Fractional diffusion equation and relaxation in complex viscoelastic materials

Optimal error bound and modified kernel method for a space-fractional backward diffusion problem

“…In recent decades, fractional operators have been playing more and more important roles in science and engineering [1], e.g., mechanics, biochemistry, electrical engineering, and medicine, see [2][3][4][5][6][7][8]. These new fractional-order models are more adequate than the integer-order models, because the fractional order derivatives and integrals enable to describe the memory and hereditary properties of different substance [9].…”

Solution of fractional partial differential equations using iterative method

“…Both second and fourth order diffusion-wave equations appear in diffusion of heat, deflection and vibration of plates and membranes and propagation of waves. Also such equations appear in relaxation phenomena in complex viscoelastic material [15], propagation of mechanical waves in viscoelastic media [24], [25], [26], non-Markovian diffusion process with memory [27], electromagnetic acoustic and mechanical responses [28]. Roman and Alemany [33] investigated a continuous time random walks on fractals by studying diffusion-wave equations.…”

Simultaneous Inversion for Space-Dependent Diffusion Coefficient and Source Magnitude in the Time Fractional Diffusion Equation

“…In recent two decades, some researches indicated that the classical model was inadequate to simulate many real situations, where a particle plume spreads slower than predicted by the integer-order diffusion equation. See, e.g., Adams and Gelhar (1992), Berkowitz, Scher, and Silliman (2000), Giona, Gerbelli, and Roman (1992), Hatano, Y. and Hatano, N. (1998), Nigmatullin (1986), Xiong, Huang, and Huang (2006), Zhou and Selim (2003), and the references therein. Such slow diffusion is called anomalous subdiffusion, and one model equation is the time fractional diffusion equation given as…”