The analysis of such PIDE can be found in-,‖. INTROODUCTION PARTIAL integro differential equations are used in various fields of science and engineering. Keywords: Maclaurin's Series, Modified differential transform method, Partial integro-differential equation, Two Dimensional Differential Transform method.--1. Finally, performance and accuracy of both the methods are discussed and reveals that MDTM is very effective, convenient and reduces a lot of computational work and time than two dimensional DTM. The results obtained by MDTM and Two Dimensional DTM are compared. The concept of two dimensional DTM and MDTM are briefly explained. In this paper, linear partial integro-differential equations (PIDE) with convolution kernel are solved using Modified Differential Transform Method (MDTM) and compared with Two Dimensional Differential Transform Method (DTM). The convergence results obtained in this work improve the known existing ones for PDM-FEM for parabolic equations which state the convergence towards (Ñu(t) u(t)) in the discrete norms of L¥(L2(W)d) L¥(L2(W)), see. Rn h r(tn) n is of order k+hl+1 in the discrete norms of L¥(Hdiv(W)) W1 ¥(L2(W)) underĪssumption that the solution u is satisfying u 2 C 3 Hl+3(W). We applied these results to the particular case when the spaces (Vdiv h Wh) are defined using the well-known Raviart-Thomas spaces of of order l 0 and we justified that the error These results are obtained thanks to some new well developed discrete a priori estimates. L¥(L2(W))), under assumption that the exact solution is smooth. ut (tn) 1unh n, where 1 denotes a discrete time derivative) in L¥(Hdiv(W)) (resp. The new convergence results which state the optimal estimate for the error Ñu(tn) pnh n We justify rigorously the existence and uniqueness of the discrete solution. The discrete unknowns of the considered scheme are the set of couples rn h := (pnh unh) 2 Vdiv h Wh which are expected to approximate r(tn) := (Ñu(tn) u(tn)) where (tn)n are the mesh points of the time discretization and u is the exact The scheme is formulated in a general setting in which the pairs of finite element spaces (Vdiv h Wh) in Hdiv(W) x L2(W) are arbitrary but satisfy the inf sup hypothesis and another known condition. The time discretization is performed using a uniform mesh. We consider a fully discrete implicit scheme in which the discretization in space is performed using the PDMFEM (Primal-Dual Mixed Finite Element Method) defined in and for the heat equation as a model of parabolic equations in any space dimension.
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