Numerical Solution for Linear Fredholm Integro-Differential Equation Using Touchard Polynomials

A new method based on the Touchard polynomials (TPs) was presented for the numerical solution of the linear Fredholm integro-differential equation (FIDE) of the first order and second kind with condition. The derivative and integration of the (TPs) were simply obtained. The convergence analysis of the presented method was given and the applicability was proved by some numerical examples. The results obtained in this method are compared with other known results.


Introduction:
Integral and integro-differential equations are originated in many scientific and engineering applications. In particular Fredholm integral equation and (FIDE) can be derived from boundary value problem. The (IDE) contains both differential and integral signs and the derivative of the unknown variable may appear to any order. (FIDE) is an equation derived from the boundary value problem with given initial boundary condition, where both the differential and integral signs appeared together in the same equation. In addition, limits of the integration are constants. The (FIDE) of the first order and second kind contains the unknown variable and its derivative inside and outside the integral sign respectively. It is noted that initial condition should be given for (FIDE) to find the particular solution (1(. (FIDEs) often come in applications being the mathematical models of processes in biological problems, physics, economy and chemistry, etc. (2). (FIDEs) are difficult to be solved analytically, so it requires effective numerical methods (3). For these reasons, many scientists have been encouraged to study many numerical methods to solve (FIDEs). All methods have pros or cons but that hasn't stopped scientists from developing various methods such as Bernstein collocation matrix method (4), the well-posedness method (5), reproducing kernel method (6), exponential spline method (7), improved reproducing kernel method (8), priori Nystrom method (9), and Fibonacci polynomials method (10).
The general form of the linear (FIDE) of 1 st order and 2 nd kind is given by (1,11): ∈ [a 1 , a 2 ], … (1) with initial boundary condition X(a 1 ) = X 0 , … (1a) where a 1 , a 2 and η are constants, T(γ, u) is a known function of the variables γ and u, called the nucleus (kernel) of the Integral equation. The unknown function X(γ) will be calculated, which exists inside and outside the integral sign. h(γ) is a given function, X ′ (γ) =
The approximate solutions and absolute error were compared in Tables 1 and 2, respectively, showing that the accuracy of the results increases as n increases. In Fig.1, the exact solution was compared with Touchard solution for n = 3.  where h(γ) = γ e γ + e γ − γ , η = 1, T(γ, u) = γ , Χ(0) = 0 and the exact solution is Χ(γ) = γ e γ . The approximate numerical results are obtained for n = 1, 3 and 4, respectively:  Table 3 shows the absolute errors for n=4, and compares with methods included in (16 and 17). In Fig. 2, the exact solution is compared with Touchard solution for n = 4.  γ , η = 1, T(γ, u) = γu , Χ(0) = 0 and the exact solution is Χ(γ) = γ. By applying suggested method for this example, for n = 5, the Touchard solution is: X 5 (γ) = (−1)I 0 (γ) + (1)I 1 (γ) + 0 = X(γ) = γ. In Table 4, the absolute error in the current method is compared with those in (11, 16 and 17), and it is found that the absolute error in the current method is the highest accuracy. In Fig. 3 Table 5, and shows that the current method for n = 1, 2 and 3 has a much higher accuracy than those in (18) for n = 5, 9 and 17. In Fig. 4, for n = 2 and 3, the Touchard solutions were compared with the exact solution.

Conclusions and Recommendations:
In this study, numerical solutions are obtained for linear (FIDEs) of the first order and second kind under condition, using Touchard polynomials, and different degrees for purpose of comparing. This method reduces the (FIDEs) into a set of algebraic equations. It's worth noting that one of the important features of this method is that the Touchard coefficients of the solutions are found easily by using PC programs. Also, another advantage is the obtaining solution is polynomials of the degree equal or less than selected n. However, the solution converges rapidly to the exact solution when n increases. The comparison between the absolute errors for four test examples and those methods included in (11, 16, 17 and 18), shows that the accuracy of the current method is almost similar or better than those of the existing methods. As a future work, the current method can also applied to the system of linear (FIDEs), because it is effective and applicable for the linear and nonlinear for these kinds of equations and the results obtained support this claim. Because the solutions obtained here are approximate solutions, it is expected in some examples that the absolute error increases when γ approaches 1 in the interval [0, 1] as in examples 1 and 2.
All the methods referred to in the introduction to this study are approximate numerical methods that have been used to solve the Fredholm integrodifferential equations that are difficult to solve analytically. These methods have been used to solve them numerically. The pros of these methods are to obtain approximate solutions and the possibility of writing algorithms for solutions in these methods. Programming these algorithms on personal computers by writing computer programs to identify unknown values and then the possibility of comparing the results obtained in these methods by graphs. The cons of these methods are the existence of errors in the accuracy of the results in reaching the approximate solutions.