The regular system of differential equations with convolution terms solved by Sumudu transform.

1. Introduction

A differential equation by itself is inherently underconstrained in the absence of initial values as well as boundary conditions. It is also well known that a differential equation along with the initial values or boundary conditions can be represented by an integral equation by using this integral representation, it becomes possible to solve the problem. However one of the most important achievements, and applications of integral transform methods is solving the partial differential equations (PDEs) of second order. For this purpose recently a new integral transform, which is called Sumudu transform, was introduced by Watugala [1, 2] and used by Weerakoon [3] for partial derivatives of Sumudu transform, provided the complex inversion formula in order to solve the differential equations in different applications of system engineering, control theory and applied physics. The convolution theorem of Sumudu transform was proved by Asiru in [4]. This new transform was applied to the solution of ordinary differential equations and control engineering problems; see [1, 5]. In [6], some fundamental properties of the Sumudu transform were established. In [7], this new transform was applied to the one-dimensional neutron transport equation. In fact the relationship between double Sumudu and double Laplace transforms were studied in [8, 9]. Furthermore in [10], the Sumudu transform was extended to the distributions and some of their properties were also studied. Thus, there have been several works on Sumudu transform and applied to different kind of problems.

In this paper, we prove Sumudu transform of convolution for the matrices and use to solve the regular system of differential equations.

Throughout the paper we use a square matrix, of regular system having size of polynomials and the associated determinant, . If is not the zero polynomial (which we write as ), we have where is the degree of the polynomials in the regular matrix The case of equality is so important that we make the following statement. We say that is regular if and the condition

where considered as the highest power of the variable term that occurs in the th column of matrix , that is,

Next we extend the result given in [4] as follows. For each and we define to be the matrix of polynomials given by the matrix product

where the are the coefficients of and . In terms of (7.1) in [4] we have

where the number of zero indicated is We define to be matrix of polynomials and having size defined in terms of the array of matrices:

For each complex number define a linear mapping of into . If any is zero, is the empty matrix for all and the corresponding column of matrices in is absent. If for all is defined to be the unique linear mapping of into its matrix representation is then the empty matrix. In particular consider

Then we have and Thus is the matrix computed as

In general, if is a sequence of functions on    with each being an integer and being times differentiable on , we will write, using the notation in [4],

whenever the limits exist. If any is zero, the corresponding string is absent. If for all we define The following proposition was proved in [4].

Proposition 1.1 (Sumudu transform of higher derivatives). Let be times differentiable on and let for Suppose that Then for and, for any polynomial of degree for . In particular (with here written as a column vector). For we have

Now we want to extend the above proposition to the system of differential equations as follows.

Proposition 1.2. Let  =  be functions and let =  be a sequence of integers such that a matrix of polynomial satisfies . Furthermore for each we let be times differentiable on and we let for . Suppose that for each Then for and, we have,

In the following theorem we discuss the Sumudu transform of convolution of matrices.

Theorem 1.3. Let and be Sumudu Transformable. Then where is the set of matrices for whose entries are integrable.

Proof. Suppose that and by induction.
In case we have the following matrices: as Sumudu transformable, then Sumudu transform of the above matrices are given by We have where Similarly, in case it is also true that Assuming that is true for the case we have the following matrices: having Sumudu transforms given by Similarly if then we have where Thus, The proof completes.

The inverse of will exist provided that is not a root of the equation ; hence let denote the adjugate matrix of by elementary matrix theory we have

Proposition 1.4 (Solution of homogeneous equation of regular system). Let be regular and let be times differentiable on and zero on and suppose that then is given (except at ) by the formula

Proof. By using (1.12) and assuming that is Sumudu transformable, we have Equation (1.26) becomes using (1.25) Finally by taking the inverse Sumudu transform of the above equation we have where we assume that the inverse transform exists.

The next proposition was proved in [4] for the single differential equation, and we extend it to the regular system of differential equations.

Proposition 1.5. Let be regular and let be the greatest of the real parts of the roots of the equation if   . Let be continuous on and zero on locally integrable, and Sumudu transformable and furthermore suppose that then we have for in

Note that the most general system of equations can be written in the matrix form as

which denotes the system

Here the are polynomials and if we let , respectively, and the vectors with components, then the above equation can be written in the form of

under the initial condition

Since is locally integrable, thus, Sumudu transformable for and for every such then Sumudu transform of (1.37) is given by

where is a matrix defined by

and by (1.41).

In order to find the solution of (1.37), first of all we multiply (1.39) by the inverse matrix then we get

Now, by taking the inverse Sumudu transform for both sides of (1.41)

provided that the inverse exists for each term in the right-hand side of (1.42).

To illustrate our method, we give the following example.

Example 1.6. Solve for the system of two equations

The matrix

and we have which has degree 4 = Thus regular. Now by applying Sumudu transform to the above system we have

where is given by

On using (1.25) we obtain then equation (1.45) becomes

finally, by taking inverse Sumudu transform equation (1.48) we obtain the solution of the system as follows

Thus based on the above discussions we note that the Sumudu transform can be applied for system of differential equations thus can be used in many engineering problems.


The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme  05-01-09-0720RU and Fundamental Research Grant Scheme  01-11-09-723FR. The authors also thank the referee(s) for very constructive comments and suggestions.