Dirichlet Boundary Value Problem Research Articles

Finite-dimensional perturbation of the Dirichlet boundary value problem for the biharmonic equation

Abstract The biharmonic equation is one of the important equations of mathematical physics, describing the behaviour of harmonic functions in higher-dimensional spaces. The main purpose of this study was to construct a finite-dimensional perturbation for the Dirichlet boundary value problem associated with the biharmonic equation. The methodological basis for this study was an integrated approach that includes mathematical analysis, algebraic methods, operator theory, and the theorem on the existence and uniqueness of a solution for a boundary value. The main tool is a finite-dimensional perturbation, which allows for examining the properties and behaviour of boundary value problems in as much detail as possible. In the study, descriptions of correctly solvable internal boundary value problems for a biharmonic equation in non-simply connected domains were considered in detail. The study is also devoted to the search for solutions and the analytical representation of resolvents of boundary value problems for a biharmonic equation in multi-connected domains. Within the framework of the study, theorems and their consequences were proved, and a finite-dimensional perturbation was constructed for the Dirichlet boundary value problem. Analytical representations of resolvents of boundary value problems for a biharmonic equation in multi-connected domains were also obtained. The examination of a finite-dimensional perturbation of the Dirichlet boundary value problem for a biharmonic equation has expanded the understanding of the properties of this equation in various contexts.

Initial-boundary value problem for transport equations driven by rough paths

In this paper, we are interested in the initial Dirichlet boundary value problem for a transport equation driven by weak geometric Hölder p p -rough paths. We introduce a notion of solutions to rough partial differential equations with boundary conditions. Consequently, we will establish a well-posedness for such a solution under some assumptions stated below. Moreover, the solution is given explicitly.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

On a Class of Nonlinear Elliptic Equations with General Growth in the Gradient

In this paper, we prove an existence and uniqueness result for a class of Dirichlet boundary value problems whose model is −Δpu=β|∇u|q+c|u|p−2u+finΩ,u=0on∂Ω, where Ω is an open bounded subset of RN, N≥2, 1<p<N, Δpu is the so-called p-Laplace operator, and p−1<q<p. We assume that β is a positive constant, c and f are measurable functions belonging to suitable Lorentz spaces. Our approach is based on Schauder fixed point theorem.

On relation-theoretic F−contractions and applications in F−metric spaces

Abstract The aim is to introduce some relation theoretic variants of F−contraction in an F−metric space endowed with a binary relation R and to prove results for its fixed point. In the sequel, several classes of contractions are sharpened, generalized, and improved. Numerical examples are presented to illustrate the theoretical conclusions. As applications of the main results, we solve a Dirichlet-Neumann initial value problem and two Dirichlet boundary value problems.

On the Dirichlet problem for A-harmonic functions

We study the Dirichlet boundary value problem with continuous boundary data for the A-harmonic equations div[A grad u] = 0 in an arbitrary bounded domain D of the complex plane £ with no boundary component degenerated to a single point. We provide integral criteria, including the BMO and FMO criteria expressed in terms of A (z), for the existence of weak solutions to the problem. We also discuss the connections between A-harmonic functions and potential theory.

Interaction of Boundary Singular Points in an Elliptic Boundary Value Problem

The paper continues the construction of the Lp-theory of elliptic Dirichlet and Neumann boundary value problems with discontinuous piecewise constant coefficients in divergent form for an unbounded domain R2 with a piecewisesmooth noncompact Lipschitz boundary and C1 smooth discontinuity lines of the coefficients. An earlier constructed Lp-theory is generalized to the case of different smallest eigenvalues corresponding to a finite and an infinite singular point, and the effect of their interaction is further studied in the class of functions with first derivatives from Lp( ) in the entire range of the exponent p (1, )..

Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator

ABSTRACT The Dirichlet boundary value problem for elliptic equations involving the (p(x), q(x))-Laplacian operator with a reaction term depending on the gradient and on two real parameters is considered in this paper. Using the topological degree theory for a class of demicontinuous operators of generalized (S +) and the theory of the variable exponent Sobolev spaces, we prove the existence of at least one weak solution of such problem.

On Simultaneous Determination of Thermal Conductivity and Volume Heat Capacity of Substance

The study of nonlinear problems associated with heat transfer in substance is important for practice. Earlier, the authors proposed an efficient algorithm for determining the thermal conductivity from experimental observations of the dynamics of the temperature field in an object. In this work, we explore the possibility of extending the algorithm to the numerical solution of the problem of simultaneous identification of the temperature-dependent volume heat capacity and the thermal conductivity of the substance under study. The consideration is based on the Dirichlet boundary value problem for the one-dimensional nonstationary heat equation. The coefficient inverse problem in question is reduced to a variational problem, which is solved by applying gradient methods based on the fast automatic differentiation technique. The uniqueness of the solution to the inverse problem is analyzed.

Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems

Abstract In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel’skiĭ’s fixed point theorem in cones and a weak Harnack-type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel’skiĭ’s theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.

Narrow escape in composite domains forming heterogeneous networks

Cellular networks are often composed of thin tubules connecting much larger node compartments. These structures serve for active or diffusion transport of proteins. Examples are glial networks in the brain, the endoplasmic reticulum in cells or dendritic spines. In this latter case, a large ball forming the head is connected to the dendrite by a narrow passage. In all cases, how the transport of molecules, ions or proteins is regulated determines the time scale of chemical reactions or signal transduction. In the present study, based on modeling diffusion in three dimensions, we compute the mean time for a Brownian particle to reach a narrow target inside such a composite network made of tubules connected to spherical nodes. We derive asymptotic formulas by solving a mixed Neumann–Dirichlet boundary value problem with small Dirichlet part. We first consider the general case of a network domain organized in a 2-D lattice structure that consists of spherical ball compartments connected via narrow cylindrical passages. For a single target located on the boundary of one of the spherical domains, we derive a sparse linear system of equations for each Mean First Passage Time (MFPT) averaged over the different compartments. We then consider a composite domain consisting of a spherical head-like domain, with narrow absorbing targets on its boundary, that is connected to a large cylinder via a narrow cylindrical neck. For Brownian particles starting within the neck, we derive an asymptotic formula for the MFPT. When diffusing particles can be absorbed upon hitting additional absorbing boundaries of the large cylindrical compartment, we derive asymptotic formulas for the splitting probability and conditional MFPT to reach a target on the spherical head. We compare these formulas with numerical solutions of the mixed boundary value problem and with Brownian simulations, allowing to explore geometrical parameter ranges. To conclude, the present analysis reveals that the mean arrival time, driven by diffusion in heterogeneous networks, is controlled by the targets and tubules sizes, as well as the size of the nodal compartments.

A monotone numerical flux for quasilinear convection diffusion equation

We propose a new numerical 2-point flux for a quasilinear convection–diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results.

Multiplicity of solutions for nonlinear coercive problems

We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.

On Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

The paper gives an explicit representation of the Green’s function of the Dirichlet boundary value problem for the polyharmonic equation in the unit ball. The solution of the homogeneous Dirichlet problem is found. An example of solving the homogeneous Dirichlet problem with the simplest polynomial right-hand side of the polyharmonic equation is given.

Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function.

Monte Carlo Algorithms for the Solution of Quasi-Linear Dirichlet Boundary Value Problems of Elliptical Type

Monte Carlo Algorithms for the Solution of Quasi-Linear Dirichlet Boundary Value Problems of Elliptical Type

Existence theory for multiple solutions to second-order singular Dirichlet boundary value problem modeling the Antarctic Circumpolar Current

In the article, we present multiple solutions for a second-order singular Dirichlet boundary value problem that arises when modeling the ocean flow of the Antarctic Circumpolar Current. The main tools of the proof are the Leray–Schauder nonlinear alternative principle and a well-known fixed point theorem in cones.

Implicit-Explicit Finite Difference Approximations of a Semilinear Heat Equation with Logarithmic Nonlinearity

Abstract We formulate an initial and Dirichlet boundary value problem for a semilinear heat equation with logarithmic nonlinearity over a two-dimensional rectangular domain. We approximate its solution by employing the standard second-order finite difference method for space discretization, and a linearized backward Euler method, or, a linearized BDF2 method for time stepping. For the linearized backward Euler finite difference method, we derive an almost optimal order error estimate in the discrete L t ∞ ⁢ ( L x ∞ ) L^{\infty}_{t}(L^{\infty}_{x}) -norm without imposing mesh conditions, and for the linearized BDF2 finite difference method, we establish an almost optimal order error estimate in the discrete L t ∞ ⁢ ( H x 1 ) L^{\infty}_{t}(H^{1}_{x}) -norm, allowing a mild mesh condition to be satisfied. Finally, we show the efficiency of the numerical methods proposed, by exposing results from numerical experiments. It is the first time in the literature where numerical methods for the approximation of the solution to the heat equation with logarithmic nonlinearity are applied and analysed.

Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method

The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level.

Linear parabolic equation with Dirichlet white noise boundary conditions

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation dudt=Au with strongly elliptic operator A on bounded and unbounded domains with white noise boundary data. Our main assumption is that the heat kernel of the corresponding homogeneous problem enjoys the Gaussian type estimates taking into account the distance to the boundary. Under mild assumptions about the domain, we show that A generates a C0-semigroup in weighted Lp-spaces where the weight is an appropriate power of the distance to the boundary. We also prove some smoothing properties and exponential stability of the semigroup. Finally, we reformulate the Cauchy-Dirichlet problem with white noise boundary data as an evolution equation in the weighted space and prove the existence of Markovian solutions and invariant measures.