This article is an attempt to obtain the numerical solution of Functional linear Voltrra two-dimensional integral equations using Radial Basis Function (RBF) interpolation which is based on linear composition of terms. By using RBF in Functional integral equation, first a linear system 􀀀, C = G will be achieved, then the coefficients vector is defined, and finally the target function will be approximated. In the end, the validity of the method is shown by a number of examples.

These days, integral equations govern a wide range of physical processes, including those related to electricity, mechanics, fluid dynamics, ecology, soil moisture dynamics, shallow water wave propagation, and chemical science. As a result, creating new techniques and putting them into practice for solving integral equations becomes more crucial. In this paper, the Upadhyaya integral transform is employed for determining the analytical solutions of the non-linear Volterra integral equations (NVIEs) of the first kind. Four examples suggest the effectiveness of the Upadhyaya integral transform, particularly for NVIEs of the first kind. The calculation results suggest that the proposed method provides accurate solutions to the original problem and this method is a valuable tool for researchers and scientists working on the broader range of problems involving NVIEs of the first kind.

In this study, a new application of Taylor expansion is considered to estimate the solution of Volterra-Fredholm integral equations (VFIEs) and systems of Volterra-Fredholm integral equations (SVFIEs). Our proposed method is based upon utilizing the nth-order Taylor polynomial of unknown function at an arbitrary point and employing integration method to convert VFIEs into a system of linear equations with respect to unknown function and its derivatives. An approximate solution can be easily determined by solving the obtained system. Furthermore, this method leads always to the exact solution if the exact solution is a polynomial function of degree up to n. Also, an error analysis is given. In addition, some problems are provided to demonstrate the validity and applicability of the proposed method.

This paper describes an approximating solution, based on Lagrange interpolation and spline functions, to treat Functional integral equations of Fredholm type and Volterra type.This method can be extended to Functional differential and integro-differential equations. For showing efficiency of the method we give some numerical examples.

In this paper‎, ‎a special system of non-linear Abel integral equations (SNAIEs) is studied which arises in astrophysics‎. ‎Here‎, ‎the well-known collocation method is extended to obtain approximate solutions of the SNAIEs‎. ‎The existence and uniqueness conditions of the solution are investigated‎. ‎Finally‎, ‎some examples are solved to illustrate the accuracy and efficiency of the proposed method.‎