Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions

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.

Automated summary

This paper investigates the solvability of boundary value problems for a nonlinear integro-differential equation. A suitable transformation is used to convert the problem into an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE). The existence and uniqueness of continuous solutions for the NVFIE are studied under certain given conditions using the Krasnoselskii fixed point theorem and Banach contraction principle. Finally, the NVFIE is numerically solved and the rate of convergence is studied using modified Adomian decomposition method and Liao's homotopy analysis method. Some examples are provided to support the findings.