Direct computation method for solving a general nonlinear Fredholm integro-differential equation under the mixed conditions: Degenerate and non-degenerate kernels

• A general form of the nonlinear Fredholm integro-differential equations under the mixed conditions is considered.
• Direct computation method is equipped with interpolation and Taylor series.
• A new error estimate is proved.
• Numerical and analytical results are compared with the some recent results.

In this paper, the direct computation method is employed to handle the general nonlinear Fredholm integro-differential equations under the mixed conditions. For a non-degenerate kernel, we use a degenerate approximation of the kernel by using Taylor series and Lagrange interpolation. The error and convergence of the algorithm are given strictly. The efficiency of the approach will be shown by applying the procedure on some ill-posed and well-posed prototype examples.