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A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations

机译:一种求解多项时间分数阶扩散波方程的半解析搭配Trefftz方案

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This study presents a semi-analytical boundary-only collocation technique for solving multi-term time-fractional diffusion-wave equations. In the present collocation technique, the Laplace transformation is first implemented to convert time-fractional diffusion-wave equation to a series of time-independent nonhomogeneous equations in Laplace domain. Then the composite multiple reciprocity method (CMRM) is applied to construct a high-order homogeneous equation, which has the same solution with one of time-independent nonhomogeneous equations in Laplace domain. The collocation Trefftz scheme with high-order T-complete functions is used to obtain the semi-analytical solution of high-order homogeneous equation with boundary-only collocation in Laplace domain. Finally the numerical Laplace inversion scheme is introduced to invert the Laplace domain solutions back into the time-dependent solutions of time-fractional diffusion-wave equations. The proposed method is easy-to-implement and flexible for irregular domain problems. It evades costly convolution integral calculation in time fractional derivation approximation, and avoids the effect of time step on numerical accuracy and stability. Error analysis and numerical investigations show that the proposed collocation method is highly accurate, computationally efficient, and numerically stable for multi-term time fractional diffusion-wave equations.
机译:这项研究提出了一种半解析的仅边界搭配技术,用于解决多项式时间分数阶扩散波方程。在当前的配置技术中,首先实现拉普拉斯变换,以将时间分数阶扩散波方程转换为一系列在拉普拉斯域中与时间无关的非齐次方程。然后,采用复合多重互易方法(CMRM)构造一个高阶齐次方程,该方程与拉普拉斯域中与时间无关的非齐次方程之一具有相同的解。使用具有高阶T-完成函数的搭配Trefftz方案来获得在Laplace域中仅边界搭配的高阶齐次方程的半解析解。最后,引入了数值拉普拉斯反演方案,以将拉普拉斯域解反演为时间分数阶扩散波方程的时间相关解。所提出的方法易于实现并且对于不规则域问题具有灵活性。在时间分数导数近似中,它避免了昂贵的卷积积分计算,并且避免了时间步长对数值精度和稳定性的影响。误差分析和数值研究表明,对于多项时间分数阶扩散波方程,该配置方法具有很高的精确度,计算效率和数值稳定性。

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