Types Of Boundary Value Problems Research Articles

A Numerical Method for Singularly Perturbed Second Order Coupled System of Convection–Diffusion Robin Type Boundary Value Problems with Discontinuous Source Term

In this paper, a numerical method for a weakly coupled system of singularly perturbed convection–diffusion second order ordinary differential equations with discontinuous source term subject to Robin type boundary conditions is presented. A fitted mesh method has been used to obtain the difference scheme to solve the system of equations on a piecewise uniform Shishkin mesh. An error estimate is derived to show that the method is uniformly convergent with respect to singular perturbation parameter. Numerical results are provided to illustrate the theoretical results.

Degenerate scales for boundary value problems in anisotropic elasticity

Degenerate scales usually refer to a size effect which causes non-unique solutions of boundary integral equations for certain type of boundary value problems with a unique solution. They are closely connected to the presence of a logarithmic function in the integral kernel of the single-layer potential operator. The equations of the elasticity theory provide one of the known application fields where degenerate scales appear. The paper discusses conditions and formula for controlling and detection of the degenerate scales in the case of fully anisotropic analysis. No restrictions are considered for the material, only the loading should cause two-dimensional deformation of the anisotropic body. A technique for the evaluation of the degenerate scales is discussed and tested. The examples provide results of special simple cases and demonstrate suitability of the proposed technique in relation to calculation of degenerate scales by numerical solution of pertinent boundary integral equation by the boundary element method.

ON THREE-POINT HADAMARD-TYPE FRACTIONAL BOUNDARY VALUE PROBLEMS

In this paper, we study the existence of solutions for a fractional bound- ary value problem involving Hadamard type fractional differential equations and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko's fixed point theorem. As an application, we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our re- sults are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

MAXIMAL USING OF SPECIFICS OF SOME BOUNDARY PROBLEMS IN POTENTIAL THEORY AFTER THEIR NUMERICAL ANALYSIS

Some typical problems in the numerical analysis of certain types of boundary value problems of the potential theory in substantially spatial formulation are considered. On the basis of the integral equation method (IE) an approximate scheme of solving one model example is built and investigated. It is also considered that the doubly connected open surface where boundary conditions are set obtains the Abelian group of symmetry of the eighth order. This article shows how using the apparatus of the group theory it is possible to solve an initial problem by the help of the sequence of the eight independent IEs, where the integration is realized only on one of the congruent constituents of the surface. It creates the conditions for two parallel processes of problem solution in general. The collocation method for obtaining approximate values of needed “density of charge distribution” in the particular two-dimensional integral equations is used. To take into account the singular way of solving the problem in the circuit of the open surface the a posteriori method of error evaluation is created and the procedure of integrating clarification of solving the task in the mesh node is implemented. To prove the reliability and estimation of the technique efficiency the number of numerical experiments is carried out including the use of so called “plane” approximation of the examined spatial problem.

Positive solutions of periodic type boundary value problems for first order singular functional differential equations

Abstract For first order singular functional differential equations, optimal sufficient conditions for the existence of positive solutions of periodic type boundary value problems are established.

The exact boundary behavior of solutions to singular nonlinear Lane–Emden–Fowler type boundary value problems

In this paper we analyze the exact boundary behavior of solutions to singular nonlinear Dirichlet problems −△u=b(x)g(u)+λa(x)f(u),u>0,x∈Ω,u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, λ≥0, g∈C1((0,∞),(0,∞)), lims→0+g(s)=∞, b,a∈Clocα(Ω), are positive in Ω, may be vanishing or singular on the boundary, and f∈C([0,∞),[0,∞)). We reveal that the nonlinear term λa(x)f(u) does not affect the first expansion of the classical solutions near the boundary to the problem for several kinds of functions a and b.

Over-determined boundary value problems for PDE and their application in the wave propagation theory

The solvability conditions of the over-determined boundary value problems for PDE with constant coefficients in the form of connection between Fourier transforms of boundary functions are obtained. The jump problem on the hyperplane for PDE is formulated and investigated. Two-dimensional Helmholtz equation is considered as an example. It is proved that the mixed type conjugation problem for Helmholtz equation, to which the diffraction problem of the electromagnetic wave on thin conducting ribbons is reduced, is equivalent to one-sided mixed type boundary value problem. The uniqueness theorem is proved for this problem.

Boundary value problems with eigenvalue-dependent boundary and transmission conditions

In this paper the operator-theoretical method to investigate a new type boundary value problems consisting of a two-interval Sturm-Liouville equation together with boundary and transmission conditions dependent on eigenparameter is developed. By suggesting our own approach, we construct modified Hilbert spaces and a linear operator in them in such a way that the considered problem can be interpreted as a spectral problem for this operator. Then we introduce so-called left- and right-definite solutions and give a representation of solution of the corresponding nonhomogeneous problem in terms of these one-hand solutions. Finally, we construct Green’s vector-function and investigate some important properties of the resolvent operator by using this Green’s vector-function.

A novel analytical method for heat conduction in convectively cooled eccentric cylindrical annuli

Heat conduction equation through a heat generated eccentric cylindrical annulus with the inner surface kept at a constant temperature and the outer surface subjected to convection is analytically solved in bipolar coordinates using the Green's function method. Since it is not possible to find an analytical Green's function to the conduction equation in bipolar coordinates for an eccentric annulus subject to boundary condition(s) of third type (convection), a novel method treating the same problem as a second type boundary value problem is devised. The method has first been applied to heat generating eccentric annuli and results have been compared to the results of computational fluids dynamics (CFD) code FLUENT. Perfect agreement was observed for various geometrical configurations and a wide range of Biot number. Then, heat transfer through eccentric annuli without heat generation was considered. Variation of heat dissipation with radii ratio was studied and a very good agreement with the literature has been observed. A simple approximate analytical expression for the heat transfer rate is derived using first term (zero-order) approximation. It has been demonstrated that this expression gives very accurate results for a wide range of geometrical configurations and Biot number.

A note on elliptic type boundary value problems with maximal monotone relations

In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as “coefficients”. A reformulation of the respective problems is constructed such that they turn out to be unitarily equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of “tailor‐made” distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach (De Gruyter, 2011)]. The abstract framework is illustrated by various examples.

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.

On Neumann type problems for nonlocal equations set in a half space

We study Neumann type boundary value problems for nonlocal equations related to Levy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of reflection we impose on the outside jumps. To focus on the new phenomenas and ideas, we consider different models of reflection and rather general non-symmetric Levy measures, but only simple linear equations in half-space domains. We derive the Neumann/reflection problems through a truncation procedure on the Levy measure, and then we develop a viscosity solution theory which includes comparison, existence, and some regularity results. For problems involving fractional Laplacian type operators like e.g.$(-\Delta)^{\alpha/2}$, we prove that solutions of all our nonlocal Neumann problems converge as alpha goes to 2 to the solution of a classical Neumann problem. The reflection models we consider include cases where the underlying Levy processes are reflected, projected, and/or censored upon exiting the domain.

Effect of injection screen slot geometry on hydraulic conductivity tests

Summary Hydraulic conductivity and its spatial variability are important hydrogeological parameters and are typically determined through injection tests at different scales. For injection test interpretation, shape factors are required to account for injection screen geometry. Shape factors act as proportionality constants between hydraulic conductivity and observed ratios of injection flow rate and injection head at steady-state. Existing results for such shape factors assume either an ideal screen (i.e., ignoring effects of screen slot geometry) or infinite screen length (i.e., ignoring effects of screen extremes). In the present work, we investigate the combined effects of circumferential screen slot geometry and finite screen length on injection shape factors. This is done in terms of a screen entrance resistance by solving a steady-state potential flow mixed type boundary value problem in a homogeneous axi-symmetric flow domain using a semi-analytical solution approach. Results are compared to existing analytical solutions for circumferential and longitudinal slots on infinite screens, which are found to be identical. Based on an existing approximation, an expression is developed for a dimensionless screen entrance resistance of infinite screens, which is a function of the relative slot area only. For anisotropic conditions, e.g., when conductivity is smaller in the vertical direction than in the horizontal, screen entrance losses for circumferential slots increase, while they remain unaffected for longitudinal slots. This work is not concerned with investigating the effects of (possibly turbulent) head losses inside the injection device including the passage through the injection slots prior to entering the porous aquifer.

Потенциальные поля свободной энергии на стадиях упрочнения и разупрочнения среды Генки при неположительности объемной деформации

The Hencky medium with softening under isothermal and quasistatic deformation is considered. It is believed that volume deformation is not positive. In this case softening is characterized by the part of union curve with negative slope. For aforementioned conditions function of free energy is presented. For all stages of deformation in the space “volume deformation - intensity of shear’s deformation” level lines of the free energy are constructed. It is established that level lines are ellipses in hardening and function of free energy increases with distance from their centers, while in softening hyperbolas are level lines and function of free energy decreases with distance from their centers. Obtained results indirectly confirm the change in type of boundary value problem from elliptic to hyperbolic under material transition to the softening.

Multidimensional integral transform involving I-function of several complex variables as kernel

The theory of integral transforms is very useful in solving various types of boundary value problems. By giving various values to the kernel k(x; s) and considering the interval (0, 1) generally, a number of integral transforms have been introduced and studied by several authors from time to time. In the present paper we have introduced multidimensional I-function transform involving I-function of r variables as kernel. In our earlier paper we have established the multiple Mellin transform of product of two I-functions. With the help of that result, in the present paper we have discussed some theorems on multidimensional I-function transform. Special cases include the results involving H-function of several complex variables, I-function of two variables and Hfunction of two variables. The special case also includes the results given by V.C.Nair

An extended magnetostatic Born-Infeld model with a concave lower order term

This paper concerns an extended Born-Infeld model with a concave lower order term for the magnetostatic case. Three types of boundary value problems are considered: the boundary condition prescribing the tangential component of A, the natural boundary condition, and the boundary condition prescribing the tangential component of \documentclass[12pt]{minimal}\begin{document}$\operatorname{curl}\mathbf {A}$\end{document}curlA. In each case we obtain existence and regularity of solutions for small boundary data.

Generalized quantum mechanical two-Coulomb-center problem (Demkov problem)

We present a new exactly solvable quantum problem for which the Schrödinger equation allows for separation of variables in oblate spheroidal coordinates. Namely, this is the quantum mechanical two-Coulomb-center problem for the case of an imaginary intercenter parameter and complex conjugate charges are considered. Since the potential is defined by the two-sheeted mapping whose singularities are concentrated on a circle rather than at separate points, there arise additional possibilities in the choice of boundary conditions. A detailed classification of the various types of boundary-value problems is given. The quasi-radial equation leads to a new type of boundary value problem which has never been considered before. Results of the numerical calculations, which allow conclusions to be drawn about the structure of the energy spectrum, are shown. Possible physical applications are discussed.

Comparison between Visibility and Diffraction Criteria on SIF and -Integral Value for Mode Crack Using RKPM

Recently, mesh-free methods are increasingly utilized in solving various types of boundary value problems. Much research has been done on mesh-free methods for solving differential equation problems including crack and also obtained satisfactory results. Among these methods, reproducing kernel particle method (RKPM) has been used increasingly in fracture mechanic problems. The -integral and the stress intensity factor (SIF) are the most important parameters for crack problems. In this study -integral has been used to calculate the SIF in the crack tip. The mode SIF at the crack tip in a work-hardening material is obtained for various dilation parameters using RKPM. A comparison between two conventional treatments, visibility and diffraction on SIF and -integral value, is conducted. Visibility and diffraction methods increase the accuracy of RKPM results and effect on the -integral results at the crack tip. In comparing between the visibility and diffraction methods to modify the shape functions, the diffraction criterion seems to have better results for the -integral and SIF value.

A New Method for Solving Boundary Value Problem of Composite Hermit Equations

The paper is devoted to studying the solution for boundary value problem of composite Hermit equations. Based on the obtained solution with similar structural formula, a new method which is called as Similar Constructive Method (shortened as SCM) was put forward for solving this type of boundary value problem. The SCM is a simply and practical method, which provides convenience for solving boundary value problems of composite differential equations.

NUMERICAL METHOD FOR A 2NTH-ORDER BOUNDARY VALUE PROBLEM

In this paper, a finite difference scheme for a two-point boundary value problem of 2nth-order ordinary differential equations is presented. The convergence and uniqueness of the solution for the scheme are proved by means of theories on matrix eigenvalues and norm. Numerical examples show that our method is very simple and effective, and that this method can be used effectively for other types of boundary value problems.