Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions北大核心CSCD

对Robin边界条件时间分数阶扩散方程的源项辨识问题进行了研究。这类问题是不适定的,因此提出了一种迭代型正则化方法,得到了源项辨识问题的正则近似解。给出了先验和后验参数选取规则下正则近似解和精确解之间的误差估计,数值算例验证了该方法的有效性。     

Analytical solutions for a time-fractional axisymmetric diffusion–wave equation with a source term

In this paper, a time-fractional axisymmetric diffusion–wave equation with a source term is considered in cylindrical coordinates. The analytical solution is obtained with the help of an integral transform method and some properties of special functions. In addition, we discuss two kinds of different boundary conditions and different forms of the source term. Finally, we analyze the effects of the fractional derivative on the solutions by using numerical results and find that sub-diffusion phenomena and oscillations exist.     

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.

     

Energy decay of variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term

In this paper, we consider a variable‐coefficient wave equation with nonlinear acoustic boundary conditions and source term. Using the Riemannian geometry method, we prove the general energy decay of the system corresponds to the ordinary differential equation (ODE), which certainly is stable under some suitable assumptions.     

Semi-analytical method for solving nonlinear heat diffusion problems in spherical medium

A semi-analytical methodology, based on the finite integral transform technique, is proposed to solve the heat diffusion problem in a spherical medium subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The method proceeds by treating the nonlinearity term in the boundary condition as a source in the differential equation and keeping other conditions unchanged. The results obtained from this semi-analytical solutions are compared with those obtained from a numerical solution developed using an explicit finite difference method, which showed very good agreement.     

SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY APPROXIMATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS

This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem.The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media,where some additional boundary measurements are required.An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost,where a regularization term is employed to eliminate the oscillations of the noisy data.Moreover,an efficient algorithm is presented and tested for some numerical examples.     

Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions.     

Singular limiting solutions for 2-dimensional semilinear elliptic system of Liouville type adding singular sources terms given by Dirac masses

We study the existence of singular limit solutions for a nonlinear elliptic system of Liouville type with a singular source term given by Dirac masses and Dirichlet boundary conditions. We use the nonlinear domain decomposition method.     

Symmetry reductions and solutions for pollutant diffusion in a cylindrical system

This study is devoted to investigating transient coupled fluid flow and mass transfer partial differential equations (PDEs) describing pollutant transport in cylindrical coordinates. Symmetry analysis of the system of coupled PDEs is performed and some large Lie algebras are obtained for some special cases of the arbitrary and special choices of constants, and the source term. Optimal systems are constructed for all the admitted symmetries. We perform reductions for different choices of the source term. In some cases invariant solution is sought, however some cases resulted in coupled systems of highly nonlinear ordinary differential equations (ODEs). Imposing realistic boundary conditions and considering a constant source term, we then use the Adomain decomposition techniques to solve the boundary value problem.     

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

Asymptotic behavior of a generalized Cahn–Hilliard equation with a mass source

We consider in this article a generalized Cahn–Hilliard equation with mass source (nonlinear reaction term) which has applications in biology. We are interested in the well-posedness and the study of the asymptotic behavior of the solutions (and, more precisely, the existence of finite-dimensional attractors). We first consider the usual Dirichlet boundary conditions and then Neumann boundary conditions. The latter require additional assumptions on the mass source term to obtain the dissipativity. Indeed, otherwise, the order parameter u can blow up in finite time. We also give numerical simulations which confirm the theoretical results.     

Global solution and blow-up solution for a nonlinear damped beam with source term

A nonlinear damped system with boundary input and output, which also has source term, is studied in this paper. It is proved that under some conditions the system has global solution and blow-up solution.     

On the wave equation with semilinear porous acoustic boundary conditions

The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

A coupled nonlinear hyperbolic-parabolic system

A boundary initial value problem for a quasi-linear hyperbolic system in one space variable is coupled to a boundary initial value problem for a quasi-linear parabolic equation in two space variables. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the parabolic equation. Such a coupling can arise in the consideration of gas flowing in a porous medium and out of that medium via a pipe. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, Bernstein's estimates for parabolic partial differential equations, and the contracting mapping theorem.     

The boundaries of the solutions of the linear Volterra integral equations with convolution kernel

Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002.

In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used.

     

Blow-up rate for a nonlinear diffusion equation

In this work we study the blow-up rate for a nonlinear diffusion equation with an inner source and a nonlinear boundary flux, which is equivalent to a porous medium equation with convection. Depending upon the sign of a parameter included, the source can be positive or negative (absorption). By the scaling method, we obtain that the blow-up rate is independent of a negative source, while for the situation with a positive source, the blow-up rate is determined by the interaction between the inner source and the boundary flux. Comparing with the previous results for the porous medium model without convection, we observe that the gradient term included here does not affect the blow-up rates of solutions.     

Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time-fractional diffusion equation from boundary measured data. The uniqueness for the inverse source problem is proved by the Laplace transformation method.     

GENERAL DECAY FOR A VISCOELASTIC EQUATION OF VARIABLE COEFFICIENTS WITH A TIME-VARYING DELAY IN THE BOUNDARY FEEDBACK AND ACOUSTIC BOUNDARY CONDITIONS

A variable coefficient viscoelastic equation with a time-varying delay in the boundary feedback and acoustic boundary conditions and nonlinear source term is considered.Under suitable assumptions, general decay results of the energy are established via suitable Lyapunov functionals and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and the elements of the matrix A and the kernel function g.     

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.