In the present paper, a new efficient technique is described for solving nonlinear mixed partial integrodifferential equations with continuous kernels. Using the separation of variables, the nonlinear mixed partial integrodifferential equation is converted to a nonlinear Fredholm integral equation. Then, using different numerical methods, the Bernoulli polynomial method and the Chebyshev polynomials of the sixth kind, the nonlinear Fredholm integral equation has been reduced into a system of nonlinear algebraic equations. The Banach fixed-point theory is utilized in order to have a conversation about the nonlinear mixed integral equation’s solution, namely, its existence and uniqueness. In addition, we talk about the convergence and stability of the solution. Finally, a comparison between the two different methods and some other famous methods is presented through various examples. All the numerical results are calculated and obtained using the Maple software.

The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple.

This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

In this paper, we have investigated the two-dimensional fractional nonlinear Fredholm integro-differential equation (TDFNFI-DE). These equations are used in many fields, including particle dynamics in physics, biology, and control theory. We have developed an effective combined approach in our work that uses Homotopy analysis (HAM) and the Adomian decomposition method (ADM) to solve fractional integro-differential equations in two dimensions. Numerical experiment results show the effectiveness of our recently created technique. We prove the existence and uniqueness of the exact solution. To illustrate the numerical effectiveness of the suggested approach, we provide a number of numerical examples. The suggested approach is accurate and applicable to various nonlinear issues in science, according to theoretical and numerical results.

This research investigates the numerical analysis of a two-dimensional (2D) delay mixed integral equation (DMIE) of the second kind with a continuous kernel. The existence, uniqueness, stability, and convergence of the solution are rigorously established in the functional space $C\left(\right[a,b]\times [0,T\left]\right)$, providing a solid theoretical foundation. A collocation method based on Chebyshev and Legendre polynomials is applied to obtain accurate numerical approximations, leading to a system of algebraic equations applicable to both linear and nonlinear cases. Numerical simulations are performed using MATLAB to evaluate the method’s performance and calculate the results, including error behavior, convergence, and stability. Several illustrative applications are presented to demonstrate the method’s numerical stability and versatility in handling both linear and nonlinear problems. The results underscore the efficiency, accuracy, and robustness of the proposed numerical schemes in solving complex 2D DMIE, highlighting the novelty of the framework and its strong potential for broad applications in contemporary scientific and engineering problems.