• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.233-245
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• 2003

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 ${\alpha}$(0 < ${\alpha}$ $\leq$ 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.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.339-350
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• 2005

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha{\in}(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta{\in}0,2]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.

• Journal of Korea Water Resources Association
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• v.32 no.6
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• pp.607-616
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• 1999

A fractional step finite difference model for the longitudinal dispersion of nonconservative contaminants is developed. It is based on splitting the longitudinal dispersion equation into a set of three equations each to be solved over a one-third time step. The fourth-order Holly-Preissmann scheme, an analytic solution, and the Crank-Nicholson scheme are used to solve the equations for the pure advection, the first-order decay, and the diffusion, respectively. To test the model, it is applied to simulate the longitudinal dispersion of continuous source released into a nonuniform flow field as well as the dispersion of an instantaneous source in a uniform flow field. The results are compared with the exact solution and those computed by an existing model. Compared to the existing model which uses Euler method for the first-order decay equation, the present model yield more accurate results as the decay coefficient increases.

• Water for future
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• v.29 no.4
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• pp.187-198
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• 1996

A numerical model for unsteady dispersion of horizontal line source in turbulent shear flow is developed. A fractional step finite difference method is used which splits the unsteady two-dimensional advective diffusion equation into the longitudinal advection and the vertical diffusion equations, and solves them alternately for half time intervals by the Holly-Preissmann scheme and the Crank-Nicholson scheme, respectively. The developed numerical model is verified using a semi-analytic solution for steady dispersion in turbulent shear flow. Dispersion of an instantaneous plane source in turbulent shear flow is analyzed using the model. The degree of mixing at the same dimensionless time is almost the same regardless of the friction factor, and the travel distance required to reach a certain degree of mixing is inversely proportional to the square root of the friction factor.