1. Introduction

Integro-Differential Equations have been discussed in many applied fields such as Biology, Physical and Engineering problems. These equations are either Fredholm or Volterra or Fredholm- Volterra Integro Differential Equations are contained in many mathematical formulations of physical phenomena.Therefore, their numerical treatment is desired. .Although, it is generally difficult in finding exact solutions of Fredholm- Volterra Integro Differential Equations (FVIDE) especially when it is nonlinear. Several numerical methods have been used to solve FVIDE among them are Approximation Method (Ezzati and Najafalizadeh, 2011) and (Salih and Mehemet, 2000). Meanwhile, in recent years, the matrix method has been developed to solve Fredholm - Volterra Integral Equations For example, the method is used to solve system of Differential Equations (Akyuz and Sezer, 2003) and Differential Algebraic Equation (Karamete and Sezer, 2002). Other methods include Block Pulse function and Operational Matrices (Leyla, Bijan and Mohammed, 2011), Chebyshev polynomials approach is used in (Yuksel, Gulsu and Sezer, 2012) to mention just a few. In this research work, the work and idea of (Aliyu, 2012) is revisited and modified as our new proposed numerical method to solve the nth order linear Fredholm-Volterra Integro-Differential Equations. The general nth order linear Volterra - Fredholm Integro Differential Equations considered in this work are in two types given as:

(1)

And,

(2)

Thus, equation (1) contains disjoint integrals while equation (2) contains mixed integrals. Here, both equations (1) and (2) are subjected to the conditions

(3)

In equations (1), (2) and (3), are real numbers, and are known given smooth functions and is the unknown function to be determined.

For the purpose of our discussion, we let

denote the indefinite integration applied to k-times and is denoted by

(4)

2. Description of Numerical Method of Solution

In this section, we discussed Standard Integral Collocation Method to solve equations (1) and (2) using the following basis functions:

* Power Series

* Chebyshev Polynomials

* Legendre's Polynomials

This method used the idea reported in (Aliyu, 2012) as applied to solve non over determined differential equation. The method was proposed and used by (Taiwo and Abubakar, 2013) to find numerical solutions to first and second orders linear and non-linear integro-differential equations.

2.1. Method of Solution by Power Series

In order to apply this method, we integrated both sides of equation (1) to have:

(5)

This implies

(6)

We assumed an approximate solution of the form

(7)

Thus, equation (7) is substituted into equation (6) to have

(8)

This implies

(9)

Thus, we collocated equation (9) at the point to obtain

(10)

Where,

(11)

Thus, equation (11) gives (N+1) algebraic linear system of equations in (N+1) unknown constants . These (N+1) algebraic linear system of equations are then solved by Gaussians Elimination Method to obtain the unknown constants which are then substituted back into equation (7) to obtain the approximate solution for case N. (the degree of the approximant)

REMARK: 1:

We demonstrated this method further by choosing n = 1, thus, putting n = 1 in equation (1), we have:

(12)

We integrated equation (12) to have

(13)

This implies

After simplification, we have

(14)

Substituting the approximate solution given in equation (7) into (14), we have

This implies,

Further simplification gives

This implies,

(15)

Where,

And,

Thus from equation (14), we have

(16)

Thus, equation (16) is collocated at the point to have

(17)

Where

Thus, equation (17) is put in matrix form as

Where,

Where,

Remark 2:

The above matrix is solved by Gaussian Elimination Method to obtain the unknown

Constants which are then substituted back into the approximate

Solution given in equation (7) to obtain the approximate solution for case N.

2.2. Method of Solution by Chebyshev Polynomials

We demonstrated this method by assuming the approximate solution of the form

(18)

Where, are unknown constants and are the Chebushev Polynomial of degrees r of first the kind which is valid in the interval and is defined by

Here,

(19)

The recurrence relation is given by

(20)

(21)

and this satisfies the recurrence relation

(22)

Substituting equation (18) into equation (6), we have

(23)

Hence, we collocated (23) at the point to have

(24)

Where

Thus, (24) gives (N + 1) algebraic linear system of equations in (N + 1) unknown constants . The (N + 1) algebraic system of linear equations are then solved by Gaussian elimination method to obtain the unknown constants which are then substituted back into equation (18) to obtain the approximate solution for case N.

2.3. Method of Solution by Legendre’s Polynomials

The Legendre’s Polynomial is denoted and defined by

(25)

(26)

With,

In this method, we assumed an approximate solution of the form

(27)

This method is similar to the two methods discussed above (the same procedure discussed as in sections (2.1.1) and (2.1.2)). We have (N + 1) algebraic linear system of equations in (N + 1) unknown constants . The (N + 1) algebraic system of linear equations are then solved by Gaussian elimination method to obtain the unknown constants which are then substituted back intoequation (27) to obtain the approximate solution for case N.

3. Numerical Experiment

In this section, we have demonstrated the Standard Integral Collocation Approximation Method on first, second and third orders Fredholm - Volterra Integro-Differential Equations using Power series, Chebyshev and Legendre's Polynomials as the basis functions. The examples are solved to illustrate the accuracy, efficiency and time of execution of the method.

We have defined absolute error as

Example 1.

We considered first order FVIDE given as

With the mixed condition

The exact solution is given as

Example 2.

We considered second order FVIDE given as

With the conditions

The exact solution is given as

Example 3.

We considered the third order FVIDE given as

With the conditions

The exact solution is given

4. Table of Results

Table 1. Numerical results for example 1

Table 2. Numerical results for example 2

Table 3. Numerical results for example 3

5. Conclusions

We have demonstrated Standard Integral Collocation Approximation Method to solve special higher orders linear Fredholm - Volterra Integro Differential Equations by Power Series, Chebyshev and Legendre's Polynomials as the basis functions of approximations. The results obtained by Chebyshev and Legendres's Polynomials as basis functions proved superior than that of power series.

However, this method yields the desired accuracy when the results obtained are compared with the exact solutions. The simplicity is an added advantage to the method and hence, it is reliable and a powerful tool for the classes of the problems considered.