A STRONG POSITIVITY PROPERTY AND A RELATED INVERSE SOURCE PROBLEM FOR MULTI-TERM TIME-FRACTIONAL DIFFUSION EQUATIONS

In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Diffusion Equations of the Electric Charges and Magnetic Flux

Innovative definitions of the electric and magnetic diffusivities through conducting mediums and innovative diffusion equations of the electric charges and magnetic flux are verified in this article. Such innovations depend on the analogy of the governing laws of diffusion of the thermal, electrical, and magnetic energies and newly defined natures of the electric charges and magnetic flux as energy, or as electromagnetic waves, that have electric and magnetic potentials. The introduced diffusion equations of the electric charges and magnetic flux involve Laplacian operator and the introduced diffusivities. Both equations are applied to determine the electric and magnetic fields in conductors as the heat diffusion equation which is applied to determine the thermal field in steady and unsteady heat diffusion conditions. The use of electric networks for experimental modeling of thermal networks represents sufficient proof of similarity of the diffusion equations of both fields. By analysis of the diffusion phenomena of the three considered modes of energy transfer;the rates of flow of these energies are found to be directly proportional to the gradient of their volumetric concentration, or density, and the proportionality constants in such relations are the diffusivity of each energy. Such analysis leads also to find proportionality relations between the potentials of such energies and their volumetric concentrations. Validity of the introduced diffusion equations is verified by correspondence their solutions to the measurement results of the electric and magnetic fields in microwave ovens.

Exact Solutions of a Generalized Multi-Fractional Nonlinear Diffusion Equation in Radical Symmetry

This paper is devoted to investigating exact solutions of a generalized fractional nonlinear anomalousdiffusion equation in radical symmetry.The presence of external force and absorption is also considered.We firstinvestigate the nonlinear anomalous diffusion equations with one-fractional derivative and then multi-fractional ones.Inboth situations,we obtain the corresponding exact solutions,and the solutions found here can have a compact behavioror a long tailed behavior.

Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.

THE LARGE TIME BEHAVIOR OF SPECTRAL APPROXIMATION FOR A CLASS OF PSEUDOPARABOLIC VISCOUS DIFFUSION EQUATION

The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d（AN,A） → 0 are proved.

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article,we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation.This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem.We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule.Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable.

IDENTIFYING AN UNKNOWN SOURCE IN SPACE-FRACTIONAL DIFFUSION EQUATION

In this paper, we identify a space-dependent source for a fractional diffusion equation. This problem is ill-posed, i.e., the solution （if it exists） does not depend continuously on the data. The generalized Tikhonov regularization method is proposed to solve this problem. An a priori error estimate between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained, Numerical examples are presented to illustrate the validity and effectiveness of this method.

FILTERING OF MEDICAL ULTRASONIC IMAGES BASED ON A MODIFIED ANISTROPIC DIFFUSION EQUATION

Speckle noise reduction is a key problem of the image analysis of medical UltraSound images. In this paper, two important improvements have been developed to a fast anisotropic diffusion algorithm for speckle noise reduction. The Gaussian filter is firstly used before gradient calculation, and then the adaptive algorithm of the factor k is proposed. Numerous experimental results show that the proposed model is superior to other methods in noise removal, fidelity and edge preservation. It is suitable for the preprocessing of a great number of medical UltraSound images, such as three dimen- sional reconstruction.

SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION

In this paper we discuss the existence and nonexistence of singular solutions for a porous medium equations with convection and absorption terms.

Balance of Datum Land Prices Among Cities Based on the City Gravitation Model and Stochastic Diffusion Equation

A balance of urban datum land prices is achieved to harmonize regional land prices and make the prices truly reflect different economic development levels and land prices among cities. The current piecewise linear interpolation balance method widely used has two drawbacks that always lead to an unsatisfactory balance among some cities. When the excess of land price in the central city to the surrounding zone reaches a certain degree, land price in the circumjacent city is not only consistent with the local land grade and land use level, but also influenced by the diffusion of land price in the central city. Thus, a new balanced scheme of datum land prices based on the city gravitation model and stochastic diffusion equation is brought forward. Finally, the new method is examined in the practice of datum land price balance in Hubei Province, China.

Superconvergence Study of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations

In this paper,we apply the Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin(DDG)method(Liu and Yan in SIAM J Numer Anal 47(1):475-698,2009),the DDG method with the interface correction(DDGIC)(Liu and Yan in Commun Comput Phys 8(3):541-564,2010),the symmetric DDG method(Vidden and Yan in Comput Math 31(6):638-662,2013),and the nonsymmetric DDG method(Yan in J Sci Comput 54(2):663-683,2013).We also include the study of the interior penalty DG(IPDG)method,due to its close relation to DDG methods.Error estimates are carried out for both P2 and P3 polynomial approximations.By investigating the quantitative errors at the Lobatto points,we show that the DDGIC and symmetric DDG methods are superior,in the sense of obtaining(k+2)th superconvergence orders for both P2 and P3 approximations.Superconvergence order of(k+2)is also observed for the IPDG method with P3 polynomial approximations.The errors are sensitive to the choice of the numerical flux coefficient for even degree P2 approximations,but are not for odd degree P3 approxi-mations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.

SOME RESULTS ON A DEGENERATE AND SINGULAR DIFFUSION EQUATION

In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.

Maximum modulus principle estimates for one dimensional fractional diffusion equation

We present scheme I for solving one-dimensional fractional diffusion equation with variable coefficients based on the maximum modulus principle and two Grunwald approxima- tions. Scheme II is obtained by using classic Crank-Nicolson approximations in order to improve the time convergence. Schemes are proved to be unconditionally stable and second-order accuracy in spatial grid size for the problem with order of fractional derivative belonging to [（√17- 1）/2, 2] using the maximum modulus principle. A numerical example is given to confirm the theoretical analysis result.

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A（u, Ux）Uxx ＋ B（u, ux） is studied by using the conditional Lie-Blicklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie--Biicklund symmetries, are characterized. To construct functionally gener- alized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.