We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

中文翻译：

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多项式时间分数扩散和扩散波方程的近似解

我们开发了一种数值方案，用于考虑光滑和非光滑解，为一维和二维多维时间分数扩散和扩散波方程找到近似解。长期时间分数导数的概念通常在Caputo观点中定义。在目前的研究中，进行了勒让德配位谱方法的收敛性分析。因此，在几个基准示例上测试了光谱搭配方法，以验证准确性并确认所提出方法的有效性。该方法的主要优点是仅需要少量移位的Legendre多项式即可获得准确而有效的结果。