Numerical method for determining the inhomogeneity boundary in the Dirichlet problem for Laplace’s equation in a piecewise homogeneous medium

The Dirichlet problem for Laplace’s equation in a two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary from additional information on the solution of the Dirichlet problem is considered. A numerical method based on the linearization of the nonlinear operator equation for the unknown boundary is proposed for solving the inverse problem. The results of numerical experiments are presented.     

Solution of boundary value problems for Laplace’s equation in a piecewise homogeneous plane with a parabolic crack (screen)

Boundary value problems for Laplace’s equation are considered in a piecewise homogeneous plane divided into two zones by a strongly permeable crack or a weakly permeable screen in the form of a parabola. The desired potentials have prescribed singular points (sources, sinks, etc.). Formulas are derived expressing the potentials in terms of harmonic functions that have the given singular points and describe similar processes in a homogeneous plane.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

Boundary integral operators and boundary value problems for Laplace’s equation

In this paper, we define boundary single and double layer potentials for Laplace’s equation in certain bounded domains with d-Ahlfors regular boundary, considerably more general than Lipschitz domains. We show that these layer potentials are invertible as mappings between certain Besov spaces and thus obtain layer potential solutions to the regularity, Neumann, and Dirichlet problems with boundary data in these spaces.     

A harmonic polynomial method with a regularization strategy for the boundary value problems of Laplace’s equation

This paper concerns a boundary value problem of Laplace’s equation, which is solved by determining the unknown coefficients in the expansion of harmonic polynomials. A regularization method is proposed to tackle the resulting ill-posed linear system. The stability and convergence results are provided and a validating numerical experiment is presented.     

A boundary value problem for Poisson’s two-media equation inL p /2 -spaces

In the paper we study a binding boundary value problem for two media for Poisson's equation μΔu=f(x) with solutions in the class , 1<p<∞, with the corresponding seminorm, where

Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

Science China Mathematics - Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q &gt; 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a...     

An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace’s equation

Basing on the fundamental ideas of Babenko, we construct a fundamentally new, unsaturated, numerical method for solving the axially symmetric exterior Neumann problem for Laplace’s equation. The distinctive feature of this method is the absence of the principal error term enabling us to automatically adjust to every class of smoothness of solutions natural in the problem.     

Mixed initial–boundary value problem for the Benjamin–Ono equation

We consider the mixed initial–boundary value problem for the Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.     

Approximate solution to a singular,plane mixed boundary-value problem for Laplace’s equation in a curved rectangle

In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curved side.After expressing the solution as a finite linear combination of harmonic trial functions, the usual BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner.The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved by QR factorization method.Numerical results are presented and discussed, in order to assess the efficiency of the proposed method.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p$p$-laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.     

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

In this paper, we consider the Cauchy problem of Laplace’s equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.     

An application of Levy-Caccioppoli’s global inversion theorem to a boundary value problem for the Soret effect

The Levy-Caccioppoli’s global inversion theorem is used to prove existence and uniqueness for a problem of heat and mass transfer. The relevant boundary value problem is first transformed in a suitable two-point problem for a first order differential equation.     

Neumann initial–boundary value problem for Benjamin–Ono equation

We consider the Neumann initial–boundary value problem for Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

Inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.