Introduction: In this paper, we have introduced a new method for solving a class of the PARTIAL INTEGRO-DIFFERENTIAL EQUATION with the singular kernel by using the finite difference method. One of the best subjects in the numerical analysis is a finite difference method (FDM). We used (FDM) to solve problems in mathematical physics, integral EQUATIONs, and engineering, such as electromagnetic potential, fluid flow, radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral EQUATIONs, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace EQUATIONs and the Poisson EQUATIONs are describe by (FDM). In real life most phenomena are modelled by PARTIAL differential EQUATIONs. Material and methods: First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline method. Results and discussion: In addition, solving some examples demonstrates the validity and applicability of the approached method, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the method. Conclusion: The following conclusions were drawn from this research. Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced. The numerical results obtained here can be compared with the cubic B-spline method The assurance of having an approximate solution are obtain by proving the stability of the method.

The main issue in any image zooming techniques is to preserve the structure of the zoomed image. The zoomed image may suffer from the discontinuities in the soft regions and edges; it may contain artifacts, such as image blurring and blocky, and staircase effects. This paper presents a novel image zooming technique using PARTIAL Differential EQUATIONs (PDEs). It combines a non-linear Fourth-order PDE method with the Locally Adaptive Zooming (LAZ) algorithm. The proposed method uses high-resolution image obtained from LAZ algorithm to construct zoomed image by Fourth-order PDE. This proposed method preserves edges and minimizes blurring and staircase effects in the zoomed image. In order to evaluate image quality obtained from the proposed method, this paper focuses on both subjective and objective assessments. The results of these measures on a variety of images show that the proposed method is superior over the other image zooming methods.

In this paper, an extension of Differential Transformation Method (DTM) which is an analytical-numerical method for solving the fuzzy PARTIAL differential EQUATION (FPDE) by using the strongly generalized differentiability concept is investigated. The proposed algorithm is illustrated by numerical example.