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June 2017 - August 2017

## Publications

Publications (80)

This article deals with numerical solution and identification of the fractional orders for the generalized nonlocal elastic model. Based on the collocation-finite difference scheme for the forward operator, a regularized method is proposed for solving of the forward problem with Tikhonov regularization, which gives a feasible approach to numerical...

A simplified linear time-fractional SEIR epidemic system is set forth, and an inverse problem of determining the fractional order is discussed by using the measurement at one given time. By the Laplace transform the solution to the forward problem is obtained, by which the inverse problem is transformed to a nonlinear algebraic equation. By choosin...

A simplified linear time-fractional SEIR epidemic system is set forth, and an inverse problem of determining the fractional order is discussed by using the measurement at one given time. By the Laplace transform the solution to the forward problem is obtained, by which the inverse problem is transformed to a nonlinear algebraic equation. By choosin...

A fractal mobile-immobile (MIM in short) solute transport model in porous media is set forth, and an inverse problem of determining the fractional orders by the additional measurements at one interior point is investigated by Laplace transform. The unique existence of the solution to the forward problem is obtained based on the inverse Laplace tran...

This article deals with an inverse problem of identifying the fractional order in the 1D time fractional diffusion equation (TFDE in short) using the measurement at one space-time point. Based on the expression of the solution to the forward problem, the inverse problem is transformed to a nonlinear algebraic equation. By choosing suitable initial...

A fractal mobile-immobile (MIM in short) model for solute transport in heterogeneous porous media is investigated from numerics. An implicit finite difference scheme is set forth for solving the coupled system, and stability and convergence of the scheme are proved based on the estimate of the spectral radius of the coefficient matrix. Numerical si...

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

Inverse problems of determining time-dependent coefficients in partial differential equations are difficult to deal with in general cases. The variational iteration method is introduced to determine the time-dependent coefficient in the fractional diffusion equation as well as the solution of the forward problem. By utilizing the additional conditi...

Inverse problems of determining time-dependent coefficients in partial differential equations are difficult to deal with in general cases. The variational iteration method is introduced to determine the time-dependent coefficient in the fractional diffusion equation as well as the solution of the forward problem. By utilizing the additional conditi...

This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Neumann) boundary condition. We first prove the wellposedness of the forward problem by means of eigenfunction expansion, which ensures that the weak solution has the classical derivatives. We next give a Harnack type inequality of the so...

This paper deals with the 1D time-fractional diffusion equations with nonlinear boundary condition. We first give an integral equation of the solution via the theta-function and eigenfunction expansion and establish the short time asymptotic behavior of the solution. We then verify the uniqueness of the inverse problem in determining the fractional...

This paper deals with an inverse problem of simultaneously determining the space-dependent diffusion coefficient and the fractional order in the variable-order time fractional diffusion equation by the measurements at one interior point. Numerical solution to the forward problem is given by the finite difference scheme, and the homotopy regularizat...

In this article, for an advection-diffusion equation we study an inverse problem for restoration of source temperature from the information of final temperature profile. The uniqueness of this inverse problem is established by taking an integral transform and using Liouville's theorem (complex analysis).

This article deals with an inverse problem of determining the diffusion coefficients in 2D fractional diffusion equation with a Dirichlet boundary condition by the final observations at the final time. The forward problem is solved by the alternating direction implicit finite-difference scheme with the discrete of fractional derivative by shift Grü...

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to...

This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations. The inversion problem is of instability, but it is uniquely solvable based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function. From view...

This article deals with an inverse problem of determining a linear source term in the multidimensional diffusion equation using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and a conditional uniqueness for the inverse source problem is proved by the app...

This article deals with an inverse problem of determining the space-dependent diffusion coefficient and the source coefficient simultaneously in the multi-term time fractional diffusion equation (TFDE in short) using measurements at one inner point. From a view point of optimality, solving the inverse problem is transformed to minimize an error fun...

This paper deals with numerical solution for the space-time fractional diffusion equation with variable diffusion coefficient, and numerical inversion for the space-dependent diffusion coefficient by the homotopy regularization algorithm. An equivalent system to the forward problem is deduced by utilizing properties of the fractional derivatives, a...

This paper deals with numerical solution to the multi-term time fractional diffusion equation in a finite domain. An implicit finite difference scheme is established based on Caputo's definition to the fractional derivatives, and the upper and lower bounds to the spectral radius of the coefficient matrix of the difference scheme are estimated, with...

This paper deals with an inverse problem of determining a space-dependent source
coefficient in the 2D/3D advection-dispersion equation with final observations using the variational adjoint method.
Data compatibility for the inverse problem is analyzed by which an admissible set for the unknowns is induced.
With the aid of an adjoint problem, a bil...

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward prob...

In this study, Mg–Al-layered double hydroxide (LDH) and montmorillonite (M) were mixed (mass ratio = 1:1) with high-shear action to prepare a mineral composite (LDH–M). The structure, morphology, and textural properties of LDH and LDH–M were investigated via X-ray diffraction, field-emission scanning electron microscopy, Fourier transform infrared...

This paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald–Letnikov's definition of the fractional derivative, and i...

This paper deals with an inverse problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equation with Robin’s boundary condition. Data compatibility for the inverse problem is analyzed by which an admissible set for the
unknown is set forth. Furthermore, with the help of an integral identity, a conditi...

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem,...

We deal with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the source magnitude in the time fractional diffusion equation from viewpoint of numerics. Such simultaneous inversion problem is often of severe ill-posedness as compared with that of determining a single coefficient function. The forward pr...

This paper deals with an inverse problem of simultaneously determining the dispersion coefficients and the space-dependent source magnitude in 2D advection dispersion equation with finite observations at the final time. The forward problem is solved by using the alternating direction implicit (ADI) finite difference scheme, and then the optimal per...

This paper deals with an inverse problem for identifying multiparameters in 1D
space fractional advection dispersion equation (FADE) on a finite domain with
final observations. The parameters to be identified are the fractional order,
the diffusion coefficient, and the average velocity in the FADE. The forward
problem is solved by a finite differen...

This paper deals with numerical inversion for space-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation with additional boundary measurement. An implicit difference scheme for the forward problem is presented based on discretization of Caputo fractional derivative, and numerical stability and convergence of the l...

This paper deals with data reconstruction problem for a real disturbed
soil-column experiment using an optimal perturbation regularization algorithm. A
purpose of doing the experiment is to simulate and study transport behaviors of
Ca2+, Na+, Mg2+, K+, SO4 2−, NO3 −, HCO3 −, and Cl− when they vertically penetrating through
sandy soils. By data anal...

This paper deals with an inverse problem for determining a time-dependent reaction coefficient in one-dimensional advection-dispersion equation by the homotopy regularization algorithm. Numerical inversions are carried out by choosing homotopy parameters with two different methods respectively, and inversion results are also presented by utilizing...

This paper deals with an inverse problem of determining a source term in the one-dimensional fractional advection–dispersion equation (FADE) with a Dirichlet boundary condition on a finite domain, using final observations. On the basis of the shifted Grünwald formula, a finite difference scheme for the forward problem of the FADE is given, by means...

The presentation is mainly devoted to the research on the regularized BEM formulations for homogeneous anisotropic potential problems. Based on a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indi...

This paper deals with an inverse problem of determining a time-depen -dent reaction coefficient arising from a disturbed soil-column infiltrating experiment based on measured breakthrough data. A purpose of doing such experiment is to simulate and study transport behaviors of contaminants when they vertically penetrating through the soils. Data com...

A real undisturbed soil-column infiltrating experiment in Zibo, Shandong, China, is investigated, and a nonlinear transport model for a solute ion penetrating
through the column is put forward by using nonlinear Freundlich's adsorption isotherm.
Since Freundlich's exponent and adsorption coefficient and source/sink terms in the model cannot be mea...

Mineral components of atmospheric fine dust of Zibo City have been investigated using XRD technique qualitatively and semi-quantitatively. The results showed that the mineral assemblages of the dusts keep stable. Quartz, albite and clay minerals, which originated from geological sources, amounted to 59.1%-95.1% weight percent in the dust. Gypsum an...

We consider the numerical solution of the first kind Fredholm integral equations. Such integral equations occur in signal processing and image recovery problems among others. The common used methods to solve it are Tikhonov regularization and the iterative Tikhonov regularization methods. In this paper, we proposed an improved iterative Tikhonov re...

To begin with, a regularized boundary integral equation with indirect formulation is adopted to deal with the singular integrals and the boundary unknown quantities can be calculated accurately. When it comes to the physical quantities at the interior points, an efficient non-linear transformation is utilized to evaluate the nearly singular integra...

This paper deals with an inverse problem of identifying a nonlinear source term g=g(u)g=g(u) in the heat equation ut-uxx=a(x)g(u)ut-uxx=a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M>0M>0, with which an optimal perturbation algorithm is applied to determine the source f...

This paper deals with numerical inversions for a diffusion coefficient in 1-D transport model by an optimal perturbation regularization algorithm. Several factors affecting the algorithm' realization are discussed which are basis functions, numerical differential steps, initial iterations, and data noises. The numerical inversion results show that...

An undisturbed soil-column infiltrating experiment is investigated, and a mathematical model describing multi-components solutes transport behaviors in the column is put forward by combing hydro-chemical analysis with advection dispersion mechanisms, which is a group of advection-dispersion-reaction partial differential equations. Since the model i...

In this paper, an undisturbed soil-column experiment and its mathematical model are investigated, and an inverse problem of determining the unknown source/sink parameter and adsorption coefficient simultaneously is considered with the additional breakthrough data. By applying an optimal perturbation algorithm, jiont inversions for the source/sink t...

Based on a mathematical model for multi-components solutes transportation in porous medium, an inverse problem of determining multi-parameters is put forward, and the inverse problem is transformed to a minimization problem. Furthermore, a modified optimal iteration algorithm is applied to determine the unknown model parameters and numerical simula...

In this article, we put forward a non-linear transport model for an undisturbed soil-column experiment with the non-linear Freundlich's isotherm. As compared with ordinary models based on linear adsorption, the retardation factor is a non-linear functional of solute concentration, and the production term is a non-linear term related with the solute...

In order to study the environmental geochemical characteristics of urban soils in Qingdao, the authors carried out an extensive soil geological survey in the Shinan, Shibei, Sifang, Licang and Chengyang districts of Qingdao City. A total of 319 surface soil samples (at 0-10 cm depth) were taken with a density of 1 sample per 1 km2. The concentratio...

In this paper, a new gradient regularization algorithm is introduced and applied to solve an inverse problem of determining source terms in one-dimensional advection–dispersion equation with final observations. By functional approximations, the algorithm is reduced to find an optimal perturbation for a given source parameter involving computations...

Solute transport in porous media involving physical/chemical reactions can always be expressed by one-dimensional advection dispersion-reaction diffusion equation with nonlinear source (sink) terms, but the source (sink) coefficient which reflects solute adsorption/degeneration capabilities is often unknown. This paper deals with an inverse problem...

In this paper, we deal with an inverse problem of data reconstruction for an undisturbed soil-column infiltrating experiment. By combing hydrochemistry analysis with advection dispersal principles, a mathematical model describing multi-component solutes transport behaviors in the soil-column dominated by sulfates is put forward. By applying an opti...

This paper deals with an inverse problem of identifying a source coefficient in the process of solute transport in 1D groundwater flow. With the help of an adjoint problem, the solute concentration can be made to be monotone in time by some constraints on the known data and unknown parameters. The integral identity connecting varies of the known da...

One-dimensional equilibrium soil-column experiment models with source (sink) reaction terms are discussed in this paper. In the case of occurring high-order chemical reactions, the zero production term in traditional models should be modified to a nonlinear term related with time (or space) and solute concentration, and then a mathematical model wi...

This article deals with an inverse problem of determining source functions in an advection–dispersion equation under final observations. By using integral identity methods, a new approach which can be called data compatibility analysis methodology is presented and applied to solve the inverse source problem. By this method, the unknown is confined...

In this paper, we are concerned to cope with a conditional stability for an inverse problem of deciding source terms in a 1-D advection-dispersion equation. The inverse problem here is based on a mathematical model derived from a real case in a geological region in Shandong Province, China. With aids of an integral identity and analysis for a norma...

An inverse problem of determining a nonlinear source term in a heat equation via final observations is investigated. By applying integral identity method, data compatibilities are obtained with which the inverse source problem here is proved to be solvable. Furthermore, with aid of an integral identity that connects unknown source terms with the kn...

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator....

This paper deals with an inverse problem of determining a
nonlinear source term in a quasilinear diffusion equation with
overposed final observations. Applying integral identity methods,
data compatibilities are deduced by which the inverse source
problem here is proved to be reasonable and solvable.
Furthermore, with the aid of an integral identit...

An inverse problem of determining a nonlinear source term in a heat equation u t -u xx =f(u), with boundary-value condition u x (0,t)=F(t,u(0,t)) is investigated. At first, the problem is equally reduced to a group of integral equations; and secondly, using Sobolev compact imbedding theorem and some skills of integral estimate, the source term’s ex...

In this paper we are concerned with a quasilinear parabolic equation with homogeneous Cauchy and non-homogeneous Neumann conditions arising from combustion theory. By using the Schauder fixed point theorem and Green function of the second homogeneous boundary value problem, we give a local existence result to the solution of an inverse problem defi...

With aid of the regularizing filters, a new class of regularization methods which called generalized Tikhonov regularization for solving the first kind equations with perturbed operators and noisy data is constructed. Applying singular systems of compact operators, the convergence and optimum asymptotic order of the regularized solution is obtained...

A new class of improved regularization methods, called modified Tikhonov regularization for solving ill-posed problems of the first kind of operator equation with noisy data is constructed. By a priori choosing a regularization parameter, optimal convergence order of the regularized solution is obtained. As compared with ordinary Tikhonov regulariz...

A mollification method for numerical differentiation is constructed by introducing a new family of mollifiers. In the case of L 2 , the higher convergence order or the mollified solution is obtained.