Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

In this paper, we have implemented Sinc collocation method (SCM) to solve linear systems of higher order Fredholm Volterra integro–differential equations (FVIDEs) with variable coefficients. This method transforms the system FVIDEs into algebraic equations. Two examples are included to illustrate the accuracy and success of the proposed method. We also point out that, as the number of dimensions increases we get satisfactory results, as explained in the figures and tables


Introduction
Systems of integro-differential equations (IDEs) are widely related to the physical and engineering problems of life such as the ripple of winds in the desert, nano-hydrodynamics, oceanography, modelling, and population growth model [2], [5].There are many numerical methods in which systems of IDEs.can be solved even if an exact solution can't be found, so knowing the accurate and appropriate method to solve such problems is the most important.Many authors have used different numerical methods to solve these systems such as Darvishi and others [6], by implementing a semi-analytical technique to give an approximate solution of systems of FVIDEs and prove the efficiency of the method by adding a higher number of terms.In [7], [13], Keramati et al. have employed Taylor method for solving linear and nonlinear systems of FVIDEs.In addition, Basirat and others [8] suggested another method that combines the Bernstein matrices and the collocation technique to improve and solve the system of FVIDEs.In [9] Attary introduced an approximation to a system of FVIDEs with variable coefficients through the Shannon technique.There are also many methods to approximate the solution of the proposed system, such as Bezier control method [10], Chebyshev polynomial method [11], Modified decomposition method [14], Shifted Chebyshev polynomial method [12], Lagrange method [15].In [1], Holmaker proved the stability in solving systems of IDEs that describe the construction of liver zones.In this paper, we will use SCM to solve linear systems of higher order FVIDEs with variable coefficients, that is, where N K P K L L L j j t t P t ( , ), ( , ), ( ) are known functions, and U t ι ( ) are the unknown functions of equation ( 1).This method reduces the solution of system (1) to a solution of an algebraic system by extending the solution U t ι ( ) into two functions based on the interpolating points and unknown coefficients resulting from the Sinc basis.
We should mention that the time and effort to calculate the solutions, when using this method, were very reasonable.

SCM for the Approximate Solution of Linear Systems of FVIDEs
The collocation method is most applied to gain the numerical solution of ordinary differential equations, partial differential equations and integral equations.
If t ≠ 0, then the sinc function is given by [3] where the translated sinc function is defined by For the interpolating points t lh l = , the Sinc function is given by [4]: ∈ * and m t ( ) is a function defined on ( , ) f f , then the Whittaker expansion of this function is defined by To get an approximation on ( , ) appointed.This map transforms the eye-shaped region onto the infinite strip: The basis functions for z B F ∈ on ( , ) 1 v h are chosen as: where S a h t S a h t ( , ) ( ) = ( , )( ( )). δ δ The interpolation points { } t l are then obtained by: be defined as Theorem 2.2.[3] If δ is a conformal one to one map of a simply connected domain B F onto B F , then Consider the FVIDEs (1) with ICs (2) and let U t L B F L W ( ) ( ) .In the beginning we suppose that the unknown functions U t ι ( ) are approximated as follows: where with p a are unknown coefficients and w t t t ( ) = ( 1) Apply ³ dK to (10), we get: and by applying 0 t d ³ K to (10), we get: By differentiating both sides of ( 10) with respect to t, we get where By substituting ( 12), ( 13) and ( 14) into (1) and find the results of Sinc points t e e l lh lh = 1 + , where , as well as using Theorems (2.1) and (2.2), we will obtain a system of algebraic equations in which the sum of number of t l 's and number of initial conditions is equal to the sum of number of p a 's and number of x j 's, which can be solved to get { } = p a a M M L and { } ,

Numerical Examples
Now we will present an approximate solution to some systems of FVIDEs with variable coefficients using SCM.
Tables 1 and 2 contain the solution by SCM.Tables 3 and 4 show the error for SCM with M = 3, M = 4.        Example 3.2.Consider the following FVIDEs x)U 2 (t) dt+  Example 3.2.Consider the following FVIDEs x)U 2 (t) dt+ The analytical solution of ( 17) is Now we apply Algorithm 1 to solve the system (17).Table 5 contains the solution by SCM.Table 6 shows the error for SCM with M = 5, M = 6. Figure 4.(A,B) shows the solution by SCM with M = 5. Figure 5.(A,B) shows the solution by SCM with M = 6. Figure 6 shows the error by SCM.x

Conclusion
In this article, an accurate algorithm (Sinc Algorithm) was constructed to solve linear systems of Fredholm Volterra integro-differential equations (FVIDEs), also we noticed that the results obtained using the proposed algorithm were very satisfactory compared with analytical solutions.In addition, we observed that when the dimension is increased, the absolute error is reduced and gives results that achieve high accuracy with analytical solutions.

Conclusion
In this article, an accurate algorithm (Sinc Algorithm) was constructed to solve linear systems of Fredholm Volterra integro-dierential equations (FVIDEs), also we noticed that the results obtained using the proposed algorithm were very satisfactory compared with analytical solutions.In addition, we observed that when the dimension is increased, the absolute error is reduced and gives results that achieve high accuracy with analytical solutions.
the number of t l 's = 2( 1) M A + , the number of initial conditions = α A , the number of p a 's = (2 1) M A + , and the number of x j 's = ( 1) D A .The SCM is illustrated by Algorithm 1 LINEAR SYSTEMS OF FREDHOLM VOLTERRA INTEGRO DIFFERENTIAL EQUATIONS 5 Example 3.1.Consider the following FVIDEs

Figure 1 .
(a,b,c) shows the solution by SCM with M = 3.

Figure 2 .
(a,b,c) shows the solution by SCM with M = 4.

Table 1 :
Solution of U 1 and U 2 for system (16).