Types Of Boundary Value Problems Research Articles

A remark on elliptic differential equations on manifold

For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

Abstract The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 {\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍ p s ⁢ ( 𝒞 ) {\mathbb{H}^{s}_{p}(\mathcal{C})} , 1 p < s < 1 + 1 p {\frac{1}{p}<s<1+\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2 × 2 {2\times 2} systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ + {\mathbb{R}^{+}} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].

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.

Hexagonal grid approximation of the solution of the heat equation on special polygons

We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles alpha _{j}pi , j=1,2,ldots,M, where alpha _{j}in { frac{1}{2},frac{1}{3},frac{2}{3} } . To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order O ( h^{2}+tau ^{2} ) and O ( h^{4}+tau ) respectively, where h and frac{sqrt{3}}{2}h are the step sizes in space variables x_{1} and x_{2} respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.

An efficient numerical technique for Lane–Emden–Fowler boundary value problems: Bernstein collocation method

In this paper, we propose an efficient numerical technique for numerical solutions of the equivalent integral form of Emden–Fowler type boundary value problems (BVPs), which model many phenomena in mathematical physics and astrophysics. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. The iterative method is applied to solve the system numerically. The error analysis of the proposed method is provided. Several examples are provided to demonstrate the accuracy, applicability, and efficiency of the present method.

Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation

We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.

Haar wavelet quasilinearization method for numerical solution of Emden–Fowler type equations

In this paper, an efficient method for solving the nonlinear Emden–Fowler type boundary value problems with Dirichlet and Robin–Neumann boundary conditions is introduced. The present method is based on the Haar-wavelets and quasilinearization technique. The quasilinearization technique is adopted to linearize the nonlinear singular problem. Numerical solution of linear singular problem is obtained by the Haar wavelet method. The numerical study is further supported by examining thoroughly the convergence of the Haar wavelet method and the quasilinearization technique. In order to check the accuracy of the proposed method, the numerical results are compared with both existing methods and exact solutions.

A Locally-Continuous Meshless Local Petrov-Galerkin Method Applied to a Two-point Boundary Value Problem

Abstract In recent years, the Meshless Local Petrov-Galerkin (MLPG) Method has attracted the attention of many researchers in solving several types of boundary value problems. This method is based on a local weak form, evaluated in local subdomains and does not require any mesh, either in the construction of the test and shape functions or in the integration process. However, the shape functions used in MLPG have complicated forms, which makes their computation and their derivative's computation costly. In this work, using the Moving Least Square (MLS) Method, we dissociate the point where the approximating polynomial's coefficients are optimized, from the points where its derivatives are computed. We argue that this approach not only is consistent with the underlying approximation hypothesis, but also makes computation of derivatives simpler. We apply our approach to a two-point boundary value problem and perform several tests to support our claim. The results show that the proposed model is efficient, achieves good precision, and is attractive to be applied to other higher-dimension problems.

Regions of existence for a class of nonlinear diffusion type problems

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) ?S??(x)- ns?(x)- m x s?(x) = f(x,s), m > 0, n ? R, x ? (0,1), s?(0) = 0, a1s(1) + a2s?(1) = C, where a1 > 0, a2 ? 0, C ? R. These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see the existing literature of this paper). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where ?f/?s ? 0 and ?f/?s ? 0. Theoretical methods are applied for a set of problems which arise in real life.

The Solvability of First Type Boundary Value Problem for a Schrödinger Equation

Abstract The paper presents an first type boundary value problem for a Schrödinger equation. The aim of paper is to give the existence and uniqueness theorems of the boundary value problem using Galerkin’s method. Also, a priori estimate for its solution is given.

On the existence of positive solutions for generalized fractional boundary value problems

The existence of positive solutions is established for boundary value problems defined within generalized Riemann–Liouville and Caputo fractional operators. Our approach is based on utilizing the technique of fixed point theorems. For the sake of converting the proposed problems into integral equations, we construct Green functions and study their properties for three different types of boundary value problems. Examples are presented to demonstrate the validity of theoretical findings.

Rigidity theorem for harmonic maps with complex normal boundary conditions

Let [Formula: see text] be the open unit ball in [Formula: see text] and let [Formula: see text] be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map [Formula: see text] satisfying the boundary condition that [Formula: see text] on [Formula: see text]. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on [Formula: see text].

Theoretical Justification and Error Analysis for Slender Body Theory

AbstractSlender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well‐posed boundary value problem for the Stokes equations in ℝ3. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well‐posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry. © 2019 Wiley Periodicals, Inc.

Ultimate load of nonhomogeneous slopes determined by using the method of characteristics

Most prior studies on nonhomogeneous slopes have focused on the spatial variation of the cohesion, while the influence of a variable friction angle on slope stability is difficult to investigate. This study adopts the method of characteristics to generate a statically admissible failure mechanism in order to address the ultimate load of nonhomogeneous slopes. Three types of boundary value problems are used to generate discretized grid points in the plastic zone in order to build a stress field. Hence, it is easy to account for variations in the shear strength in a slip line field, especially for variations in the friction angle. Considering the edge distance of a surcharge loading, face and toe failure mechanisms are proposed to determine the influence of the variable shear strength. The proposed approach is compared with the upper bound method and numerical simulations. The good agreement shows that the novel approach is reliable and effective. Then, the ultimate load of nonhomogeneous slopes is calculated considering that the cohesion and friction angle vary linearly with depth. Moreover, the ultimate load obtained by the proposed method can be regarded as a lower bound solution, which supplements the limit analysis of nonhomogeneous slopes.

Numerical Approximation of Plane Deformation for the Indentation of Granular Material by a Smooth Rigid Wedge Punch

In this paper, a numerical approximation of the stress equation for the indentation of granular materials by a smooth rigid wedge is presented. Plane strain conditions are assumed, and the materials obey the Mohr-Coulomb yield condition. This method determines the deformation of granular material under a smooth rigid wedge punch and construction of stress field in the deforming region which are presented by using MATLAB programme. The granular material is assumed to be in dense, solid like state. The solution only refers to the initial motion after the punch. This was then applied on one type of boundary value problem. By using MATLAB, the value of each point (x,y) and the stress variables (p,ψ) which construct the deformation field are calculated. This method provides simple and reliable algorithms for the solution of the deformation problems involving the stress variables. The results will consequently help in the improvement of the existing labs and experimental facilities in the industries and will eventually increase its efficiency.

Three-dimensional explicit finite element formulation for shear localization with global tracking of embedded weak discontinuities

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.

Representation and uniqueness for boundary value elliptic problems via first order systems

Given any elliptic system with t -independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann types with area integral control or non-tangential maximal control. The trace spaces are obtained in a natural range of boundary spaces which is parametrized by properties of some Hardy spaces. This implies a complete picture of uniqueness vs solvability and well-posedness.

Special functions, integral transforms with applications

In this study, the author used integral transforms and a method of the exponential nature to deal with the families of fractional differential equations, Stieltjes type singular integral equations and boundary value problems. Certain integrals involving special functions are evaluated. Constructive examples are also provided throughout the paper. The main purpose of this article is to present mathematical results that are useful to researchers in a variety of fields.

A stabilized RBF collocation scheme for Neumann type boundary value problems

The numerical solutionof partial dif- ferential equations (PDEs) with Neumann bound- ary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neu- mann BC requires the approximation of the spa- tial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. In- creased accuracy of the spatial derivative approx- imation can be achieved by h-refinement reduc- ing the spacing between discretization points or byincreasingthemultiquadricshape parameter, c. Increasing the MQ shape parameter is very com- putationally cost effective, but leads to increased ill-conditioning. We have implemented an im- proved version of the truncated singular value de- composition (IT-SVD) originated by Volokh and Vilnay (2000) that projects very small singular values into the null space, producing a well con- ditioned system of equations. To assess the pro- posed refinement scheme, elliptic PDEs with dif- ferent boundary conditions are analyzed. Com- parisons that made with analytical solution reveal superior accuracy and computationalefficiency of the IT-SVD solutions.