A collocation procedure is developed for the linear AND nonlinear FREDHOLM AND VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS, using the globally defined B-spline AND auxiliary basis functions. The solution is collocated by cubic B-spline AND the integrAND is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods to illustrate the accuracy AND the implementation of our method.

In this article we use discrete collocation method for solving FREDHOLM–VOLTERRA INTEGRO–DIFFERENTIAL EQUATIONS, because these kinds of integral EQUATIONS are used in applied sciences AND engineering such as models of epidemic diffusion, population dynamics, AND reaction–diffusion in small cells. Also the above integral EQUATIONS with convolution kernel will be solved by discrete collocation method. In this method we approximate solution of problem by no smooth piecewise polynomial. Numerical results show a high accuracy AND validity discrete collocation method.

In this paper, we apply Newton's method to solve a class of INTEGRO-DIFFERENTIAL EQUATIONS of the VOLTERRA-FREDHOLM type with nonlocal characteristics, involving almost sectorial operators AND Hilfer fractional derivatives. Since these EQUATIONS play a key role in mathematics, engineering, biology, physics, chemistry, control theory AND  economy, finding an appropriate solution is important. By Newton's method, we linearize a nonlinear VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL equation which is equivalent to a category of VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS with nonlocal properties, incorporating almost sectorial operators AND Hilfer fractional derivatives AND nearly sectorial operators. This technique has been shown to be an effective tool for the numerical solution of initial value problems in nonlinear INTEGRO-DIFFERENTIAL EQUATIONS. The convergence analysis is also investigated.

In this paper, we introduce the Fibonacci polynomials (FPs) AND approximate functions using them. Furthermore, several lemmas AND corollaries present the properties of FPs. Also, we derive the Fibonacci polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before.  As applications of the Fibonacci polynomials operational matrix, we solve the system of VOLTERRA FREDHOLM INTEGRO-fractional DIFFERENTIAL EQUATIONS. In this scheme, we approximate one AND two variable functions based on Fibonacci basis. Then by applying Fibonacci polynomials operational matrix, the system of VOLTERRA FREDHOLM INTEGRO-fractional DIFFERENTIAL EQUATIONS is reduced to a system of algebraic EQUATIONS that is easily solvable with the help of a software (version 13 of the Mathematica software). The obtained results are in good agreement with the exact solutions AND with those in literature. As anticipated, the solutions converge to classical solutions as the fractional derivative order approaches integer values.

This paper demonstrates a study on some significant latest innovations in the approximated techniques to find the approximate solutions of Caputo fractional VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS. To this aim, the study uses the modified Adomian decomposition method (MADM) AND the modified variational iteration method (MVIM). A wider applicability of these techniques are based on their reliability AND reduction in the size of the computational work. This study provides an analytical approximate to determine the behavior of the solution. It proves the existence AND uniqueness results AND convergence of the solution. In addition, it brings an example to examine the validity AND applicability of the proposed techniques.