An asymptotic solution of the Cauchy problem for the Davey-Stewartson-I equation

We construct and asymptotic solution of the Cauchy problem for the Davey-Stewartson-I equation as t→∞. The solution, which is of order t⁻¹, is rapidly oscillating. The envelope of the oscillations can be defined by an integral equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 1, pp. 104–114, January, 1998.     

Regularization of the Cauchy problem for the Laplace equation

We suggest a method for regularizing the solution of the Cauchy problem for the Laplace equation by introducing the biharmonic operator with a small parameter. We show that if there exists a solution of the original problem, then the difference between the spectral expansions of solutions of the original and regularized equations tends to zero in the space of square integrable functions as the regularization parameter tends to zero. If the original solution belongs to a Sobolev class, then we use results of Il’in’s spectral theory to derive an estimate for the rate of the convergence of the regularized solution to the exact solution.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Wavelets and regularization of the Cauchy problem for the Laplace equation

In this paper, a Cauchy problem for two-dimensional Laplace equation in the strip 0<x?1 is considered again. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data, a small perturbation in the data can cause a dramatically large error in the solution for 0<x?1. The stability of the solution is restored by using a wavelet regularization method. Moreover, some sharp stable estimates between the exact solution and its approximation in Hr(R)-norm is also provided.     

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.     

Criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation

In the present paper, we prove a necessary and sufficient condition for the well-posedness of the problem indicated in the title in the space L ₂(Ω). To this end, we use expansions in the eigenfunctions of the mixed Cauchy problem for the Laplace equation with a deviating argument.     

Stable numerical solution of the Cauchy problem for the Laplace equation in irregular annular regions

This article is mainly concerned with the numerical study of the Cauchy problem for the Laplace equation in a bounded annular region. To solve this ill‐posed problem, we follow a variational approach based on its reformulation as a boundary control problem, for which the cost function incorporates a penalized term with the input data. The cost function is minimized by a conjugate gradient method in combination with a finite element discretization. In the case where the input data is noisy, some preliminary error estimates, show that the penalization parameter may be chosen like the inverse of the level of noise. Numerical solutions in simple and complex domains show that this methodology produces stable and accurate solutions.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1799–1822, 2017     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

Stability of the solution of the Cauchy problem for the Laplace equation on a weak compactum

The paper investigates the stability of the Cauchy problem for the Laplace equation under the a priori assumption that the solution is bounded. A special metrization of the weak topology in the space L₂ and the standard Fourier series technique are applied to obtain stability bounds for the solution of the Cauchy problem on the class of absolutely bounded functions.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 44–50, 1985.     

A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].     

An iterative method for a Cauchy problem for the heat equation

^** Email: tomjo{at}itn.liu.se An iterative method for reconstruction of the solution to aparabolic initial boundary value problem of second order fromCauchy data is presented. The data are given on a part of theboundary. At each iteration step, a series of well-posed mixedboundary value problems are solved for the parabolic operatorand its adjoint. The convergence proof of this method in a weightedL²-space is included.     

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)     

Cauchy problem for the generalized Kadomtsev-Petviashvili equation

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 160–172, January–February, 1992.     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

The a posteriori Fourier method for solving the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data

In the present paper, the Cauchy problem for the Laplace equation with nonhomogeneous Neumann data in an infinite “strip” domain is considered. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A conditional stability result is given. A new a posteriori Fourier method for solving this problem is proposed. The corresponding error estimate between the exact solution and its regularization approximate solution is also proved. Numerical examples show the effectiveness of the method and the comparison of numerical effect between the a posteriori and the a priori Fourier method are also taken into account.     

The Cauchy problem for the Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces ${H^s(\mathfrak{R})}$ with ${s < \frac{3}{2}}$ in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in ${H^{s}(\mathfrak{R})}$ with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in ${H^{s}(\mathfrak{R})}$ with s > 3/2.