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An efficient numerical treatment of fourth-order fractional diffusion-wave problems. (English) Zbl 1407.74099

Summary: In this paper, we consider the numerical treatment of a fourth-order fractional diffusion-wave problem. Our proposed method includes the use of parametric quintic spline in the spatial dimension and the weighted shifted Grünwald-Letnikov approximation of fractional integral. The solvability, stability, and convergence of the numerical scheme are rigorously proved. It is shown that the theoretical convergence order improves those of earlier work. Simulation is further carried out to demonstrate the numerical efficiency of the proposed scheme and to compare with other methods.

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65D07 Numerical computation using splines
35R11 Fractional partial differential equations
76Q05 Hydro- and aero-acoustics
86A15 Seismology (including tsunami modeling), earthquakes
35Q35 PDEs in connection with fluid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74D10 Nonlinear constitutive equations for materials with memory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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