Jiang H, Liu F, Meerschaert M M, McGough R J
Department of Mathematical, Qinghai Normal University, Xining 810008, China.
Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia.
Electron J Math Anal Appl. 2013 Jan;1(1):55-66.
Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, ) ( > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko's Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
具有多个时间分数阶导数项的分数阶偏微分方程，如萨博波动方程或幂律波动方程，描述了重要的物理现象。然而，这些多时间项 - 空间或时间分数阶波动方程的研究仍在发展中。本文考虑了有限域中的多时间项修正幂律波动方程。多时间项分数阶导数在卡普托意义下定义，其阶数分别属于区间(1, 2]、[2, 3)、[2, 4)或(0, )( > 2)。推导了多时间项修正幂律波动方程的解析解。这些新技术基于卢奇科定理、拉普拉斯算子的谱表示、分离变量法和分数阶导数技术。然后将这些通用方法应用于萨博波动方程和幂律波动方程的特殊情况。这些方法和技术也可以扩展到其他类型的多时间项 - 空间分数阶模型，包括分数阶拉普拉斯算子。
