Numerical solution of fractional derivative equations in mechanics: advances and problems

Numerical solution of fractional derivative equations in mechanics: advances and problems Wen Chen1, Hongguang Sun1 Summary This report is to make a survey on the numerical techniques for fractional derivative equations in mechanical and physical fields, including numerical integration of fractional time derivative and emerging approximation strategies for fractional space derivative equations. The perplexing issues are highlighted, while the encouraging progresses are summarized. We also point out some emerging techniques which will shape the future of the numerical solution of fractional derivative equations. The frequency scaling power law of fractional order appears universal in physical behaviors of complex solids, fluids and soft matter (e.g., polymers, colloids, emulsions, foams, biomaterials, rock layers, sediments, plastics, glass, rubber, oil, soil, DNA) and is considered “anomalous” compared with those of the ideal solids and fluids, in that the various gradient laws of physics, mechanics and chemistry (e.g., Fickian diffusion, Fourier heat conduction, Darcy’s law) are broken. The particular examples are viscous dampers in seismic isolation of buildings, anomalous diffusions in porous media and turbulence, power law dissipation of medical ultrasonic imaging, inelastic dissipative vibration of polymers and soil, just to mention a few. The standard mathematical modeling approach using integer-order timespace derivatives can not accurately reflect fractional power law, while the fractional derivatives are instead found an irreplaceable modeling approach [1]. For example, anomalous diffusion equation of fractional derivatives has been recognized as a master equation in nature for multidisciplinary applications (e.g., transport, relaxation, heat conduction) and is stated as [2]