Abstract
We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.
Highlights
Fractional differential equations play a significant role in modeling the so-called anomalous transport phenomena and in the theory of complex systems
The books [1,2,3,4] are completely devoted to different applications of fractional differential equations in many areas, such as engineering, physics, chemistry, astrophysics, and other sciences and historical summaries of the development of fractional calculus
Fractional kinetics systems are widely applied to describe anomalous diffusion or advectiondispersion processes [5, 6]. For processes lacking such scaling the corresponding description may be given by distributedorder fractional partial differential equations [7]
Summary
Fractional differential equations play a significant role in modeling the so-called anomalous transport phenomena and in the theory of complex systems. Kochubei [12] considered the time distributedorder equation and developed a mathematical theory of this equation and studied the derivatives and integrals of distributed order This equation is applied in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement. Nowadays the studies of the dissipative and dispersive properties in diffusion equation with fractional order in time and/or space domain in anelastic and dielectric media have been spread to many phenomena from nonlinearity to statistical mechanics and memory formalisms, to represent the diversified forms of deviations from the classic constitutive laws and several complex mathematical methods. Hu et al [27] considered a new time distributed-order and two-side space-fractional advectiondispersion equation and a time distributed-order diffusion model, respectively They discretized the distributed-order equation into a multiterm fractional partial differential equation. We give two examples to illustrate the behavior of our numerical methods and demonstrate the effectiveness of our theoretical analysis
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