An ε -Approximate Approach for Solving Variable-Order Fractional Differential Equations

Fractal and Fractional

article

Yahong Wang¹  Wenmin Wang¹  Liangcai Mei²  Yingzhen Lin²  Hongbo Sun² 

[1] School of Computer Science and Engineering, Macau University of Science and Technology;Zhuhai Campus, Beijing Institute of Technology

关键词: VO-FDEs;    VO-FIDEs;    Sobolev space;    scale functions;    ε-approximate;   

DOI  :  10.3390/fractalfract7010090

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

As a mathematical tool, variable-order (VO) fractional calculus (FC) was developed rapidly in the engineering field due to it better describing the anomalous diffusion problems in engineering; thus, the research of the solutions of VO fractional differential equations (FDEs) has become a hot topic for the FC community. In this paper, we propose an effective numerical method, named as theε -approximate approach, based on the least squares theory and the idea of residuals, for the solutions of VO-FDEs and VO fractional integro-differential equations (VO-FIDEs). First, the VO-FDEs and VO-FIDEs are considered to be analyzed in appropriate Sobolev spacesH 2 n [ 0 , 1 ]and the corresponding orthonormal bases are constructed based on scale functions. Then, the spaceH2 , 02 [ 0 , 1 ]is chosen which is just suitable for one of the models the authors want to solve to demonstrate the algorithm. Next, the numerical scheme is given, and the stability and convergence are discussed. Finally, four examples with different characteristics are shown, which reflect the accuracy, effectiveness, and wide application of the algorithm.

【 授权许可】

CC BY