In this paper, the existence and uniqueness of solution of the Fredholm-Volterra integral equation (F-VIE) with a generalized singular kernel, are discussed and proved in the space 2 ( ) (0, ). L C T  The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of

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... tions by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. Numerical examples are considered when the generalized kernel takes the following forms: Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.