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Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients. (English) Zbl 1185.65146
The authors study a non homogeneous evolution equation with variable coefficients and fractional diffusion term of the form $$d_{+}(x)\frac{\partial^{\alpha}u(x,t)}{\partial_{+}x^{\alpha}}+ d_{-}(x)\frac{\partial^{\alpha}u(x,t)}{\partial_{+}x^{\alpha}},$$ where $1\leq\alpha\leq2,d_{+}(x)\geq0,d_{-}(x)\geq0$. Using a weighted finite difference method and the Grünwald formula for the fractional derivative, the authors obtain an approximate scheme for the initial value problem with the truncation error $ R_{j}^{n}(u)=O(\tau^{2}+h)$ for $r=1/2$ and $R_{j}^{n}(u)=O(\tau+h)$ for $ r\neq1/2,0\leq r\leq 1$, where $r$ is a weighted parameter. Some numerical examples are given.

MSC:
65M06Finite difference methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35R11Fractional partial differential equations
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Full Text: DOI
References:
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