Symmetry is presented in many works involving differential and integral equations. Whenever a human is involved in the design of an integral equation, they naturally tend to opt for symmetric features. The most common examples are the Green functions and linguistic kernels that are often designed symmetrically and regularly distributed over the universe of discourse. In the current study, the authors report a study on boundary value problem (BVP) for a nonlinear integro Volterra–Fredholm integral equation with variable coefficients and show the existence of solution by applying some fixed-point theorems. The authors employ various numerical common approaches as the homotopy analysis methodology established by Liao and the modified Adomain decomposition technique to produce a numerical approximate solution, then graphical depiction reveals that both methods are most effective and convenient. In this regard, the authors address the requirements that ensure the existence and uniqueness of the solution for various variations of nonlinearity power. The authors also show numerical examples of how to apply our primary theorems and test the convergence and validity of our suggested approach.

In this paper, we analyze a boundary value problem (BVP) for a nonlinear integro-Volterra-Fredholm integral equation with variable coefficients. We employ common numerical methods such as the homotopy analysis technique developed by Liao and the modified Adomain decomposition technique to construct an approximate numerical solution. Graphical representations show that these methods are the most efficient and convenient. Additionally, we examine the conditions that guarantee the existence and distinctiveness of the solution for different types of kernels and levels of non-linearity. We provide numerical examples to demonstrate the applicability of our main theorems, and to determine the convergence and accuracy of our proposed technique.