A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options

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A high-accuracy numerical approach for a nonhomogeneous time-fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time-fractional derivative is described in the sense of Riemann-Liouville and discretized by the backward Euler scheme. A fourth-order optimal cubic B-spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx 4 + Δt 2 −) convergence, where ∈ (0,1). Moreover, the impact of fractional-order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B-spline collocation method. The CPU time for present numerical method and the method based on cubic B-spline collocation method are provided.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

Roul

Goura

2020

Self Cite

Orthonormal discrete Legendre polynomials for stochastic distributed‐order time‐fractional fourth‐order delay sub‐diffusion equation

Heydari,

Razzaghi

2024

In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples.

An efficient computational approach and its analysis for the Caputo time‐fractional convection reaction–diffusion equation in two‐dimensional space

Roul,

Yadav,

Kumari

2024

This work concerns with a novel computational approach to solve a two‐dimensional Caputo time‐fractional convection reaction–diffusion (CTFCRD) equation with the weak singularity at the initial time. In this method, the ‐scheme on graded mesh is used to approximate the Caputo temporal derivative, while the spatial derivatives are approximated by using a compact finite difference method (CFDM) coupled with alternating direction implicit (ADI) scheme. We present a methodology to examine the optimal error estimates of the proposed scheme, in terms of ‐norm. Convergence and stability results are proved. It is shown that the suggested method yields an optimal order, that is, min(2‐ , , 2 +1) in time direction, where is the order of time fractional derivative and is the grading parameter and has a fourth‐order of convergence in space direction. Finally, two numerical examples are provided to verify the theoretical estimates and to illustrate the accuracy and efficiency of the method.