In this paper, we investigate the well-posedness and the long-time asymptotic behavior for the initial-boundary value problem for multi-term time-fractional diffusion equations, where the time differentiation consists of a finite summation of Caputo derivatives with decreasing orders in (0, 1) and positive constant coefficients. By exploiting several important properties of multinomial Mittag-Leffler functions, various estimates follow from the explicit solutions in form of these special functions. Then the uniqueness and continuous dependency upon initial value and source term are established, from which the continuous dependence of solution of Lipschitz type with respect to various coefficients is also verified. Finally, by a Laplace transform argument, it turns out that the decay rate of the solution as t → ∞ is dominated by the minimum order of the time-fractional derivatives.

In this paper, we first establish a weak unique continuation property for timefractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.2 BANGTI JIN, RAYTCHO LAZAROV, YIKAN LIU, AND ZHI ZHOU space and slow asymptotic decay in time [30], which in turn also impacts related numerical analysis [12] and inverse problems [14,30].The model (1.1) was developed to improve the modeling accuracy of the single-term model (1.3) for describing anomalous diffusion. For example, in [31], a two-term fractionalorder diffusion model was proposed for the total concentration in solute transport, in order to distinguish explicitly the mobile and immobile status of the solute using fractional dynamics. The kinetic equation with two fractional derivatives of different orders appears also quite naturally when describing subdiffusive motion in velocity fields [26]; see also [16] for discussions on the model for wave-type phenomena.There are very few mathematical studies on the model (1.1). Luchko [23] established a maximum principle for problem (1.1), and constructed a generalized solution for the case f ≡ 0 using the multinomial Mittag-Leffler function. Jiang et al [9] derived analytical solutions for the diffusion equation with fractional derivatives in both time and space. Li and Yamamoto [20] established existence, uniqueness, and the Hölder regularity of the solution using a fixed point argument for problem (1.1) with variable coefficients {bi}. Very recently, Li et al [19] showed the uniqueness and continuous dependence of the solution on the initial value v and the source term f , by exploiting refined properties of the multinomial Mittag-Leffler function.The applications of the model (1.1) motivate the design and analysis of numerical schemes that have optimal (with respect to data regularity) convergence rates. Such schemes are especially valuable for problems where the solution has low regularity. The case m = 0, i.e., the single-term model (1.3), has been extensively studied, and stability and error estimates were provided; see [21,35] for the finite difference method, [18,34] for the spectral method, [25,27,28,12,11,10] for the finite element method, and [3,7] for meshfree methods based on radial basis functions, to name a few. In particular, in [10,11,12], the authors established almost optimal error estimates with respect to the regularity of the initial data v and the right hand side f for a semidiscrete Galerkin scheme. These studies includ...

The complex [Cu(en) 2 (H 2 O)](sy) 2 (en)(H 2 O) 2 has been synthesized and characterized by its electronic and vibrational spectra. The molecular structure of the complex has been determined by X-ray diffraction methods. The complex crystallizes in the orthorhombic space group Pnma with unit-cell parameters a = 10.7236 (5), b = 20.4660(10), c = 14.4523(11)Å and Z = 4. In the cation, the Cu(II) ion has a distorted square pyramidal coordination with two bidendate (en) ligands forming the basal plane and a H 2 O molecule in the apical position. The complex cations and syringate anions constitute chains along the b axis in -A-B-A-fashion. The members of the chains are linked by through N-H···O hydrogen bonds. The (en) molecules are responsible for connecting adjacent layers.

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying properties of solutions. This chapter surveys works on such inverse problems for fractional diffusion equations.