On Dirichlet-type problems for the Lavrent’ev-Bitsadze equation

The existence and uniqueness issues are discussed for several boundary value problems with Dirichlet data for the Lavrent’ev-Bitsadze equation in a mixed domain. A general mixed problem (according to Bitsadze’s terminology) is considered in which the Dirichlet data are relaxed on a hyperbolic region of the boundary inside a characteristic sector with vertex on the type-change interval. In particular, conditions are pointed out under which the problem is uniquely solvable for any choice of this vertex.     

On the solvability of the tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions

We study the existence of a regular (classical) solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions. We find conditions under which the homogeneous problem has only the zero solution and give an example in which the homogeneous Tricomi problem has a nonzero solution. We also study the solvability of the inhomogeneous Tricomi problem.     

On a nonlocal problem of A. A. Dezin for the Lavrent’ev-Bitsadze equation

In a special rectangular domain, for a second-order linear equation of mixed type with discontinuous coefficients and with the Lavrent’ev-Bitsadze operator in the leading part, we prove an extremum principle and existence and uniqueness theorems for the solution of a nonlocal problem stated by A.A. Dezin in his report at the Joint Soviet-American Symposium on Partial Differential Equations (Novosibirsk, 1963).     

Properties of solutions of the Tricomi problem for the Lavrent’ev-Bitsadze equation at corner points

We consider the Tricomi problem for the Lavrent’ev-Bitsadze equation for the case in which the elliptic part of the boundary is part of a circle. For the homogeneous equation, we introduce a new class of solutions that are not continuous at the corner points of the domain and construct nontrivial solutions in this class in closed form. For the inhomogeneous equation, we introduce the notion of an n-regular solution and prove a criterion for the existence of such a solution.     

The boundary-value problem for the Lavrent’ev-Bitsadze equation with unknown right-hand side

We study the inverse problem for the Lavrent’ev-Bitsadze equation in a rectangular domain. We construct its solution as a series of eigenfunctions for the corresponding problem on eigenvalues and establish a criterion for its uniqueness. We also prove the stability of the obtained solution.     

Eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation

We determine eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation.     

Nonlocal problem for the Lavrent’ev-Bitsadze equation and its analogs in the theory of equations of mixed parabolic-hyperbolic type

We study a nonlocal interior-boundary value problem with an Erdelyi-Kober operator for the Lavrent’ev-Bitsadze equation and its analogs in the theory of equations of mixed parabolic-hyperbolic type.     

Inverse problem for equations of mixed type with Lavrent’ev-Bitsadze operator

For the equation of mixed elliptic-hyperbolic type $u_{xx} + (\operatorname{sgn} y)u_{yy} - b^2 u = f(x)$ in a rectangular domainD = {(x, y) | 0 < x < 1, ?α < y < β}, where α, β, and b are given positive numbers, we study the problem with boundary conditions $\begin{gathered} u(0,y) = u(1,y) = 0, - \alpha \leqslant y \leqslant \beta , \hfill \\ u(x,\beta ) = \phi (x),u(x,\alpha ) = \psi (x),u_y (x, - \alpha ) = g(x),0 \leqslant x \leqslant 1. \hfill \\ \end{gathered} $ . We establish a criterion for the uniqueness of the solution, which is constructed as the sum of the series in eigenfunctions of the corresponding eigenvalue problem and prove the stability of the solution.     

Correctness of the Dirichlet problem for the Lavrent’ev–Bitsadze equation

It is shown that the Dirichlet problem in a multidimensional domain for the Lavrent’ev–Bitsadze equation is uniquely solvable. A criterion of the uniqueness of the solution is obtained.     

A criterion for the unique solvability of the dirichlet spectral problem in a cylindrical domain for a multidimensional Lavrent’ev-Bitsadze equation

We obtain a criterion for the unique solvability of the spectral problem in a cylindrical domain for a multidimensional Lavrent’ev-Bitsadze equation.     

Inverse problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator and with a nonlocal boundary condition

For an equation of the mixed elliptic-hyperbolic type, we study the inverse problem with a nonlocal condition relating the derivatives of the solution on the elliptic and hyperbolic parts of the boundary. We prove a uniqueness criterion and construct the solution in the form of a Fourier series.     

On the convergence of the Lavrent’ev method for an integral equation of the first kind with involution

The convergence in the mean-square metric of the Lavrent’ev regularizationmethod for an integral equation with involution is established. The proof of the convergence is based on studying the behavior of the resolvent of a certain integro-differential equation related to the original equation.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

Discretized Lavrent’ev regularization for the autoconvolution equation

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.     

A problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation

In this paper we study a problem with conditions analogous to Frankl’ and Bitsadze-Samarskii ones for the Gellerstedt equation. We prove that the stated problem is well-posed.     

Tricomi problem for a functional-differential advanced-retarded Lavrent’ev–Bitsadze equation

We study a boundary value problem for the Lavrent’ev–Bitsadze equation with functional delay and advance. The general solution is constructed. The problem is uniquely solvable.     

On the uniqueness of the solution of dual equation of a singular Sturm–Liouville problem

In this paper we consider differential systems having a singularity and one turning point. First, by a replacement, we transform the system to a linear second-order equation of Sturm–Liouville type with a singularity. Using the infinite product representation of solutions provided in [8], we obtain the dual equation, then we investigate the uniqueness of the solution for the dual equation of the inverse spectral problem of Sturm–Liouville equation. This result is necessary for expressing inverse problem of indefinite Sturm–Liouville equation.     

Asymptotics at t→∞ of the solution of the Cauchy problem for the Landau-Lifshitz equation

Conclusions The formulas obtained in the present paper for the leading term in the asymptotic behavior of the solution of the Cauchy problem for the LL equation subject to the boundary conditions L₃1, x± describe the solitonless sector. The transition to the general case, which takes into account the presence in the solution of soliton formations, can be made on the basis solely of algebraic considerations that use the procedure of soliton dressing developed in [17, 18] for the LL equation. In particular, applying to the obtained asymptotic formulas the procedure for a dressing of domain wall type (see [17]), we arrive at formulas that describe the asymptotic solution of the Cauchy problem for the LL equation with boundary conditions of the form L₃±1, x±.Leningrad State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 1, pp. 3–17, July, 1988.     

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplaces equation in R² is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poissons equation, which the derivatives of particular solutions are estimated under the L² norm. The convergent order of solving the Dirichlet problem of Poissons equation by the method of particular solution and method of fundamental solution is derived. Dedicated to Charles A. Micchelli with esteem on the occasion of his 60th birthdayAMS subject classification 35J05, 31A99