Computational Block-Pulse Functions Method for Solving Volterra Integral Equations with Delay

The aim of this work is to present method of the Block-pulse function approach to numerical solution of Volterra integral equations with delay. This method is used to obtain numerical solution. Moreover, programs for his method is written in MATLAB language. An error analysis is worked out and applications demonstrated through illustrative examples.


1-Introduction
Delay Volterra integral differential equations arise in many areas of mathematical modelling: for example, population dynamics (taking into account the gestation times), infectious diseases (accounting for the incubation periods), physiological and pharmaceutical kinetics (modelling, for example, the body's reaction to CO2, etc. in circulating blood) and chemical kinetics (such as mixing reactants), the navigational control of ships and aircraft (with respectively large and short lags), and more general control problems [1].The analysis to the Volterra integral equations with delays dates back to the works in [2] and [3].Some more recent results on this subject can be found in [4] and [5].They must be solved successfully with efficient numerical approaches.
The numerical solutions of integral equations with delays have also been discussed by several authors such as [6] and [7].Block-Pulse functions (BPFs) have been used by many authors and applied for solving various problems, for example see Steffens [8] and [9].
The main objective of the current study is to implement the BPFs has been used to solve Volterra integral equations with constant time delay of the form The layout of this paper is as follows: In the next section, we discuss BPFs, their properties and function approximation by them.In Section 3, we give the description and development of the method for solving Volterra integral equations with time delay.Error estimation and rate of convergence of the method is discussed in Section 4. In Section 5 for showing the efficiency of this method, numerical example is presented.Finally, in Section 6 is devoted to the conclusion of this paper.

2-Review of Block Pulse Functions
The purpose of this section is to interpose definition and properties of BPFs.Function approximation using BPFs and operational matrix associated with BPFs have been discussed briefly.

Definition of BPFs and their properties
An M-set of Block-Pulse function is defined over the interval [0, T) as interval [0, T) is defined as with a positive integer value for k.In this paper, it is assumed that T = 1, so BPFs are defined over [0,1).There are some main properties for BPFs, the most important of these properties disjointness, orthogonality and completeness that can be expressed as follows [10] and [11]: Disjointness.The BPFs are disjoint with each other, i.e., where ,  = 1, … ,  and   is Kronecker delta.
Orthogonality.The BPFs are orthogonal with each other, i.e., it is clear that Completeness.For every  ∈ ℒ 2 ([0,1)) approaches to the infinity, Parseval's identity holds, that is where

Vector Form and BPFs Expansion
The set of BPFs is written as Also,  is a m-vector given by where   is the block pulse coefficient with respect to the ith BPF    ().

Operational matrix of integration
The integral ∫    ()   0 is follows As represented in [11]: where Υ is called operational matrix of integration which can represented by So, the integral of every function () can be approximated as follows: For approximate a function with time delay, we consider a block pulse function containing time delay  = ( + )ℎ with a nonnegative integer p and 0 ≤  < 1 that can be expressed as a vector form: to avoid the expression   () in the above equation, we expand the function   () ( − ) into its block pulse series : where   , ,  = 1,2, … .,  are block pulse coefficients given by Therefore, the block pulse series in a vector form : ( − ) = ((1 − )  +   +1 ) () … … … (2.14) the matrix (1 − )  +   +1 is called the block pulse operational matrix for time delay as: Thus, the block pulse series of a function containing time delay  can be expressed as follows: ( − ) ≃   ( − ) =   ((1 − )  +   +1 ) () … … … (2.16)

Application of the Method
In this section, we applied the BPFs method to solve (1.1).Now, we approximate (), (), (, ) with respect to BPFs as follows : Now, assume that   be the ith row of the constant matrix   ,   be the ith row of the matrix Υ, and    be a diagonal matrix with   as its diagonal entries, we will have, where

Numerical Implementation
In this section, to achieve the validity, the accuracy and support our theoretical discussion of the proposed method, we give some computational results.The computations, associated with the example, are performed by MATLAB 7. Let   denote the Block pulse coefficient of exact solution of the given example, and let   be the Block pulse coefficient of computed solutions by the presented method.The error is defined as The exact solutions are () =  2  0 ≤  ≤ 1 .The numerical results obtained with BPFs are presented in Tables 1-2 and Figures 1-4.

Conclusion
This method is more efficient and more accurate than some other methods for solving this class of integral equations.On the other hand, the benefit of this method is low cost of computing operations.The applied method transforms the singular integral equation into triangular linear algebraic system that can be solved easily.

Figure 1 .
Figure 1.Plots of the exact solution against the computed solution when m = 32.

Figure 2 .
Figure 2. Plots of the exact solution against the computed solution when m = 64