A Comprehensive Study of Nonlinear Mixed Integro-Differential Equations of the Third Kind for Initial Value Problems: Existence, Uniqueness and Numerical Solutions

Nonlinear mixed integro-differential equations (NM-IDEs) of the third kind present a complex challenge during solving initial value problems (IVPs), particularly after converting them from standard forms. In this work, we address the existence and uniqueness of a type of NM-IDEs employing Picard’s method. Additionally, we estimate the solution using the homotopy analysis method (HAM) and analyze the convergence of the approach. To demonstrate the credibility of the theoretical results, various applications are given, and the numerical results are displayed in a group of figures and tables to highlight that solving IVPs by first converting them to NM-IDEs and using the HAM is a computationally efficient approach.