Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation.     

On the dirichlet problem for an improperly elliptic equation

We consider the problem of solvability of an inhomogeneous Dirichlet problem for a scalar improperly elliptic differential equation with complex coefficients in a bounded domain. A model case where the unit disk is chosen as the domain and the equation does not contain lower terms is studied. We prove that the classes of Dirichlet data for which the problem has a unique solution in the Sobolev space are spaces of functions with exponentially decreasing Fourier coefficients.     

The Dirichlet problem for the Laplace equation in a starlike domain of a Riemann surface

We consider the Dirichlet problem for the Laplace equation in a starlike domain, i.e. a domain which is normal with respect to a suitable polar co-ordinates system. Such a domain can be interpreted as a non-isotropically stretched unit circle. We write down the explicit solution in terms of a Fourier series whose coefficients are determined by solving an infinite system of linear equations depending on the boundary data. Numerical experiments show that the same method works even if the considered starlike domain belongs to a two-fold Riemann surface.     

On the optimal reconstruction of a solution of the Poisson equation

We consider the problem of optimal reconstruction of a solution of the generalized Poisson equation in a bounded domain Q with homogeneous boundary conditions for the case in which the right-hand side of the equation is fuzzy. We assume that right-hand sides of the equations belong to generalized Sobolev classes and finitely many Fourier coefficients of the right-hand sides of the equations are known with some accuracy in the Euclidean metric. We find the optimal reconstruction error and construct a family of optimal reconstruction methods. The problem on the best choice of the coefficients to be measured is solved.     

Convergence of the Neumann Series in BEM for the Neumann Problem of the Stokes System

A weak solution of the Neumann problem for the Stokes system in Sobolev space is studied in a bounded Lipschitz domain with connected boundary. A solution is looked for in the form of a hydrodynamical single layer potential. It leads to an integral equation on the boundary of the domain. Necessary and sufficient conditions for the solvability of the problem are given. Moreover, it is shown that we can obtain a solution of this integral equation using the successive approximation method. Then the consequences for the direct boundary integral equation method are treated. A solution of the Neumann problem for the Stokes system is the sum of the hydrodynamical single layer potential corresponding to the boundary condition and the hydrodynamical double layer potential corresponding to the trace of the velocity part of the solution. Using boundary behavior of potentials we get an integral equation on the boundary of the domain where the trace of the velocity part of the solution is unknown. It is shown that we can obtain a solution of this integral equation using the successive approximation method.     

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.     

Controllability of an Elliptic equation and its Finite Difference Approximation by the Shape of the Domain

In this article we study a controllability problem for an elliptic partial differential equation in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution, with a given right hand side source term, into an open subdomain. The mapping that associates this trace to the shape of the domain is nonlinear. We first consider the linearized problem and show an approximate controllability property. We then address the same questions in the context of a finite difference discretization of the elliptic problem. We prove a local controllability result applying the Inverse Function Theorem together with a ``unique continuation' property of the underlying adjoint discrete system. Mathematics Subject Classification (1991):35J05, 93B03, 65M06     

On a nonclassical interpretation of the dirichlet problem for a fourth-order pseudoparabolic equation

For a fourth-order pseudoparabolic equation with nonsmooth coefficients in a rectangular domain, we consider the Dirichlet problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions and the classical boundary conditions for the case in which the solution of the problem is sought in a Sobolev space.     

On the nonuniqueness of the solution of the Tricomi problem with the generalized Frankl matching condition

We obtain an integral representation of the solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions in the elliptic part of the domain and with zero posed on one characteristic of the equation. The gradient of the solution is not continuous but satisfies some condition referred to as the “generalized Frankl matching condition.” We state theorems implying that the inhomogeneous Tricomi problem either has a unique solution or is determined modulo a solution of the homogeneous Tricomi problem.     

Solution of the Falkner–Skan equation by recursive evaluation of Taylor coefficients

We present a computational method for the solution of the third-order boundary value problem characterized by the well-known Falkner–Skan equation on a semi-infinite domain. Numerical treatments of this problem reported in the literature thus far are based on shooting and finite differences. While maintaining the simplicity of the shooting approach, the method presented in this paper uses a technique known as automatic differentiation, which is neither numerical nor symbolic. Using automatic differentiation, a Taylor series solution is constructed for the initial value problems by calculating the Taylor coefficients recursively. The effectiveness of the method is illustrated by applying it successfully to various instances of the Falkner–Skan equation.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

On a class of sectorial functional-differential operators

In a bounded domain containing the origin, we consider a partial differential equation whose leading terms contain transformations of arguments of the unknown function in the form of contractions and dilations. We study algebraic conditions under which the operator occurring in the equation satisfies the Gårding inequality. A criterion obtained earlier for constant coefficients cannot be generalized to the case of variable coefficients. We suggest a new approach to the solution of the problem in the case of variable coefficients based on the pseudodifferential operator calculus.     

A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains

A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Analytic and Experimental Studies of the Errors in Numerical Methods for the Valuation of Options

The value of a European option satisfies the Black-Scholes equation with appropriately specified final and boundary conditions.We transform the problem to an initial boundary value problem in dimensionless form.There are two parameters in the coefficients of the resulting linear parabolic partial differential equation.For a range of values of these parameters,the solution of the problem has a boundary or an initial layer.The initial function has a discontinuity in the first-order derivative,which leads to the appearance of an interior layer.We construct analytically the asymptotic solution of the equation in a finite domain.Based on the asymptotic solution we can determine the size of the artificial boundary such that the required solution in a finite domain in x and at the final time is not affected by the boundary.Also,we study computationally the behaviour in the maximum norm of the errors in numerical solutions in cases such that one of the parameters varies from finite (or pretty large) to small values,while the other parameter is fixed and takes either finite (or pretty large) or small values. Crank-Nicolson explicit and implicit schemes using centered or upwind approximations to the derivative are studied.We present numerical computations,which determine experimentally the parameter-uniform rates of convergence.We note that this rate is rather weak,due probably to mixed sources of error such as initial and boundary layers and the discontinuity in the derivative of the solution.     

Problem with displacements for the three-dimensional Bianchi equation

We consider the Bianchi equation in a rectangular 3D parallelepiped G. For this equation, we analyze the problem of finding a regular solution on the basis of three given linear relations each of which relates the values of the unknown function at 60 points lying on the faces of G and inside the domain. We obtain sufficient conditions for the unique solvability of the problem in terms of the coefficients of these relations.     

An Economical Algorithm for Numerical Solution of the Problem of Identifying the Right-Hand Side of the Poisson Equation

We propose two economical algorithms for numerical solution of the problem of identifying the right-hand side of the Poisson equation from information on the solution on the boundary of the domain. Both algorithms are based on the method of separation of variables. The method is presented on a discrete level. We use the nonuniform grids along one of the coordinates. There are possible applications for operators with variable coefficients of a special kind.     

A trace theorem for solutions of the wave equation,and the remote determination of acoustic sources

The determination of sources of acoustic wave motion in several dimensions from remote measurements is of considerable interest in many applications, and the underlying mathematical problem is quite ill-posed. We separate the source determination problem into a control problem for the wave equation and an inverse mixed initial-boundary value problem, and concentrate on the latter, in which the initial data for a solution of the wave equation are to be determined from its trace on a time-like hyperplane. Though the geometry of this problem is simple, it exhibits some of the central analytic difficulties of more complex problems. We prove a uniqueness theorem, give examples of instability, establish regularity properties of the trace, and locate noncompact classes of stable functionals. The existence of these noncompact classes shows that the problem is “partially well-posed”, i.e. that smoothing in all directions is not required to regularize the problem, and distinguishes it from most other ill-posed problems, such as backwards diffusion and analytic continuation.     

On the determination of the coefficients in the viscoelasticity equations

For the integrodifferential viscoelasticity equations, we study the problem of determining the coefficients of the equations and the kernels occurring in the integral terms of the system of equations. The density of the medium is assumed to be given. We suppose that the inhomogeneity support of the sought functions is included in some compact domain B ₀. We consider a series of inverse problems in which an impulse source is concentrated at the points y of the boundary of B ₀. The point y is the parameter of the problem. The given information about the solution is the trace of the solution to the Cauchy problem with zero initial data. This trace is given on the boundary of B ₀ for all y ∈ ?B ₀ and for a finite time interval. The main result of the article consists in obtaining uniqueness theorems for a solution to the initial inverse problem.     

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     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.