arXiv subjects

Sivaji Ganesh Sista

Publications and source records attributed to Sivaji Ganesh Sista.

6 recordsLinked to original sources

arXiv2024

Existence and regularity results for space-time fractional integro-differential equation of Kirchhoff type with memory

This paper analyses a Kirchhoff type quasilinear space-time fractional integro-differential equation with memory $(\mathcal{K}^{s}_α)$. Various a priori bounds are derived in different norms on the solution of the considered equation. Utilizing these a priori bounds, existence and uniqueness of the weak solution to the proposed model are proved. Furthermore, regularity results on the solution of $(\mathcal{K}^{s}_α)$ are established. The contribution made in this work provides a framework for further investigation of such types of partial integro-differential equation $(\mathcal{K}^{s}_α)$.

math.AP↗

arXiv2023

A Linearized L1-Galerkin FEM for Non-smooth Solutions of Kirchhoff type Quasilinear Time-fractional Integro-differential Equation

In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $α\in (0,1) $. For the semi discrete formulation of the equation under consideration, we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz-Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations (PDEs) have a weak singularity near time $t=0$. Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the considered equation on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted $H^{1}(Ω)$ norm. We prove that the developed numerical scheme has an accuracy rate of $O(P^{-1}+N^{-(2-α)})$ in $L^{\infty}(0,T;L^{2}(Ω))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(Ω))$, where $P$ and $N$ are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples.

math.NA↗

arXiv2022

Finite Element Analysis of Time Fractional Integro-differential Equations of Kirchhoff type for Non-homogeneous Materials

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered problem. Consequently, we prove the existence and uniqueness of the weak solution to the problem under consideration. We keep the time variable continuous and discretize the space domain using a conforming FEM to obtain the semi discrete formulation of the problem. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator. To obtain the numerical solution to the problem efficiently, we develop a new linearized L1 Galerkin FEM. This numerical scheme is shown to have a convergence rate of $O(h+k^{2-α})$, where $α~ (0<α<1)$ is the fractional derivative exponent, $h$ and $k$ are the discretization parameters in the space and time directions respectively. Further, this convergence rate is improved in the time direction by proposing a novel linearized L2-1$_σ$ Galerkin FEM. We prove that this numerical scheme has an accuracy rate of $O(h+k^{2})$. Finally, a numerical experiment is conducted to validate our theoretical claims.

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arXiv2020

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the homogenization result in this case, by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

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arXiv2019

Bloch Wave Homogenization of Quasiperiodic Media

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a "cut and project" procedure. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift an almost periodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularized version of the degenerate equation. Homogenized coefficients for quasiperiodic media are determined in terms of the first Bloch eigenvalue of the regularized lifted equation. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenization limit for an equation with highly oscillating quasiperiodic coefficients.

math.AP↗

arXiv2019

Generic Simplicity of Spectral Edges and Applications to Homogenization

We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.

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