A 1 � A 11 0 0 0 and B 1 � B 11 0 0 0 . Now using Lemma 1 to matrix A 1 of (5), we can get an unimodular matrix S 2 such that where A 2 � S 2 A 1 � A 11 0 0 0 and A 11 is invertible matrix, and B 2 � S 2 B 1 � B 11 0 0 0 . Again, using Lemma 1 to matrix B 2 of (6), we can construct an unimodular matrix T 2 such that where such that the given system is in reduced form as follows: where L � SLT has the form L 11 0 0 0 and L 11 is invertible matrix differential operator; A has the form A Indeed, (i) if rank (L) < rank (A) and rank (L) < rank (B), then the reduced system of DAEs (10) has the following form: where L � L 11 z L 12 0 Hence, the system of DAEs in (3) is decomposed into two systems as follows: 2 International Journal of Mathematics and Mathematical Sciences with some necessary conditions on the right-hand side expressed by f 3 � 0.
(ii) If rank (L) � rank (A) � rank (B), then the reduced DAS (10) has the following form: where L � L 11 0 Using Lemma 1 to L, one can construct two unimodular matrices S 1 and T 1 (obtained using a basis of left null space and right null space of L ) as follows: Now the unimodular matrices S 1 and T 1 are us, multiplying operator L on the left and right by S 1 and T 1 yields the operator where International Journal of Mathematics and Mathematical Sciences Now, using Lemma 1 to matrix A 1 , we can construct an unimodular matrix S 2 , using a basis of left null space of A 1 , as follows: We have where Again, using Lemma 1 to matrix B 2 of the matrix differential operator L 2 , we can construct an unimodular matrix T 2 using a basis of right null space of B 2 as follows: We have If we denote S � S 2 S 1 and T � T 1 T 2 , then we have two unimodular matrix differential operators: We have that the given system is in reduced form as follows: where L � SLT has the form L 11 0 0 0 and is invertible matrix differential operator.
Example 2. Consider a matrix differential operator as given below: Applying the proposed algorithm in eorem 1 to matrix differential operator (28) similar to example 1, one can construct two unimodular matrix differential operators S and T as International Journal of Mathematics and Mathematical Sciences S � 1 0 0 0 such that the given system is in reduced form as follows: where L � SLT has the form L 11 0 0 0 and is invertible matrix differential operator.
Example 3. Consider the following system of differentialalgebraic equations to verify that the reduced system and the given system of DAEs have the same solution: e solution of the given system (32) is In particular, if we take u 1 � 0, then the solution becomes u 1 � 0, u 2 � (x 2 /2) + (c 2 e − x /2) + c 2 and u 3 � c 1 e − x . e operator notation Lu � f of the given system (14) is given by where L � z 0 z + 1 1 2 z 1 2 z 0 2 z + 1 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ . Now, apply the proposed algorithm to matrix differential operator L to get a reduced operator L with two unimodular matrix differential operators S and T. We get Now the reduced system of DAEs is 6 International Journal of Mathematics and Mathematical Sciences Solution of the reduced system (36), for u 1 � 0, is u 1 � 0, u 2 � (x 2 /2) + (c 1 e − x /2) + c 2 and u 3 � c 1 e − x .
One can observe that the solution of the given system of DAEs (32) and the reduced system of DAEs (36) have the same solution. We can also observe that solving the reduced system (36) (contains two equations only) is simple compared to solving the given system (32) (contains three equations).

Maple Implementation
In this section, we discuss the Maple implementation of the algorithm by creating different data types. Using the Maple package, one can obtain the two unimodular matrix differential operators S, T and the reduced matrix differential operator of the given system. In Maple implementation, x is complex variable and δ � (d/dx) is the differential operator.

Input:
A and B, the coefficient matrices of a given matrix differential operator L � A z + B. Output: L, S, and T, the reduced matrix differential operator L of a given matrix differential operator L and two unimodular matrix differential operators S and T. e following procedure is DAEs_Reduction: the reduced matrix differential operator of a given matrix differential operator with two unimodular matrix differential operators. In this procedure, δ � (d/dz) is differential operator and x is complex variable.