روش تفاضلات متناهی برای حل معادله انتگرال-دیفرانسیل با مشتقات جزئی

در این مقاله یک روش عددی بر مبنای تفاضلات متناهی برای حل مسئله انتگرالدیفرانسیل با مشتقات جزئی با هستۀ منفرد ارائه شده است. ابتدا یک الگورتیم عددی برای حل مسئله براساس طرح کرانکنیکلسون با شرایط داده شده ارائه و سپس گسسته‌سازی انتگرال منفرد را برای حل این مسئله به‌کار می‌بریم. در ادامه برای نشان دادن کارایی روش بیان شده با مقایسۀ جواب تقریبی و دقیق، با روش بی‌اسپلاین مکعبی نتیجه می‌گیریم که روش ارائه شده از دقت و کارائی لازم برخوردار است. در ادامه شکل تقریبی نیز رسم شده است. سرعت بالای محاسبات، سهولت در به‌دست آمدن و اطمینان از داشتن جواب تقریبی به‌دلیل اثبات پایداری از مزایای این روش است.

Finite Difference Method for Solving Partial Integro-Differential Equations

IntroductionIn this paper, we have introduced a new method for solving a class of the partial integrodifferential 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 boundarylayer 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 methodsFirst, we employing an algorithm for solving the problem based on the CrankNicholson 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 Bspline method.Results and discussionIn 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.ConclusionThe following conclusions were drawn from this research.Coefficients of the approximate function via CrankNicholson scheme are found very easily and therefore many calculations are reduced.The numerical results obtained here can be compared with the cubic Bspline methodThe assurance of having an approximate solution are obtain by proving the stability of the method../files/site1/files/61/8.pdf