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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:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
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