Regularization of the solution of the cauchy problem for the generalized Moisil-Theodoresco system

We consider the problem of analytic continuation of a solution of the generalized Moisil-Theodoresco system in a spatial domain on the basis of its values on part of the boundary of this domain, that is, the Cauchy problem. We construct an approximate solution of this problem on the basis of the Carleman matrix method.     

On a numerical solution of the Cauchy problem for the Laplace equation by the fundamental solutions method

We discuss a numerical method to solve a Cauchy problem for the Laplace equation in the two-dimensional annular domain. We consider the case that the Cauchy data is given on an arc. We develop an approximation method based of the fundamental solutions method using the least squares method with Tikhonov regularization. The effectiveness of our method is examined by a numerical experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.     

On the uniqueness of the solution of nonlinear differential-algebraic systems

We consider the Cauchy problem for a system of nonlinear ordinary differential equations unsolved for the derivative of the unknown vector function and identically degenerate in the domain. We prove a theorem on the coincidence of two smooth solutions of the considered problem. We show that, under some additional assumptions, the above-mentioned problem cannot have classical solutions with less smoothness. We obtain conditions under which the problem has a fixed finite number of solutions.     

First mixed problem for a nonstrictly hyperbolic equation of the third order in a bounded domain

We study the classical solution of a boundary value problem for a nonstrictly parabolic equation of the third order in a rectangular domain of two independent variables. We pose Cauchy conditions on the lower base of the domain and the Dirichlet conditions on the lateral boundary. By the method of characteristics, we obtain a closed-form analytic expression for the solution of the problem. The uniqueness of the solution is proved.     

Reconstruction of solutions to a generalized Moisil-Theodorescu system in a spatial domain from their values on a part of the boundary

In this paper we consider the problem of reconstructing solutions to a generalized Moisil-Theodorescu system in a spatial domain from their values on a part of the domain boundary, i.e., the Cauchy problem. We construct an approximate solution to this problem with the help of the Carleman matrix method.     

Carleman inequalities for the heat equation in singular domains

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we show a result of null controllability using Carleman estimates.     

Missing boundary data reconstruction by the factorization method

We consider the data completion problem for the Laplace equation in a cylindrical domain. The Neumann and Dirichlet boundary conditions are given on one face of the cylinder while there is no condition on the other face. This Cauchy problem has been known since Hadamard (1953) to be ill-posed. Here it is set as an optimal control problem with a regularized cost function. We use the factorization method for elliptic boundary value problems. For each set of Cauchy data, to obtain the estimate of the missing data one has to solve a parabolic Cauchy problem in the cylinder and a linear equation. The operator appearing in these problems satisfy a Riccati equation that does not depend on the data. To cite this article: A. Ben Abda et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Dirichlet Problem for the Stokes Flow Function in a Simply-Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for the Stokes flow function in a simply-connected domain of the meridian plane to the Cauchy singular integral equation. For the case where the boundary of the domain is smooth and satisfies certain additional conditions, the regularization of the indicated singular integral equation is carried out.     

On The Trace Problem For Solutions Of The Vlasov Equation

We study tlie trace problem for weak solutions of the Vlasov equation set in a domain. When the force field has Sobolev regularity, we prove the existence of a trace on the boundaries, which is defined thanks to a Green formula, and we show that the trace can be renormalized. We apply these results to prove existence and uniqueness of tlie Cauchy problem for the Vlasov equation witli specular reflection at the boundary. We also give optimal trace theorems and solve the Cauchy problem witli general Dirichlet conditions at the boundary     

Dirichlet Problem for an Axisymmetric Potential in a Simply Connected Domain of the Meridian Plane

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.     

Application of the fundamental principle to complex Cauchy problem

In this paper we give an explicit formula for the solution of the non-homogeneous complex Cauchy problem with Cauchy data given on a bounded smooth strictly convex domain in a non-characteristic hyperplane. These formulas are obtained using the explicit version of the fundamental principle given in terms of residue currents; moreover, we characterize the domain of definition of the solution and we generalize these techniques to the non-homogeneous Goursat problem.     

Cauchy Problem for Elliptic Systems in the Space ℝ m

In this paper, we investigate the analytic continuation of the solution of the system of Lamé equations in a bounded space domain from the values of the solution and the stress values on part of the boundary of this domain, i.e., a Cauchy problem is studied. We construct an approximate solution of this problem based on the Carleman matrix method.     

A Carleman Function and the Cauchy Problem for the Laplace Equation

We suggest an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit continuation formula and a regularization procedure for a solution to the Cauchy problem for the Laplace equation.     

On Harmonic Continuation of Differentiable Functions Defined on Part of the Boundary

We establish an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit solution to the Cauchy problem for the Laplace equation.     

On Harmonic Continuation of Differentiable Functions Defined on Part of the Boundary

We establish an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit solution to the Cauchy problem for the Laplace equation.     

Trefftz energy method for solving the Cauchy problem of the Laplace equation

The inverse Cauchy problem of Laplace equation is hard to solve numerically, since it is highly ill-posed in the Hadamard sense. With this in mind, we propose a natural regularization technique to overcome the difficulty. In the linear space of the Trefftz bases for solving the Laplace equation, we introduce a novel concept to construct the Trefftz energy bases used in the numerical solution for the inverse Cauchy problem of the Laplace equation in arbitrary star plane domain. The Trefftz energy bases not only satisfy the Laplace equation but also preserve the energy, whose performance is better than the original Trefftz bases. We test the new method by two numerical examples.     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.