arXiv subjects

Pari J. Kundaliya

Publications and source records attributed to Pari J. Kundaliya.

5 recordsLinked to original sources

arXiv2023

$α$-Robust Error Analysis of $L2$-$1_σ$ Scheme on Graded Mesh for Time-fractional Nonlocal Diffusion Equation

In this work, a time-fractional nonlocal diffusion equation is considered. Based on the $L2$-$1_σ$ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete $L2$-$1_σ$ finite element method is investigated for a time-fractional nonlocal diffusion problem. We prove the existence and uniqueness of fully-discrete solution. The $α$-robust error bounds are derived, i.e. bounds remain valid as $α$ $\rightarrow {1}^{-},$ where $α\ \in (0,1)$ is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.

math.NA↗

arXiv2023

Fractional Crank-Nicolson Galerkin finite element analysis for coupled time-fractional nonlocal parabolic problem

In this article we propose a scheme for solving the coupled time-fractional nonlocal diffusion problem. The scheme consist of fractional Crank-Nicolson method with Galerkin finite element method (FEM) and Newton's method. We derive \emph{a priori} error estimates for fully-discrete solutions in $L^2$ and $H^1_0$ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates.

math.NA↗

arXiv2023

$L2$-$1_σ$ Scheme on a Graded Mesh for a Multi-term Time-fractional Nonlocal Parabolic Problem

In this article, we propose numerical scheme for solving a multi-term time-fractional nonlocal parabolic partial differential equation (PDE). The scheme comprises $L2$-$1_σ$ scheme on a graded mesh in time and Galerkin finite element method (FEM) in space. We present the discrete fractional Gr$\ddot{o}$nwall inequality for $L2$-$1_σ$ scheme in case of multi-term time-fractional derivative, which is a multi-term analogue of~\cite[Lemma 4.1]{[r16]}. We derive \textit{a priori} bound and error estimate for the fully-discrete solution. The theoretical results are confirmed via numerical experiments. We should note that, though the way of proving the discrete fractional Gr$\ddot{o}$nwall inequality is similar to~\cite{[r5]}, the calculation parts are more complicated in this article.

math.NA↗

arXiv2023

Symmetric fractional order reduction method with $L1$ scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in time. We derive the $α$-robust \textit{a priori} bound and \textit{a priori} error estimate for the fully-discrete solution in $L^{\infty}\big(H^1_0(Ω)\big)$ norm, where $α\in (1,2)$ is the order of time fractional derivative. Finally, we perform some numerical experiments to verify the theoretical results.

math.NA↗

arXiv2022

L1 scheme on graded mesh for subdiffusion equation with nonlocal diffusion term

The solution of time fractional partial differential equations in general exhibit a weak singularity near the initial time. In this article we propose a method for solving time fractional diffusion equation with nonlocal diffusion term. The proposed method comprises L1 scheme on graded mesh, finite element method and Newton's method. We discuss the well-posedness of the weak formulation at discrete level and derive \emph{a priori} error estimates for fully-discrete formulation in $L^2(Ω)$ and $H^1(Ω)$ norms. Finally, some numerical experiments are conducted to validate the theoretical findings.

math.NA↗