Noncharacteristic Cauchy Problem Research Articles

A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions

A variational regularization method for solving the non-characteristic Cauchy problem in multiple dimensions

Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

A modified regularization method for a Cauchy problem for heat equation on a two-layer sphere domain

In this paper, we study a non-characteristic Cauchy problem for a radially symmetric inverse heat conduction equation in a two-layer domain. This is a severely ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. It is well-known that the classical Tikhonov regularization solutions are too smooth and the approximate solutions may lack details that might be contained in the exact solutions. Combining Fourier transform technique with a modified version of the classical Tikhonov regularization, we obtain a regularized solution which is stably convergent to the exact solution with a sharp error estimate.

Determining surface heat flux for noncharacteristic Cauchy problem for Laplace equation

In this paper, the noncharacteristic Cauchy problem for the Laplace equation {wxx+wyy=0x∈(0,1),y∈R,w(0,y)=g(y)y∈R,wx(0,y)=h(y)y∈R, is investigated, where the Cauchy data is given at x=0 and the heat flux is sought in the interval 0<x≤1. This problem is severely ill-posed: the solution (if it exists) does not depend continuously on the given data. A modified regularization method is used to solve this problem. Furthermore, some error estimates for the heat flux between the regularization solution and the exact solution are given. Finally, a numerical example shows that the proposed method works well.

Stability estimate and the modified regularization method for a Cauchy problem of the fractional diffusion equation

In this paper we investigate a non-characteristic Cauchy problem for a fractional diffusion equation. Using the Fourier transformation technique, we give a conditional stability estimate on the solution. Since the problem is highly ill-posed in the Hadamard sense, a modified version of the Tikhonov regularization technique is devised for stable numerical reconstruction of the solution. An error bound with optimal order is proven. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.

Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method

The sideways parabolic equation (SPE) is a model of the problem of determining the temperature on the surface of a body from the interior measurements. Mathematically it can be formulated as a noncharacteristic Cauchy problem for a parabolic partial differential equation. This problem is severely ill-posed in an $L_2$ setting. We use a preconditioned generalized minimum residual method (GMRES) to solve a two-dimensional SPE with variable coefficients. The preconditioner is singular and chosen in a way that allows efficient implementation using the FFT. The preconditioner is a stabilized solver for a nearby problem with constant coefficients, and it reduces the number of iterations in the GMRES algorithm significantly. Numerical experiments are performed that demonstrate the performance of the proposed method.

Regularization error analysis on a one-dimensional inverse heat conduction problem in multilayer domain

We investigate in this paper a non-characteristic Cauchy problem for a one-dimensional heat conduction equation in a multilayer domain, which is a well-known ill-posed problem. We first formulate the Cauchy problem in the frequency domain via Fourier transform technique. A modified version of the classical Tikhonov regularization technique together with the corresponding error estimates is presented. Several numerical examples are constructed to demonstrate the superior efficiency and accuracy of the proposed approach. Numerical results show that the modified Tikhonov regularization technique performs better than the classical Tikhonov regularization technique.

Monge-Ampère Equations on (Para-)Kähler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampere equations (MAEs) and discuss a natural class of MAEs arising in Kahler and para-Kahler geometry whose solutions are special Lagrangian submanifolds.

Existence of Classical Solutions near Characteristic Points of First Order Nonlinear Partial Differential Equations

We treat a nonlinear partial differential equation F(x,u,ux) = 0 in a neighborhood of x = 0 ∈ Rd, where F(x,u,p) is a real-valued smooth function. It is well-known that solutions are constructed by solving noncharacteristic Cauchy problem with the method of characteristics, provided Fpi(0,0,0) ≠ 0 for some i. In this paper we study the existence of a classical solution under the condition that Fpi(0,0,0) = 0 for all 1 ≤ i ≤ d.

On the problem of control the composition of material in the binary alloy solidification process

Abstract In a framework of classical one-dimensional binary alloy solidification problem with zero diffusion coefficient in the solid phase the inverse problem is posed and considered. The problem consists in finding of the temperature regime at the outer boundary of domain when the final distribution of admixture in solidified part is known. In the cases when the desired concentration is linear function of space coordinate the boundary-value problem for the system of equations can be disjoin and the inverse problem can be reduced to the successive solving of three problems. First one is a “supercooled” Stefan problem [Fasano, Primicherio, Quart. Appl. Math. 38: 439–460, 1981,Petrova, Tarzia, Turner, Advances in Math. Sciences and Applications, Gakkotosho 4: 35–50, 1994], the second one is an initial-boundary problem for heat equation in the domain with known moving boundary, and the last one is a non-characteristic Cauchy problem [Alifanov, Inverse Problems of Heat Exchange (in Russian), Nauka, 1988]. We investigate this last problem in two aspects – classical “exact” and extremal, as a problem of minimizing a functional. Besides, we consider the self-similar version of the inverse problem and construct the exact solution to the inverse problem.

On some local property of the shape of interfaces for the porous media equation

We consider the porous media equation with absorption for various conditions and prove that the shape of ist interface never becomes strongly upward convex. For this sake we derive an improperly posed estimate for solutions of the porous media equation for the non-characteristic Cauchy problem (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Singularity propagation of completely nonlinear equations in a complex domain

We study completely nonlinear and noncharacteristic Cauchy problems of order two for singular initial values in a complex domain. We shall prove that if the characteristic roots are distinctive, then the singularities of the solution propagate toward characteristic directions. For this purpose, we generalize the classical theory of hodograph transformation. We also mention applications to the theory of perfect irrotational fluids and Monge–Ampère equations.

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving − 2 π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε > 0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε . This sequence of exact solutions is analogous to that of the well-known N -soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε ↓ 0 one observes the appearance of nonlinear caustics, i.e. curves in space–time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases. In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L 1 -Sobolev initial data, and Appendix B establishes the well-posedness for L p -Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).

An improperly posed estimate for the equation |u|α ut =uxx +F(u)(0

The noncharacteristic Cauchy problem of heat equations is not well-posed. But the estimation on the continuous dependence of solutions holds under their prescribed bound and the bound of their Cauchy data. In this paper we show that a similar estimation holds also for some degenerate quasilinear parabolic equation.

Unique Continuation for fast diffusion

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation $|u|^{\alpha}u_{t}=\triangle u$, where we assume that $1/2 \lt \alpha \lt 1$. This equation is based on the fast diffusion model. And we prove the unique continuation property for the above problem.

On some Improperly Posed Problem for a Degenerate Nonlinear Parabolic Equation

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation$|u|^{\\alpha}u_t – \\Delta u – \\gamma |u|^{–\\beta} u = 0$ under some assumptions on$\\alpha, \\beta$ and $\\gamma$. The problem is improperly posed in the sense of Hadamard. We derive for such solutions an estimate in terms of the Cauchy data and a prescribed bound of the solution.

Stability results for solutions of a linear parabolic noncharacteristic Cauchy problem

Abstract - We study a noncharacteristic Cauchy problem for a linear parabolic equation in one space variable. This problem is ill-posed. We prove that solutions that are bounded in the supremum norm depend continuously upon Cauchy data and the same is true for regular level lines of such solutions.

On the Local Structure of the Klein–Gordon Field on Curved Spacetimes

This paper investigates wave equations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the non-characteristic Cauchy problem to show that a solution to a wave equation vanishing in an open set vanishes in the 'envelope' of this set, which may be considerably larger and in the case of timelike tubes may even coincide with the spacetime itself. We apply this result to the real scalar field on a globally hyperbolic spacetime and show that the field algebra of an open set and its envelope coincide. As an example, there holds an analog of Borchers' timelike tube theorem for such scalar fields and, hence, algebras associated with world lines can be explicitly given. Our result applies to cosmologically relevant spacetimes.

Regularization of a non-characteristic Cauchy problem for a parabolic equation in multiple dimensions

In this paper we consider the non-characteristic Cauchy problem where with appropriate coefficient functions a, b and c. Assuming that the Cauchy data are given inexactly by a function satisfying ||-||Hr for some r0 and that f(y,t): = u(l,y,t) exists and belongs to Hs(n-1 × ) for some s, it is desired to calculate f from the improper data . This problem is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. In this paper the following mollification method is suggested for this problem: if the Cauchy data are given inexactly then we mollify them by projection on elements of Meyers multiresolution approximation {Vj}j. Within every space Vj the solution of the above problem depends continuously on the data, and we can find a mollification parameter J depending on the noise level in the Cauchy data such that the error estimation between the exact solution and the mollified solution is of Hölder type. At the end of this paper numerical examples for our method will be given.

A sequential conjugate gradient method for the stable numerical solution to inverse heat conduction problems

In this paper we propose a new numerical method for solving noncharacteristic Cauchy problems (NCCPs) for parabolic equations, namely a time-stepping sequential variant of the conjugate gradient method (CGM). In contrast to other related papers, we do not assume that the initial condition is given. Moreover, we allow that the coefficients in the parabolic equations are nonsmooth and may depend on time. The purpose of this paper is to present the method and to show a few numerical results. A detailed analysis of the method will be carried out in a forthcoming paper. The CGM itself applied to linear ill-posed problems has optimal regularization properties as it is shown by Nemirovskii [23]. Our sequential variant of the CGM proceeds similarly to the well-known Beck method (see [3]). It computes the solution of the associated direct problems only in a short time interval of – say – r time steps but takes, as a result, only the solution at the next (discrete) time. The latter is then used as the starting valu...