Two regularization methods for a spherically symmetric inverse heat conduction problem

In this paper, we consider a spherically symmetric inverse heat conduction problem of determining the internal surface temperature of a hollow sphere from the measured data at a fixed location inside it. This is an ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Tikhonov type’s regularization method and a Fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates.     

Solutions to the seepage face model for dual porosity flows with hysteresis

This paper is concerned with theoretical analysis of the initial-boundary value problem for the dual porosity model of Darcian flows with the Preisach hysteresis operator. Due to the presence of the so-called unilateral boundary condition, the problem comes as an evolution variational inequality. We apply the penalty method to prove the global-in-time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates.     

A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates

We consider the problem of determining the temperature u(x,t)$u(x,t)$, for (x,t)∈[0,π]×[0,T)$(x,t)\in [0,\pi ]\times [0,T)$ in the parabolic equation with a time-dependent coefficient. This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. In this paper, we use a modified method for regularizing the problem and derive an optimal stability estimation. A numerical experiment is presented for illustrating the estimate.     

Nonlinear second order evolution inclusions with noncoercive viscosity term

In this paper we deal with a second order nonlinear evolution inclusion, with a nonmonotone, noncoercive viscosity term. Using a parabolic regularization (approximation) of the problem and a priori bounds that permit passing to the limit, we prove that the problem has a solution.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.     

Self-similar solutions for the LSW model with encounters

The LSW model with encounters has been suggested by Lifshitz and Slyozov as a regularization of their classical mean-field model for domain coarsening to obtain universal self-similar long-time behavior. We rigorously establish that an exponentially decaying self-similar solution to this model exist, and show that this solutions is isolated in a certain function space. Our proof relies on setting up a suitable fixed-point problem in an appropriate function space and careful asymptotic estimates of the solution to a corresponding homogeneous problem.     

Elliptic regularization for the semi-linear telegraph system

The solution of the semi-linear telegraph system is compared with the solution of an elliptic regularization, to which one associates two-point boundary conditions. An asymptotic approximation for the solution of the elliptic regularization is constructed. The method employed here is the boundary function method due to Vishik and Lyusternik. The problem is singularly perturbed of elliptic-hyperbolic type. To conduct this analysis, high regularity with respect to t for the solutions of both problems is required. Finally, the order of this approximation is found in different spaces of functions.     

Partial regularity of weak solutions for the curve shortening flow

We study the regularity of certain weak solutions for the curve shortening flow in arbitrary codimension. These solutions arise as limits of a regularization process which is related to an approach suggested by Calabi. We prove that the set of times for which such a weak solution is not smooth has Hausdorff dimension at most ?. Received: 23 May 1998 / Revised version: 7 September 1998     

Existence and uniqueness of a solution for a parabolic quasilinear problem for linear growth functionals with $L^1$ data

We introduce a new concept of solution for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. Using Kruzhkov's method of doubling variables both in space and time we prove uniqueness and a comparison principle in for these solutions. To prove the existence we use the nonlinear semigroup theory. Received: 26 October 2000 / Revised version: 1 May 2001 / Published online: 24 September 2001     

Large time behavior of free boundary for a degenerate parabolic equation with absorption

This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously.     

Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation

We consider the inverse time problem for the nonlinear heat equation in the form

     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate nonlinear parabolic equation of the form , where denotes the so-called infinity-Laplacian given by . To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results on the Cauchy–Dirichlet problem. Dedicated to the memory of our friend Kyoji Takaichi. The research of the first author was partially supported by Waseda University Grant for Special Research Projects, #2004A-366.     

A nonlinear parabolic equation backward in time: Regularization with new error estimates

Consider a nonlinear backward parabolic problem in the form

     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

A uniqueness result on the solution for a Landau–Lifshitz system with penalization

The author proves the uniqueness of the solution to an evolution Landau–Lifshitz type problem, when the parameter tends to zero. In addition, the unique solution is just a heat flow of a harmonic map. This uniqueness result is derived by establishing a uniform estimate for the solution.     

Determination of the insulated inclusion in conductivity problem and related Eshelby conjecture

In the study of composites, it is important to determine the shape of inclusions. There is an interesting case in conductivity problem that the inclusion is insulated. In present paper, we first obtain the representation formula of the solution to an exterior problem, and then prove that for any uniform loading such solution can be extended into the inclusion as an affine function if and only if the insulated inclusion is an ellipse or an ellipsoid. We also show that an analogous result holds for the elasticity problem, which is related to Eshelby conjecture. The main results in this paper are motivated by Ammari, Kang, Kim and Lee (2013), Ammari, Kang and Lim (2005), Kang and Milton (2008), and Liu (2008).     

An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options

We consider the approximation of the optimal stopping problem associated with ultradiffusion processes in the context of mathematical finance and the valuation of Asian options. In particular, the value function is characterized as the solution of an ultraparabolic variational inequality. Employing the penalty method and a regularization of the state space, we develop higher-order adaptive approximation schemes which utilize the extrapolation discontinuous Galerkin method in temporal space. Numerical examples are provided in order to demonstrate the approach.     

Homotopy perturbation-reproducing kernel method for nonlinear systems of second order boundary value problems

In this paper, based on homotopy perturbation method (HPM) and reproducing kernel method (RKM), a new method is presented for solving nonlinear systems of second order boundary value problems (BVPs). HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKM is also an analytical technique, which can solve powerfully linear BVPs. Homotopy perturbation-reproducing kernel method (HP-RKM) combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear BVPs. Three numerical examples are presented to illustrate the strength of the method.     

On a backward Cauchy problem associated with continuous spectrum operator

The nonlinear backward Cauchy problem

u_t+Au(t)=f(u(t)),u(T)=φ,