Galerkin-finite difference method for fractional parabolic partial differential equations

Abstract

The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.

Keywords: Black-scholes model; Fractional PDE; Galerkin -finite difference method; Galerkin-Finite Difference Method; Option pricing,.

Graphical abstract

Fig. 1

Absolute errors for Problem-1.

Fig. 2

Values and absolute errors for Problem-2 with $\beta =1.6$.

Fig. 3

Values and absolute errors for Problem-2 with $\alpha =0.5$.

Fig. 4

Prices and absolute errors to approximate Put option values for Problem-3.

Fig. 5

Prices and absolute errors to approximate Call option values for Problem-3.

Fig. 6

Prices and absolute errors to approximate Call option values for Problem-4 with $\beta =1.6$.

Fig. 7

Prices and absolute errors to approximate Call option values for Problem-4 with $\alpha =0.5$.

Fig. 8

Prices and absolute errors to approximate Put option values for Problem-4 with $\beta =1.6$.

Fig. 9

Prices and absolute errors to approximate Put option values for Problem-4 with $\alpha =0.5$.