Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics

Abstract

This paper is aimed at solving the nonlinear time-fractional partial differential equation with two small parameters arising from option pricing model in financial economics. The traditional reproducing kernel (RK) method which deals with this problem is very troublesome. This paper proposes a new method by adaptive multi-step piecewise interpolation reproducing kernel (AMPIRK) method for the first time. This method has three obvious advantages which are as follows. Firstly, the piecewise number is reduced. Secondly, the calculation accuracy is improved. Finally, the waste time caused by too many fragments is avoided. Then four numerical examples show that this new method has a higher precision and it is a more timesaving numerical method than the others. The research in this paper provides a powerful mathematical tool for solving time-fractional option pricing model which will play an important role in financial economics.