Chin. Phys. B, 2023, Vol. 32(3): 030202    DOI: 10.1088/1674-1056/ac9368
 GENERAL Prev   Next

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

Ming-Jing Du(杜明婧)1,†, Bao-Jun Sun(孙宝军)1, and Ge Kai(凯歌)2
1 School of Computer Information Management, Inner Mongolia University of Finance and Economics, Hohhot 010070, China;
2 School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
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.
Keywords:  time-fractional partial differential equation      adaptive multi-step      reproducing kernel method method      numerical solution
Received:  05 June 2022      Revised:  01 September 2022      Accepted manuscript online:  21 September 2022
 PACS: 02.60.Cb (Numerical simulation; solution of equations) 02.60.-x (Numerical approximation and analysis) 02.30.Jr (Partial differential equations)
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 71961022, 11902163, 12265020, and 12262024), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant Nos. 2019BS01011 and 2022MS01003), 2022 Inner Mongolia Autonomous Region Grassland Talents Project-Young Innovative and Entrepreneurial Talents (Mingjing Du), 2022 Talent Development Foundation of Inner Mongolia Autonomous Region of China (Ming-Jing Du), the Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region Program (Grant No. NJYT-20-B18), the Key Project of High-quality Economic Development Research Base of Yellow River Basin in 2022 (Grant No. 21HZD03), 2022 Inner Mongolia Autonomous Region International Science and Technology Cooperation High-end Foreign Experts Introduction Project (Ge Kai), and MOE (Ministry of Education in China) Humanities and Social Sciences Foundation (Grants No. 20YJC860005).
Corresponding Authors:  Ming-Jing Du     E-mail:  724297269@qq.com