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 paper, we use Petrov-Galerkin elements such as continuous and discontinuous Lagrange-type k-0 elements and Hermite-type 3-1 elements to find an approximate solution for linear Fredholm Integro-differential equations on [0; 1]. Also we show the efficiency of this method by some numerical examples.

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, an expansion method based on orthonormal polynomials is presented to find the numerical solution of partial fractional Fredholm Integro-differential equations (PFFIDEs). A PFFIDE is converted to a system of linear algebraic equations, which is solved for the expansion coefficients of approximate solution based on orthonormal polynomials. An estimation error is discussed and some illustrative examples are given to demonstrate the validity and applicability of the proposed 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.

A numerical scheme has been developed for solving the system of linear Fredholm Integro-differential equations subject to the mixed conditions using Laguerre polynomials. Using collocation method, the system of Fredholm Integro-differential equations has been transformed to the system of linear equations in unknown Laguerre coefficients, which leads to the solution in terms of Laguerre polynomials. Moreover, the accuracy and applicability of the scheme have been compared with Tau method and Adomian decomposition method that reveals the proposed scheme to be more efficient.