Department of Mathematics
Permanent URI for this collectionhttp://localhost:4000/handle/123456789/1920
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Item Backward euler method for 2d sobolev equation with burgers’ type non-linearity(AIP, 2023-06) Yadav, SangitaBackward Euler for two dimensional Sobolev equation is discussed in this article. We begin by obtaining some basic a priori estimates for the semi-discrete scheme and for the backward Euler approximation. It is proven that these estimations for the discrete scheme are valid uniformly in time using the discrete Gronwall’s Lemma. In addition, the presence of a discrete global attractor is established. Furthermore, optimal a priori error bounds are determined, which are time dependent exponentially. Under the uniqueness condition, these error estimates are demonstrated to be uniform in time. Finally, we establish several numerical examples that validate our theoretical approach.Item Mixed virtual element method for integro-differential equations of parabolic type(Springer, 2024-04) Yadav, SangitaThis article presents and analyzes a mixed virtual element approach for discretizing parabolic integro-differential equations in a bounded subset of , in addition to the backward Euler approach for temporal discretization. With the help of the intermediate projection along with Fortin and projections, we effectively tackle the treatment of integral terms in both the fully discrete and semi-discrete analysis. This inclusion leads to the derivation of optimal a priori error estimates with an order of for the two unknowns. Furthermore, we present a systematic analysis that outlines the step-by-step process for achieving super convergence of the discrete solution, with an order of . Several computational experiments are discussed to validate the proposed scheme’s computational efficiency and support the theoretical conclusions.Item Mixed virtual element method for linear parabolic integro-differential equations(Global Science Press, 2024) Yadav, SangitaThis article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order O(hk+1) for the two unknowns p(x,t) and σ(x,t). In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order O(hk+2). The fully discrete case has also been analyzed and discussed to achieve O(τ) in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions