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Session 8. Dynamic Systems with Fractional and Time Scale Derivatives
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Numerical solution of the 1D subdiffusion equation with two moving
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Marek Błasik, Częstochowa University of Technology, Poland
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The talk is based on the joint work with Małgorzata Klimek
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Moving boundary problems are a special case of the boundary value problems. They are often called Stefan problems and were extensively studied in the partial differential equations theory (compare monograph [1] and the
references therein). The description including moving boundaries was applied in modeling of the
formation of sedimentary ocean deltas [2] and the moisture transport such as swelling
grains or polymers [3]. The fractional extension of the dual moving boundaries problem is
used as the mathematical model of a drug release from a polymeric matrix [4].
We shall construct a numerical solution of the system of equations presented below:
\begin{eqnarray}
{}^{c}D^{\alpha}_{0+,\tau}f(X, \tau)= \frac{\partial^{2}f(X,\tau)}{\partial X^{2}},
\quad S_1(\tau)<X<S_2(\tau), \quad \tau >0\\
f(S_{1}(\tau),\tau)=1, \quad f(S_{2}(\tau),\tau)=0, \quad \tau >0\\
f(0^{+},0)=0,\quad S_1(0)=0,\quad S_2(0)=0\\
{}^{c}D^{\alpha}_{0+,\tau}S_{2}(\tau)=-\Lambda_2 \frac{\partial f(X,\tau)}
{\partial X}|_{X=S_{2}(\tau)}\\
{}^{c}D^{\alpha}_{0+,\tau}S_{1}(\tau)=\Lambda_1 \frac{\partial f(X,\tau)}
{\partial X}|_{X=S_{1}(\tau)}
\end{eqnarray}
which constitute the 1D fractional Stefan problem with two moving boundaries given as
\(S_1 (\tau)\) and \(S_2 (\tau)\). In our approach we use
new spatial variable \(u=\frac{X-S_1(\tau)}{S_2(\tau)-S_1(\tau)}\) as in this new coordinates system: \((u, \tau)\)
the boundaries are fixed.
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References- S.C. Gupta. The Classical Stefan Problem. Basic Concepts, Modeling and
Analysis. Elsevier, Amsterdam, 2003.
- J. Lorenzo-Trueba, V.R. Voller. Analytical and numerical solution of a
generalized Stefan problem exhibiting two moving boundaries with application to ocean delta
formation. J. Math. Anal. Appl. 366, 2010, 538-549.
- S.I. Barry, J. Caunce. Exact and numerical solutions to a Stefan
problem with two moving boundaries. Appl. Math. Model. 32, 2008, 83-98.
- Chen Yin, Mingyu Xu An asymptotic analytical solution to the problem of
two moving boundaries with fractional diffusion in one-dimensional drug release devices. J.
Phys. A: Math. Theor. 42, 2009, 115210.
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