分数阶偏微分方程已经在复杂系统中的反常扩散、多孔介质中的传播、电解质的极化等一系列领域中有了令人瞩目的应用。 本项目旨在研究多项时间-空间分数阶波动和扩散方程的解析解、数值解及其相关应用。详细分析多项时间-空间分数阶波动和扩散方程非齐次初边值问题经典解和弱解的存在唯一性，并讨论解的正则性、能量估计、渐近性态等。特别地，针对一维空间中多项时间-空间分数阶波动和扩散方程,如Szabo波方程、幂律波方程，给出具体的解析解表达式。另外，给出多项时间-空间分数阶波动和扩散方程的数值解，证明算法的稳定性和收敛性。最后，将上述理论和数值结果推广到二维、三维甚至更高维空间中。 本项目不仅可以促进分数阶偏微分方程相关理论的进一步发展，而且使得生物工程实际问题得以解决。

The investigation of fractional partial differential equations has recently drawn attentions from researchers and academics during the last decade. Research results indicated its potential applications in modelling various kinds of real physical phenomena such as anomalous diffusion, dispersion in porous media, and electrolyte polarization. This project aims to develop both theoretical and numerical analyses on the solutions of the multi-term time-space fractional wave and diffusion equations. The theoretical part will be focused on proving the uniqueness and existence of their solutions under standard initial/boundary conditions in the classical and weak senses. Also, the regularity, energy estimation, and asymptotic behaviors of the solutions will be investigated. For illustration, the analytical expression of the solution to a one-dimensional multi-term time-space fractional wave equation, such as Szabo wave equation and the power law wave equation, will be derived. The numerical analyses will be emphasized on the stabilities and convergences of the numerical algorithms to the solutions of the multi-term fractional wave and diffusion equations. An extension of both the theoretical and numerical results to higher dimensional space will be made. The outcome of this project will further promote the current researches on fractional partial differential equations. In particular, the analytical and numerical results on the demonstrated Szabo and power law equations will have an impact to the present study on biological engineering disciplines.