The main purpose of this article is to present a new technique for solving (1+1) mixeddimensional difference integro-differential Equations (2D-MDeIDEs) in position and time with coefficients of variables under mixed conditions. The equations proposed for the solution represent a link between time and delay in position that has not been previously studied. Therefore, the authors used the technique of separation of variables to transform the 2D-MDeIDE into one-dimensional Fredholm difference integro-differential Equations (FDeIDEs), and then using the Bernoulli polynomial method (BPM), we obtained a system of linear algebraic equations (SLAE). The other aspect of the technique of separation of variables is explicitly obtaining the necessary and appropriate time function to obtain the best numerical results. Some numerical experiments are performed to show the simplicity and efficiency of the presented method, and all results are performed by Maple 18.

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space $L_2(-1,1)\times C[0,T],(T<1).$ The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y.

In the present article, we establish the numerical solution for the mixed Volterra- Fredholm integral equation (MV-FIE) in (1+1) dimensional in the Banach space \(L_2[-1,1] \times C[0, T], T < 1.\) The Fredholm integral term is considered in the space \(L_2[-1,1]\) and it has a discontinuous kernel in position. While the Volterra integral term is considered in the class of time \(C[0, T], T < 1,\) and has a continuous kernel in time. The necessary conditions have been established to ensure that there is a single solution in the space \(L_2[-1,1] \times C[0, T], T < 1.\) By utilizing the separation of variables technique, MV-FIE is transformed to Fredholm integral equation (FIE) of the second kind with variables coefficients in time. The separation technique of variables helps the authors choose the appropriate time function to establish the conditions of convergence in solving the problem and obtaining its solution. Then, using the Boubaker polynomials method, we end up with a linear algebraic system (LAS) abbreviated. The Banach fixed point (BFP) hypothesis has been presented to determine the existence and uniqueness of the solution of the LAS. The convergence of the solution and the stability of the error are discussed. The Maple 18 software is used to perform some numerical calculations once some numerical experiments have been taken into consideration.