《走出去：Fractional Partial Differential Equations and their Numerical Solutions》 mainly concerns the partial differential equations of fractional order and their numerical solutions. In Chapter 1， we briefly introduce the history of fractional order derivatives and the background of some fractional partial differential equations， in particular， their interplay with random walks. Chapter 2 is devoted to the definition of fractional derivatives and integrals from different points of view， from the Riemann-Liouville type， Caputo type derivatives and fractional Laplacian， to several useful tools in fractional calculus， including the pseudo-differential operators， fractional order Sobolev spaces， commutator estimates and so on. In chapter 3， we discuss some partial differential equations of wide interests， such as the fractional reaction-diffusion equation， fractional Ginzburg-Landau equation， fractional Landau-Lifshitz equations， fractional quasi-geostrophic equation， as well as some boundary value problems， especially the harmonic extension method. The local and global well-posedness， long time dynamics are also discussed. Last three chapters are devoted to the numerical aspects of fractional partial differential equations， mainly focusing on the finite difference method， series approximation method， Adomian decomposition method， variational iterative method， finite element method， spectral method and meshfree method and so on.

暂无

暂无