Spectral relationships explain many physical phenomena, especially in quantum physics and astrophysics. Therefore, in this paper, we first attempt to derive spectral relationships in position and time for an integral operator with a singular kernel. Second, using these relations to solve a mixed integral equation (MIE) of the second kind in the space $L_2\left[-1,1\right]\times C\left[0,T\right],T<1.$ The way to do this is to derive a general principal theorem of the spectral relations from the term of the Volterra-Fredholm integral equation (V-FIE), with the help of the Chebyshev polynomials (CPs), and then use the results in the general MIE to discuss its solution. More than that, some special and important cases will be devised that help explain many phenomena in the basic sciences in general. Here, the FI term is considered in position, in $L_2\left[-1,1\right],$ and its kernel takes a logarithmic form multiplied by a general continuous function. While the VI term in time, in $C\left[0,T\right],T<1,$ and its kernels are smooth functions. Many numerical results are considered, and the estimated error is also established using Maple 2022.

In this paper, an infinite elastic plate weakened by two holes are considered and the complex variable method is used to derive a closed form of Gaursat functions for the first and second fundamental problems with variant time. The holes, in all previous works, are conformally mapped outside the unit circle without time. Here, the two holes are conformally mapped into the unit circle $\varpi$ in the effect of time by the generalized rational mapping function with complex constant coefficients. By using this conformal mapping function, the fundamental problems transfer to an integro-differential equation with Cauchy kernel. Then, after applying the complex variable method, one can obtain a closed form of Gaursat functions. Some applications were discussed and the time effect on the applications was studied. In addition, the different stress components in each application have also been calculated using Maple 2022.1.