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Time fractional advection-dispersion equation

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Abstract

A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order α(0<α<-1). Using variable transformation, Mellin and Laplace transforms, and properties of H-functions, we derive the complete solution of this time fractional advection-dispersion equation.

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Author information

Correspondence to F. Liu.

Additional information

Fawang Liu received his MSc from Fuzhou University in 1982 and PhD from Trinity College, Dublin, in 1991. Since graduation, he has been working in computational and applied mathematics at Fuzhou University, Trinity College Dublin and University College Dublin, University of Queensland, Queensland University of Technology and Xiamen University. Now he is a Professor at Xiamen University. His research interest is numerical analysis and techniques for solving a wide variety of problems in applicable mathematics, including semiconductor device equations, microwave heating problems, gas-solid reactions, singular perturbation problem, saltwater intrusion into aquifer systems and fractional differential equations.

Vo And received his PhD degree from the University of Tasmania, Australia, in 1978, He has been with Queensland University of Technology since 1984. His research interests include stochastic processes and random fields, fractional diffusion, environmental modelling financial modelling.

Ian Turner is a senior lecturer at the School of Mathematical Science, QUT. He has extensive experience in the solution of systems of non-linear partial differential equations using the fimite volume discretisation process and has written numerous journal publications in the field. He has been awarded outstanding paper awards from two international journals for his modelling work, with the most significantcontribution being the use of mathematical models for furthering the understanding of how microwaves interact with lossy materials during heating and drying processes. He also has considerable expertise in solving large sparse non-linear and linear systems via preconditioned Krylov based methods.

Zhuang Pinghui received his BSc and MSc from Fuzhou University in 1982 and 1988 respectively, Since graduation, he has been working in computational and applied mathematics at Xiamen University. Now he is an associate professor. His research interest is numerical analysis and techniques for solving singular perturbation problem and fractional differential equations, numerical simulation for saltwater intrusion into aquifer systems and computing.

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Liu, F., Anh, V.V., Turner, I. et al. Time fractional advection-dispersion equation. JAMC 13, 233 (2003). https://doi.org/10.1007/BF02936089

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AMS Mathematics Subject Classification

  • 26A33
  • 49K20
  • 44A10

Key words and phrases

  • time fractional advection-dispersion equation
  • Mellin transform
  • Laplace transform