Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black–Scholes

Abstract

In this paper, as far as the authors know, for the first time, a one-dimensional partial differential model is generalized using fractional differential operators and the same principle that provides the dimensional invariance of the radial basis functions methodology, resulting in a multidimensional fractional partial differential model that can be solved using a numerical scheme of radial basis functions. A radial basis functions scheme is proposed to solve numerically, on different node configurations, multidimensional fractional partial differential equations, both in space and in time. Using the QR factorization, a way to reduce the condition number of the interpolation matrices of the proposed scheme is presented, the resulting scheme is used to numerically solve the diffusion equation that may be obtained from the Black–Scholes model, as well as some generalizations of this diffusion model with fractional differential operators and multiple dimensions. The Caputo fractional derivative is discretized with an order error $${\mathcal {O}}(\mathrm{{d}}t^{n-\alpha +1})$$ , with $$(n-1)<\alpha \le n$$ . The examples of fractional partial differential equations that are presented involve the Caputo fractional operator in the temporal part due to the memory phenomenon, and the Riemann–Liouville fractional operator in the spatial part due to the property of nonlocality.