In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,ex...In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,existing numerical methods often struggle with the computational challenges posed by such equations,especially in nonlinear,multi-term formulations.This study introduces two hybrid numerical methods—the Linear-Sine and Cosine(L1-CAS)and fast-CAS schemes—for solving linear and nonlinear multi-term Caputo variable-order(CVO)fractional partial differential equations.These methods combine CAS wavelet-based spatial discretization with L1 and fast algorithms in the time domain.A key feature of the approach is its ability to efficiently handle fully coupled spacetime variable-order derivatives and nonlinearities through a second-order interpolation technique.In addition,we derive CAS wavelet operational matrices for variable-order integration and for boundary value problems,forming the foundation of the spatial discretization.Numerical experiments confirm the accuracy,stability,and computational efficiency of the proposed methods.展开更多
In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order a...In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.展开更多
基金supported by the National Research Foundation of Korea(NRF)grant funded by the Korean government(MSIT)(NRF-2021R1A2C1011817)the BK21 Program(Next Generation Education Program for Mathematical Sciences,4299990414089)funded by the Ministry of Education(MOE,Republic of Korea).
文摘In recent years,variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability tomodel complex physical phenomena withmemory and spatial heterogeneity.However,existing numerical methods often struggle with the computational challenges posed by such equations,especially in nonlinear,multi-term formulations.This study introduces two hybrid numerical methods—the Linear-Sine and Cosine(L1-CAS)and fast-CAS schemes—for solving linear and nonlinear multi-term Caputo variable-order(CVO)fractional partial differential equations.These methods combine CAS wavelet-based spatial discretization with L1 and fast algorithms in the time domain.A key feature of the approach is its ability to efficiently handle fully coupled spacetime variable-order derivatives and nonlinearities through a second-order interpolation technique.In addition,we derive CAS wavelet operational matrices for variable-order integration and for boundary value problems,forming the foundation of the spatial discretization.Numerical experiments confirm the accuracy,stability,and computational efficiency of the proposed methods.
文摘In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments.
