Modified Cubic B-spline Based Differential Quadrature Methods for Time-fractional Black-Scholes Equation

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Farook College (Autonomous) affiliated to University of Calicut, Kozhikode, 673632, India.

2 Department of Computational Science and Humanities, Indian Institute of Information Technology Kottayam, Valavoor, Kottayam 686635, Kerala, India.

3 Department of Mathematics, CUSAT (Cochin University of Science and Technology), Cochin, 682022, India.

4 Department of Mathematics, Farook College (Autonomous) affiliated to University of Calicut, Kozhikode, 673632, India

Abstract

The time-fractional Black–Scholes equation (TFBSE) is indented to price the options for which the underlying price fluctuates within a correlated fractal transmission system. Although the TFBSE is an influential approach for grasping the long-term memory traits of financial markets, the non-local nature of fractional derivatives makes significant challenges in finding an accurate solution. We perform an efficient use of the differential quadrature method (DQM) based on modified cubic B-splines to solve the TFBSE governing European options. This paper constructs an algorithm by the combination of time fractional discretization using the finite difference method L1 and space discretization using the modified cubic B-spline-based differential quadrature method. Uniform meshes are considered for the discretization of both temporal and spatial domains. Theoretical stability has been established by finding an estimate for the maximum norm of the inverse operator regardless of the involvement of the mesh parameters. We trigger the Nuemann series theorem to obtain a uniform bound for the inverse operator under reasonable conditions on the mesh parameters. The numerical illustrations
show that this implicit numerical method exhibits a fourth-order convergence in the
space direction and the order 2 − α in time. Moreover, we observe an enhancement in order of spatial convergence whenever α tends to 0. The results obtained are then compared with existing popular techniques to demonstrate the accuracy of modified cubic B-spline-based DQM.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 08 June 2026

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History

  • Receive Date: 17 February 2026
  • Revise Date: 02 June 2026
  • Accept Date: 06 June 2026

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