Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

• Published in: Applied mathematics and computation 2019-04, Vol.346, p.531-544
• Main Authors: Zhao, Dazhi, Luo, Maokang
• Format: Article
• Language: English
• Subjects: GC derivative; General fractional derivative; GRL derivative; Heat conduction; Heat waves; Memory effect
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2018.10.037

•The general fractional derivatives with memory effects named GC/GRL derivative are proposed.•The definition of new derivatives is derived from some basic principles.•The Gurtin–Pipkin theory of heat conduction is reformulated by the GC/GRL derivative.•Simple numerical schemas for the corresponding FDEs are given. Fractional derivative is a widely accepted theory to describe physical phenomena and processes with memory effect that is defined in the form of convolution with power kernel. Due to the shortcomings of power law distribution, some derivatives with other kernels are proposed, including Caputo–Fabrizio derivative, Atangana–Baleanu derivative and so on. In this paper, in order to provide some flexible and more appropriate tools which can better describe cases of the dynamics with memory effects or of nonlocal phenomena, we derive the definition of general fractional derivatives with memory effects named GC derivative and GRL derivative from some basic principles. We demonstrate that the mathematical expression of Gurtin–Pipkin theory of heat conduction with finite wave speeds is a special example of GC/GRL derivative.