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This paper studies the solvability of boundary value problems for a nonlinear integro-differential equation. Converting the problem to an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE) is driven by using a suitable transformation. To investigate the existence and uniqueness of continuous solutions for the NVFIE under certain given conditions, the Krasnoselskii fixed point theorem and Banach contraction principle have been used. Finally, we numerically solve the NVFIE and study the rate of convergence using methods based on applying the modified Adomian decomposition method, and Liao's homotopy analysis method. As applications, some examples are provided to support our work.