Laguerre and Touchard Polynomials for Linear Volterra Integral and Integro Differential Equations

In this paper, efficient numerical methods are given to solve linear Volterra integral (VI) equations and Volterra Integro differential (VID) equations of the first and second types with exponential, singular, regular and convolution kernels. These methods based on Laguerre polynomials (LPs) and Touchard polynomials (TPs) that convert these equations into a system of linear algebraic equations. The results are compared with one another method and with each other. The results show that these methods are applicable and efficient. In addition, the accuracy of solution is presented and also the calculations and Graphs are done by using matlab2018 program.

The idea of this work is to illustrate the results of the solutions for linear Volterra integral (VI) equations and linear Volterra integro differential (VID) equations in two methods using the (LPs) and (TPs). Such equations are model of problems in many applications, like, heat conduction, dynamics of viscoelastic, electrodynamics [1]. The solutions of integral and integro differential equations have an essential role in several applied areas which include "mechanics, chemistry, physics, biology, astronomy and potential theory" [2].
The general formulas of the linear (VI) equations of the 2 nd and 1 st types [3,4] respectively are defined by: Also the general formula of linear Abel's singular of the 1 st type [4, 5 and 6] is defined as follows: The general formula of the linear (VID) equation of the 1 st order and 2 nd type [4] is defined as follows: Q ᇱ (α) = w(α) + γ න Y(α, τ) Q(τ)dτ, , b ଵ ≤ α ≤ b ଶ , … with initial condition Q(0) = Q , … (3a) where Q ᇱ (α) = ୢ ୢ , b ଵ , b ଶ are constants, Q (α) is the unknown function that must be determined, γ is a known constant, it represents the physical meaning of the material, and Y (α, τ) is a kernel of the Integral equations (IEs), which is a known continuous or dis-continuous function holds characteristic or property of the material, w (α) is a known function represents the integration surface and Q(0) = Q is a constant initial condition for eq. (3).
There are many approximate numerical methods used and developed by the scientific researchers to obtain the approximate numerical solutions for the (VI) equations and (VID) equations, mentioned as follows: [7] proposed numerical methods to solve weakly (VI) equations of the 1 st type. [8] gave numerical method for the approximation of the (VI) equations with oscillatory Bessel kernels. [9] applied Chebyshev wavelet method to solve the (VI) equations with weakly singular of kernels. [10] used the standard Galerkin polynomial method to solve weakly singular kernels for the (VI) equations. [11] extended the single step pseudo spectral method to the multi step pseudo spectral method for the (VI) equations of 2 nd type. [12] applied the Galerkin weight residual method and (LPs) as a trial function for solving the (VI) equations of the 1 st , 2 nd type with singular and regular kernels. [13] used the (LPs) for solving system of generalized Abel integral equations. [14] used iterative methods to solve the (VID) equations with singular kernel. [15] applied collocation method to solve the (VID) equations. [16] applied "Galerkin the weight residual method" with the (TPs) as a trial function to get numerical solutions to (IEs).
This article is arranged as follows: Laguerre polynomials, function of approximation using the (LPs), Touchard polynomials, function of approximation using the (TPs), solution the (VI) equation using the (LPs) method, accuracy of solutions, convergence rate, illustrative examples, tables and figures are provided, summary of conclusions and recommendations. Finally the references are mentioned.

Laguerre Polynomials [12 and 13]:
This section, begin with definition of the (LPs) which was studied in 1782 by Adrien-Marie Legendre. The (LPs) consisting of the polynomial sequence of binomial type, it's defined on [0, ∞) as follows: , k = 0, 1, 2,… n and α ∈ [0, ∞ ) ⋯ (4 ) where k and s represent the degree and the index for the (LPs) respectively. The first five polynomials of the (LPs) are given below:

Function of Approximation using the (LPs):
Suppose that the function Q ୩ (α) is approximated using the (LPs) as follows: denotes the Laguerre basis polynomials of kth degree, as defined in Eq. (4). ϑ ୱ (s = 0,1,… , k) are the unknowns Laguerre coefficients that calculate later. Writing Eq. (5) as a dot product: Eq. (6) can be written in the following form: where θ (ρ = 0,1,2, … , k) are known values of the power basis that are used to find the (LPs), also the square matrix is an upper triangular and non-singular. For example, if k= 1, and 2, the operational matrices are shown as in Eqs. (8) and (9) respectively: Since the derivative of Eq. (4) is: , k = 1, 2, …n, and α ∈ [0, ∞ ) ⋯ (10 ) so, the derivative of Eqs. (7), (8) and (9) is respectively: where k and s represent the degree and the index for the (TPs) respectively. The first five polynomials of the (TPs) are written below:

Function of Approximation using the (TPs):
Suppose that the function Q ୩ (α) is approximated using the (TPs) as follows: denotes the Touchard basis polynomials of kth degree, as defined in Eq. (11). ϑ ୱ (s = 0,1,… , k) are the unknowns Touchard coefficients that determine later.
Writing Eq. (12) as a dot product: Eq. (13) can be written as follows: where ε (ρ= 0, 1, 2,…, k) are known constants of the power basis that are used to find the (TPs), also the square matrix is an upper triangular and non-singular. For instance, if k=2 and 3, the operational matrices are shown in Eqs. (15) and (16) respectively: Since, the derivative of Eq. (11) is: then, the derivative of Eqs. (14), (15) and (16) respectively is:
The same procedure can be applied to Eqs. (1a) and (2) when using the (TPs).

Solution the (VID) Equation of the 1 st order and 2 nd type using the (LPs):
In this section, the (TPs) is used to find the solutions for the (VID) equation. Since Eq. (3) is: by using Eqs. (7) and (10a), suppose that: So, after simplifying Eq. (26), the unknown Touchard coefficients (ϑ , ϑ ଵ , … , ϑ ୩ ) are obtained by selecting points α ஒ (β = 0,1, … , k) in the interval [b ଵ ,b ଶ ], with the initial condition Eq. (23a). Therefore, Eq. (26) converts to a system of (k+1) linear algebraic equations in (k+1) unknown coefficients, this system can be solved using "Gauss elimination method" to obtain theses coefficients, which have unique solutions. These coefficients are substituted into Eq. (5), to get the approximate numerical solution for Eq. (3). The same procedure can be applied when using the (TPs).

Accuracy of Solutions:
In this section, the accuracy of the proposed methods is tested.

8.1: For the (VI) equation:
Since Eq. (20) has the following formula: Since Eq. (5) has the following form: And the unknown Laguerre coefficients (ϑ ,ϑ ଵ ,… , ϑ ୩ ) were determined by using Eq. (22). Also, by using Eq. (19), we have: Then, the difference for error function AR (α ) at each point α will be smaller than any positive integer ϵ > 0. Thus, the error function AR (α) can be estimated using the relation: This procedure is suitable for Eqs. (1a) and (2). Also this procedure can be applied using the (TPs).

Convergence Rate:
In this section, the error function can be defined by the following relation [20]: where ‖AR ୩ (α)‖ is an arbitrary vector norm of error function, AR ୩ (α) = Q(α) − Q ୩ (α), where Q (α) and Q ୩ (α), are the exact and approximate numerical solutions respectively.

Illustrative Examples:
In this section, the (LPs) and (TPs) are used to solve linear (VI) and (VID) equations. These two polynomials have been applied to six numerical examples, and the convergence of solutions using the error function is given. For k = 2, 3, 4, 5 and 6, the approximate results using: 1. The (LPs) are:  Table 1, showing the (LPs) and the (TPs) methods having a higher accuracy than in [20] with the same degrees, and that both proposed methods having the same accuracy. Figure 1 shows the comparison of result for k=6 with exact solution. They seem to be identical.
where Q(α) = α ଷ − α ଶ + 1 is the exact solution.  Table 2, showing the (LPs) and (TPs) methods having a higher accuracy than in [20] with the same degrees, and that both proposed methods having the same accuracy. Figure 2 shows the comparison of result for k=4 with exact solution. They seem to be identical.  The solutions were approximated in five different degrees. The comparison of error functions of the (LPs) method and those in the (TPs) method is shown in Table 3, showing the (TPs) method having a higher accuracy than in the (LPs) method with the same degrees. Figure 3 shows the comparison of result for k=6 with exact solution. They seem to be identical.     Table 4, showing the (LPs) method having a higher accuracy than in the (TPs) method with the same degrees. Figure 4 shows the comparison of results for k=4 and 5 with exact solution. They seem to be identical.
Table4. Comparison of the Error Function of the (LPs) and (TPs) of Example 4.
Example 5: Solve the (VID) equation of the 2 nd type with constant kernel [4]: where Q(α) = 6α is the exact solution.
For k= 2, 3 and 4, the same exact solution is obtained, so, using the (LPs), we have:  The solutions were approximated in three different degrees and the exact solution was obtained the same and this shows that the error function is zero in this case. Figure 5 displays the comparison of results for k=2, 3 and 4 with exact solution. They seem to be identical. where Q(α) = 1 − α, is the exact solution.
The solutions were approximated in three different degrees and the exact solution was obtained the same and this shows that the error function is zero in this case. Figure 6 displays the comparison of results for k=2, 3 and 4 with exact solution. They seem to be identical.   Tables 1 and 2 were better than in [20]. The results of error function for example 3 in Table 3 were decreasing with increased polynomials degrees, also, the results in Table 4 have shown that the (LPs) method is better than the (TPs) method.
In examples 5 and 6, the approximate solutions were exactly the same as exact solution, so, the error functions were zero in these cases for both proposed methods. In general, all results indicate that the errors function decreasing with increasing the degree of polynomials as shown in the relevant Tables and Figures. Therefore, the methods used in this article can be applied to other types of integral equations, like, nonlinear integral and integro differential equations.