The aim of this paper is solving nonlinear Volterra-Fredholm fractional integro-differential equations with mixed boundary conditions‎. ‎The basic idea is to convert fractional integro-differential equation to a type of second kind Fredholm integral equation‎. ‎Then the obtained Fredholm integral equation will be solved with Nystr\"{o}m and Newton-Kantorovitch method‎. ‎Numerical tests for demonstrating the accuracy of the method is ‎included.‎ Manuscript profile

In this paper, we consider some initial-boundary value problems which contain one-dimensional heat equation in non-classical case. For this problem, we can not use the classical methods such as Fourier, Laplace transformation and Fourier-Birkhoff methods. Because the eigenvalues of their spectral problems are not strictly and they are repeated or we have no eigenvalue. The presentation of the solution and also satisfying the solution in the given P.D.E and satisfing the given initial and boundary conditions are established by complex analysis theory and Countour integral ‎method. Manuscript profile