In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space $L_2\left[\mathsf{\Omega }\right]\times C\left[0,T\right], T<1$. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS).

This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

In this article, the existence of a unique solution of Fredholm–Volterra integral equation of the second kind is guaranteed. The Fredholm integral term is assumed in position with bad kernel, while the Volterra integral term is considered in time with continuous kernel. Under certain conditions and new discussions, the bad kernel will tend to a logarithmic kernel. Then, using Chebyshev polynomial, a main theorem of spectral relationships of Fredholm integral equation of the first kind with logarithmic kernel multiplying by a smooth kernel is stated and used to obtain numerically the Fredholm–Volterra integral equation of the second kind. Finally, numerical results are obtained and the error, in each case, is computed

In this paper, we used the semi-analytical technique called the degenerate kernel method to examine a Fredholm linear integral equation of the second kind. The Banach fixed point theorem discusses the existence and uniqueness of the equation's solution. We insert some numerical examples. The important point here is comparing the solutions and their errors using different programs, i.e., MATLAB, Maple, and Mathematica, employed.