Approximate Numerical Solutions for Linear Volterra Integral Equations Using Touchard Polynomials

In this paper, Touchard polynomials (TPs) are presented for solving Linear Volterra integral equations of the second kind (LVIEs-2k) and the first kind (LVIEs-1k) besides, the singular kernel type of this equation. Illustrative examples show the efficiency of the presented method, and the approximate numerical (AN) solutions are compared with one another method in some examples. All calculations and graphs are performed by program MATLAB2018b.


Introduction:
In many problems, the Volterra integral equations (VIEs) arise from the real life, such as population dynamics, feedback control theory, and fluid dynamics etc. (1). Furthermore, many applications of engineering or physics can often lead to Volterra integral equations. The singular status, which appears in the physical modelling structures, has importance in mathematical physics and other sciences. The weakly singular kernel of the (VIEs) represents such phenomena, which have important applications in mathematical physics, electrochemistry, chemical reactions, superfluidity, and heat conduction (2). The standard forms of (LVIEs-2k) and (LVIEs-1k) (3,4) respectively defined as follows: and the general form of Abel's singular (LVIEs-1k) (4) is: ζ a ⊆ R … (1b) Which is derived from a concrete problem of physics without passing during differential equations (5). A(ζ) is an unknown function that must be calculated, α is a known constant, it takes physical meanings of the characteristic of the material, and G(ζ, τ) is the kernel of the Integral equation (IE), which is a known continuous or noncontinuous function ( the kernel of such equations has jump discontinuities along the continuous curve which started at the origin (6) ), holds characteristic and property of the material, k(ζ) is the function of the integration surface which is a known.
There are many numerical methods used or developed by the researchers to obtain the approximate numerical solutions for the (LVIEs), some of them are mentioned as follows: (7) gave the numerical scheme for the approximation of the (LVIEs) with highly oscillatory Bessel kernels. (8) applied the optimal homotopy asymptotic method for finding the approximate solutions of a class of the (LVIEs) with weakly singular kernels. (9) applied the standard spectral Galerkin polynomial method and a variant to solve a weakly singular for the (VIEs). (10) extended the single-step pseudospectral method for the (VIEs-2k) to the multistep pseudo-spectral method. (11) used the Galerkin weight residual numerical method with Chebyshev polynomials and Touchard polynomials as a trial function to obtain a numerical solution for the (IEs).
In this work, Touchard polynomials method is used to solve (LVIEs) numerically. The rest of the paper is arranged as follows : The solution method,  approximation function, algorithm of solution,  accuracy of the solution, convergence rate, test  illustrative examples and the corresponding Tables  and Figs. are presented. Finally, brief conclusions and references are listed.
Output: The polynomials of the degree n.
Step 1: Select a degree n for the (TPs).

Accuracy of the Solution:
To determine the error estimate if the exact (analytic) solution is known, then the absolute error must be the difference between the analytic solution A(ζ) and the approximate solution A n (ζ) defined by: E n ( ) = |A(ζ) − A n (ζ)|. Definition: With h is a real value function defined and bounded on [0, 1], let A n (h ) be the polynomial on [0, 1], that assigns to h (ζ) the following value: where A n (h ) is the nth Touchard polynomials for h (ζ) (15).
Theorem: For all functions h in C [0, 1], the sequence of A n (h ) converges uniformly to h.

Conclusions:
The prior knowledge have used in a new concept, by using the Touchard polynomials to construct matrices and get functions of the approximation. These functions are used to find the approximate numerical solutions for several kinds of mathematical equations: The integral equations, in particular (LVIEs) of all kinds. The results showed that the present method for solving (LVIEs) of the first and the second kind, also with singular kernel is very effective and their accuracy is high. The Tables and Figs. support this claim. The results indicated that when the polynomial degree n increases, the error decreases rapidly. Moreover, the presented method has been tested by six examples, and approximate numerical results have been compared with one another method. Consequently, the comparison is compatible with it or better.