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An analytic algorithm for the space-time fractional advection-dispersion equation. (English) Zbl 1217.65196
Fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first-order time derivative and first- and second-order space derivatives are replaced by Caputo derivatives of orders $$0<\alpha\leq1$$; $$0<\beta\leq1$$ and $$1<\gamma\leq2$$, respectively. We use Caputo definition to avoid (i) mass balance error, (ii) hyper-singular improper integral, (iii) non-zero derivative of constant, and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present an analytic algorithm to solve FADE based on homotopy analysis method which has the advantage of controlling the region and rate of convergence of the solution series via the auxiliary parameter $$\hbar$$ constant over two pi over the variational iteration method and homotopy perturbation method. We find that the proposed method converges to the numerical/exact solution of the ADE as the fractional orders $$\alpha, \beta, \gamma$$ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. Example 5 describes the intermediate process between advection and dispersion via Caputo fractional derivative.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations 35R11 Fractional partial differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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