arXiv subjects

Rajesh K. Pandey

Publications and source records attributed to Rajesh K. Pandey.

4 recordsLinked to original sources

arXiv2025

An Efficient Numerical Scheme for a Time-Fractional Burgers Equation with Caputo-Prabhakar Derivative

This paper presents a numerical method to solve a time-fractional Burgers equation, achieving order of convergence $(2-α)$ in time, here $α$ represents the order of the time derivative. The fractional derivative is modeled by Caputo-Prabhakar (CP) formulation, which incorporates a kernel defined by the three-parameter Mittag-Leffler function. Finite difference methods are employed for the discretization of the derivatives. To handle the non-linear term, the Newton iteration method is used. The proposed numerical scheme is proven to be stable and convergent in the $L_{\infty}$ norm. The validity of the theory is supported by two numerical examples.

math.NA↗

arXiv2022

High-order approximation to generalized Caputo derivatives and generalized fractional advection-diffusion equations

In this article, a high-order time-stepping scheme based on the cubic interpolation formula is considered to approximate the generalized Caputo fractional derivative (GCFD). Convergence order for this scheme is $(4-α)$, where $α~(0<α<1)$ is the order of the GCFD. The local truncation error is also provided. Then, we adopt the developed scheme to establish a difference scheme for the solution of generalized fractional advection-diffusion equation with Dirichlet boundary conditions. Furthermore, we discuss about the stability and convergence of the difference scheme. Numerical examples are presented to examine the theoretical claims. The convergence order of the difference scheme is analyzed numerically, which is $(4-α)$ in time and second-order in space.

math.NA↗

arXiv2022

Comparative study of Three Numerical Schemes for Fractional Integro differential Equations

This paper presents a comparative study three numerical schemes such as Linear, Quadratic and Quadratic-Linear scheme for the fractional integro-differential equations defined in terms of the Caputo fractional derivatives. The error estimates of the respective approximations are also established. Numerical tests of the discussed schemes show that all schemes work well, and when the number of terms approximating the solution are increased, the desired solution is achieved. The accuracy of the numerical schemes with respect to the step size h is analyzed and illustrated through various tables. Finally, comparative performances of the schemes are discussed.

math.NA↗

arXiv2022

High Order Numerical Scheme for Generalized Fractional Diffusion Equations

In this paper, a higher order finite difference scheme is proposed for Generalized Fractional Diffusion Equations (GFDEs). The fractional diffusion equation is considered in terms of the generalized fractional derivatives (GFDs) which uses the scale and weight functions in the definition. The GFD reduces to the Riemann-Liouville, Caputo derivatives and other fractional derivatives in a particular case. Due to importance of the scale and the weight functions in describing behaviour of real-life physical systems, we present the solutions of the GFDEs by considering various scale and weight functions. The convergence and stability analysis are also discussed for finite difference scheme (FDS) to validate the proposed method. We consider test examples for numerical simulation of FDS to justify the proposed numerical method.

math.NA↗