Fredholm-Volterra integral equation with singular kernel
In this paper, under certain conditions, a series in the Legendre polynomials form is used to obtain the solution of Fredholm-Volterra integral equation of the second kind in L"2(-1,1)xC(0,T),T<~. The uniqueness of the solution is considered. The kernel of Fredholm integral term is established in a logarithmic form in position, while the kernel of Volterra integral term is a positive continuous function in time and belongs to the class C(0,T),T<~. The solution by series leads us to obtain an infinite system of linear algebraic equations, where the convergence of this system is studied. In the end of this paper, the Fredholm-Volterra integral equation of the first kind is established and its solution is discussed.