Dirichlet-type Boundary Value Problems Research Articles

On a Dirichlet boundary value problem for an Ermakov–Painlevé I equation. A Hamiltonian EPI system

Here, a proto-type Ermakov–Painlevé I equation is introduced and a homogeneous Dirichlet-type boundary value problem analysed. In addition, a novel Ermakov–Painlevé I system is set down which is reducible by an involutory transformation to the autonomous Ermakov–Ray–Reid system augmented by a single component Ermakov–Painlevé I equation. Hamiltonian such systems are delimited

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

Explicit solutions of the BVPs of the theory of thermoelasticity for an elastic circle with voids and microtemperatures

AbstractIn the present paper, we consider an elastic circle and a plane with circular hole with voids and microtemperatures. Representations of a general solution of a system of equations for a homogeneous isotropic thermoelastic medium with voids and microtemperatures are constructed by means of the elementary (harmonic, bi‐harmonic and meta‐harmonic) functions. The Dirichlet type boundary value problems for a circle and for a plane with circular hole are solved explicitly. The obtained solutions are represented in the form of absolutely and uniformly convergent series.

Positive solutions to singular Dirichlet-type boundary value problems of nonlinear fractional differential equations

In this paper, by using the properties of the Green function, u_{0}-positive operator and Gelfand’s formula, some properties of the first eigenvalue corresponding to the relevant operator are obtained. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence and uniqueness results of positive solution are established.

Riemann and Dirichlet Type Problems for Hyperanalytic Functions in Fractal Domains

In this paper we are interested in finding solvability conditions for Riemann and Dirichlet type boundary value problems with generalized Holder-continuous boundary data in the case of hyperanalytic functions defined in a domain of the complex plane bounded by an h-summable closed curve.

Existence and multiplicity results for resonant fractional boundary value problems

<p style='text-indent:20px;'>We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory.

Existence and nonexistence of positive solutions for Dirichlet-type boundary value problem of nonlinear fractional differential equation

Existence and nonexistence of positive solutions for Dirichlet-type boundary value problem of nonlinear fractional differential equation

Three nontrivial solutions for nonlinear fractional Laplacian equations

Abstract We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

The dirichlet BVP of the theory of thermoelasticity with microtemperatures for microstretch sphere

ABSTRACTThe present article studies the equilibrium theory of thermomicrostretch elastic solids with microtemperatures. The general solution of the equations for a homogeneous isotropic microstretch thermoelastic sphere with microtemperatures is constructed and the solution of the Dirichlet-type boundary value problem for the sphere in the form of absolutely and uniformly convergent series is obtained.

The solution of the Dirichlet BVP in the fully coupled theory of elasticity for spherical layer with double porosity

The main goal of this paper is to consider the Dirichlet type boundary value problem (BVP) of the fully coupled equilibrium theory of elasticity for solids with double porosity and to construct explicitly the solution of BVP for a spherical layer in the form of absolutely and uniformly convergent series.

Boundary value problems for hyperholomorphic solutions of two dimensional Helmholtz equation in a fractal domain

A theory of quaternion-valued functions, called hyperholomorphic, of two real variables has long been established. This theory is in the same relation to the two dimensional Helmholtz equation as the usual one-dimensional complex analysis is to the Laplace equation in R2. In this work we define a new Cauchy integral for domains with fractal boundary illustrating its applications and usage to study the jump and Dirichlet type boundary value problems in a fractal domain.

On Dirichlet two-point boundary value problems for the Ermakov–Painlevé IV equation

Two-point boundary value problems of Dirichlet-type are investigated for a hybrid Ermakov–Painlevé IV equation. Existence and uniqueness results are established in terms of the Painlevé parameters. In addition, it is shown how Ermakov invariants may be used to systematically obtain solutions of a coupled Ermakov–Painlevé IV system in terms of seed solutions of the canonical integrable Painlevé IV equation.

A numerical method for the solution of singular and singularly perturbed boundary value problems

This note is concerned with a new numerical method for the solution of singular and singularly perturbed boundary value problems. The method uses Fourier sine series and is suitable for Dirichlet-type boundary value problems. It removes the singularity of the problem in a natural way. A number of numerical examples are used to study the applicability of the method. Keywords: Dirichlet-type boundary valueproblems, Singular boundary value problems, Singularly perturbed baoundary value problems.

Dirichlet type problems in Hermitian Clifford analysis

Solvability conditions for some Dirichlet type boundary value problems in the framework of Hermitian Clifford analysis are established.

Solutions to Dirichlet-Type Boundary Value Problems of Fractional Order in Banach Spaces

We consider the boundary value problems with Dirichlet-type boundary conditions of nonlinear fractional differential equation in Banach space. The existence of the solution to the boundary value problems is established. Our analysis relies on the Sadovskii fixed point theorem. As an application, we give an example to demonstrate our results.

The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems

The application of the sinc-Galerkin method to an approximate solution of second-order singular Dirichlet-type boundary value problems were discussed in this study. The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. The differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products without any numerical integration which is needed to solve matrix system. This study shows that the sinc-Galerkin method is a very effective and powerful tool in solving such problems numerically. At the end of the paper, the method was tested on several examples with second-order Dirichlet-type boundary value problems.

An efficient computer application of the sinc-Galerkin approximation for nonlinear boundary value problems

A powerful technique based on the sinc-Galerkin method is presented for obtaining numerical solutions of second-order nonlinear Dirichlet-type boundary value problems (BVPs). The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. Without any numerical integration, the differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products; therefore, the evaluation is based on solving a matrix system. The solution is obtained by constructing the nonlinear (or linear) matrix system using Maple and the accuracy is compared with the Newton method. The main aspect of the technique presented here is that the obtained solution is valid for various boundary conditions in both linear and nonlinear equations and it is not affected by any singularities that can occur in variable coefficients or a nonlinear part of the equation. This is a powerful side of the method when being compared to other models.

Multiple positive solutions to singular positone and semipositone Dirichlet-type boundary value problems of nonlinear fractional differential equations

In this paper, we establish the existence of multiple positive solutions to positone and semipositone Dirichlet-type boundary value problems of the nonlinear fractional differential equation: D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ( 1 ) = 0 by using the Leray–Schauder nonlinear alternative and a fixed-point theorem on cones, where 1 < α < 2 is a real number and D 0 + α is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0 .

The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application

In this paper, we discuss some new positive properties of the Green function for boundary value problems of nonlinear Dirichlet-type fractional differential equations. Its applications are also given.

A note on biquaternionic MIT bag model

The term ”bag model” unifies several attempts to construct a mathematical description of the phenomenon of quark confinement. Such models are often used for estimating different characteristics of a hadron. The aim of this note is to add one more characterization to the list of known characterizations of Dirac equation (see, e.g., [8], [10]). In the article, we study the Dirichlet type boundary value problems and MIT bag model in three-dimensional piece-wise Liapunov domains. Mathematical Subject Classification: Primary 81V05, 78A25; Secondary 30G35