A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

Self Cite

This article aims at developing a robust numerical method based on graded mesh for solving the multiterm time‐fractional convection‐diffusion‐reaction (TFCDR) equation whose solution very likely exhibits a weak singularity at the initial time. The time‐fractional derivative in the model problem is described in terms of Liouville–Caputo. In order to handle the weak singularity at the initial time, we use a graded mesh technique for discretization of multiterm temporal fractional derivatives. The space derivatives are approximated by means of a compact finite difference (CFD) method. The proposed method is analyzed for its stability and convergence. Two numerical examples are considered to demonstrate the applicability and accuracy of the method. It is shown that the proposed graded mesh technique provides an optimal rate of convergence in time direction for the problem with nonsmooth exact solution, while the method on the uniform mesh yields a nonoptimal rate of convergence.

Section: Introductionmentioning

confidence: 99%

An efficient numerical scheme and its analysis for the multiterm time‐fractional convection‐diffusion‐reaction equation

Roul

Rohil

2023

Self Cite

“…To mention few, the areas of applications of FDEs include signal processing, viscoelasticity, electrical network, continuum mechanics, material science and nuclear science, see in previous works. [1][2][3][4][5][6][7][8][9][10][11][12] Moreover, the FDEs were employed to model financial problems. [13][14][15] It should be noted that the classical Black-Scholes (BS) model 16,17 was derived by imposing quite restrictive assumptions, for example, stock pays no dividend, interest rates are constant, transaction costs are eliminated and the values of short-term rates are available.…”

Section: Introductionmentioning

confidence: 99%

“…This is due to their wide‐range of applications in various fields of applied science and engineering. To mention few, the areas of applications of FDEs include signal processing, viscoelasticity, electrical network, continuum mechanics, material science and nuclear science, see in previous works 1–12 . Moreover, the FDEs were employed to model financial problems 13–15 …”

Section: Introductionmentioning

confidence: 99%

Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing

Roul

2022

Self Cite

This work deals with the construction and analysis of a high‐order computational scheme for a time‐fractional Black‐Scholes model that governs the European option pricing. The time‐fractional derivative is considered in the sense of Caputo and the L1 − 2 formula is employed to approximate the Caputo temporal‐fractional derivative of order α, where α ∈ (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and false(3−αfalse)−th order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock‐out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock‐out option are investigated. The computational time for the proposed method is provided.

“…2,[4][5][6][7][8] Fractional-order reaction-diffusion models have played a significant role in studying nonlinear physics problems and have attracted the interest of many researchers in applied chemistry, engineering, economics, physics, and mathematics. Several studies focused on numerical and analytical techniques for solving time-fractional reaction-diffusion equations (TFRDEs), such as the Haar wavelets method, 9 fully discrete spectral scheme, 10 alternating direction implicit, 11 L 1 -Galerkin finite element schemes, 12 quintic B-spline basis functions, 13 the novel alternating direction implicit finite difference method, 14 Fourier spectral method, 15 shifted Legendre polynomials, 16 Mittag-Leffler kernel, 17,18 HAT method 19 and a fully discrete ADI scheme. 20 Yuan 21 used the Fourier-like spectral approach to solve fractional reaction-diffusion equations in unbounded domains.…”

Section: Introductionmentioning

confidence: 99%

Numerical method for solving two‐dimensional of the space and space–time fractional coupled reaction‐diffusion equations

Hadhoud

Rageh

Agarwal³

2022

This paper proposes the shifted Legendre Gauss–Lobatto collocation (SL‐GLC) scheme to solve two‐dimensional space‐fractional coupled reaction–diffusion equations (SFCRDEs). The proposed method is implemented by expressing the function and its spatial fractional derivatives as a finite expansion of shifted Legendre polynomials. Then the expansion coefficients are determined by reducing the SFCRDEs with their initial and boundary conditions to a system of ordinary differential equations for these coefficients. Subsequently, we applied the proposed method to discretize the temporal and spatial variables to convert the two‐dimensional spacetime fractional coupled reaction–diffusion equations (STFCRDEs) to a system of algebraic equations. Some results regarding the error estimation are obtained. Several examples are discussed to validate the capability and efficiency of the proposed scheme.