Solution of mixed integral equation in position and time using spectral relationships

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.