Kind Of Boundary Value Problem Research Articles

Common best proximity points for a pair of mappings with certain dominating property

Abstract This article introduces a type of dominating property, partially inherited from L. Chen’s, and proves an existence and uniqueness theorem concerning common best proximity points. A certain kind of boundary value problem involving the so-called Caputo derivative can be formulated so that our result applies.

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed.

Multiplicity of solutions for the Dirichlet boundary value problem to a fractional quasilinear differential model with impulses

This paper aims to consider the multiplicity of solutions for a kind of boundary value problem to a fractional quasilinear differential model with impulsive effects. By establishing a new variational structure and overcoming the difficulties brought by the influence of impulsive effects, some new results are acquired via the symmetry mountain-pass theorem, which extend and enrich some previous results.

Global classical solutions of a kind of boundary value problem for quasilinear hyperbolic systems

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global‐in‐time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string.

Jordan-form special solutions corresponding to arbitrary anti-plane shear forces in polynomial forms imposed on bimaterial interfacial crack surfaces

Bimaterial anti-plane interfacial crack problems with arbitrary distributed forces applied on crack surfaces are numerically investigated. The problems are in essence a kind of boundary value problems about Laplace's equations. Previous work of using Symplectic analytical singular element (SASE) with high-order accuracy and other singular finite elements for calculating mode III stress intensity becomes invalid when the distributed forces on each surface are different. In this paper, Jordan-form special solutions (JFSS) are proposed for the first time. We derive the JFSS for present problems and use them to improve displacement modes of SASE. The distributed forces are therefore converted into equivalent nodal forces of the nodes around SASE. The new SASE is found to be directly and conveniently used for cases where the two distributed forces are either equal (even both zero) or not equal. Numerical examples are provided to demonstrate the effectiveness and accuracy of the present method. The necessity and effectiveness of the JFSS is also validated. Moreover, the JFSS can be applied to some other physical problems represented by the Laplace's equation.

Solution of generalised type – 2 Fuzzy boundary value problem

In this article, we proposed a new representation of type-2 fuzzy numbers called generalized type-2 fuzzy numbers. Dirichlet, Neumann and Mixed kind of boundary value problem have been considered with the boundary condition as generalized type-2 fuzzy numbers. The theorems have been developed for solving generalized type-2 fuzzy boundary value problems. In every case, suitable examples have also been provided. The solutions graph for fuzzy cases has been plotted and discuss for understanding of the nature of fuzzy solutions.

Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix.

Revisiting the peridynamic motion equation due to characterization of boundary conditions

The peridynamic motion equation was investigated once again. The origin of incompatibility between boundary conditions and peridynamics was analyzed. In order to eliminate this incompatibility, we proposed a new peridynamic motion equation in which the effects of boundary traction and boundary displacement constraint were introduced. The new peridynamic motion equation is invariant under the transformations of rigid translation and rotation. Meanwhile, it also satisfies the requirements of total linear and angular momentum equilibrium. By this motion equation, three kinds of boundary value problems containing the displacement boundary condition, the traction boundary condition and mixed boundary condition are characterized in peridynamics. As examples, we calculated static tension and longitudinal vibration of a finite rod. The acquired solutions exhibit obvious nonlocal features, and the vibration has the dispersion similar to one dimensional atom chain vibration.

The Calculation Methods for Solving a Kind of Boundary Value Problems in Elasticity

This paper will study the numerical solutions for equations with a kind of boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods(MQMs) are presented to solve the nonlinear equations that the accuracy of the solutions are three order. According to the asymptotical compact convergence theory, the errors with an odd powers asymptotic expansion is obtained. Some results are shown regarding these approximations for problems by the numerical example.

New existence results for solutions of BVPs for higher order IFDEs involving Riemann-Liouville type Hadamard fractional derivatives

In this article, we present a method for converting boundary value problems for impulsive fractional differential systems involving the Riemann-Liouville type Hadamard fractional derivatives to integral systems. The existence results for solutions of this kind of boundary value problems are established. Our analysis relies on the well known fixed point theorem. Some comments on recent published papers are made at the end of the paper.

Existence of solutions of multi-term fractional differential equations with impulse effects on a half line

A class of boundary value problem for impulsive fractional differential equation on a half line is proposed. Some results on existence of solutions of this kind of boundary value problem for impulsive multi-term fractional differential equation on a half line are established by constructing a weighted Banach space, a completely continuous operator and using a fixed point theorem in the Banach space. Some unsuitable lemmas in recent published papers are pointed out. An example is given to illustrate the efficiency of the main theorems.

Non-parabolic interface motion for the one-dimensional Stefan problem: Neumann boundary conditions

In this work, we study the liquid-solid interface dynamics for large time intervals on a 1-D sample, with homogeneous Neumann boundary conditions. In this kind of boundary value problem, we are able to make new predictions about the interface position by using conservation of energy. These predictions are confirmed through the heat balance integral method of Goodman and a generalized non-classical finite difference scheme. Since Neumann boundary conditions imply that the specimen is thermally isolated, through well stablished thermodynamics, we show that the interface behavior is not parabolic, and some examples are built with a novel interface dynamics that is not found in the literature. Also, it is shown that, on a Neumann boundary value problem, the position of the interface at thermodynamic equilibrium depends entirely on the initial temperature profile. The prediction of the interface position for large time values makes possible to fine tune the numerical methods, and given that energy conservation demands highly precise solutions, we found that it was necessary to develop a general non-classical finite difference scheme where a non-homogeneous moving mesh is considered. Numerical examples are shown to test these predictions and finally, we study the phase transition on a thermally isolated sample with a liquid and a solid phase in aluminum.

Solvability of impulsive periodic boundary value problems for higher order fractional differential equations

A class of periodic boundary value problems for higher order fractional differential equations with impulse effects is considered. We first convert the problem to an equivalent integral equation. Then, using a fixed-point theorem in Banach space, we establish existence results of solutions for this kind of boundary value problem for impulsive singular higher order fractional differential equations. Two examples are presented to illustrate the efficiency of the results obtained.

An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

A kind of boundary value problem for inhomogeneous partial differential system

In this article, we first define a kind of generalized singular integral operator and discuss its properties. Then we propose a kind of boundary value problem for an inhomogeneous partial differential system in $R^{4}$ . Finally, the integral representation of the solution to a boundary value problem for the inhomogeneous partial differential system is obtained using the above singular integral operator.

Representations of super-holomorphic functions and their applications

In this paper, we give three kinds of Almansi representations for super-polyharmonic functions. As applications of these representations, we investigate two kinds of boundary value problems in superspace and construct solutions to the super-Poisson equation and the inhomogeneous super-polyharmonic equation.

A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems

Motivated by the incompleteness of the method of fundamental solution (MFS) for the interior problem, we give another meshless method both theoretically and numerically for recovering the temperature and the heat flux based on the normal derivative of the fundamental solution in this paper. Although the problems under investigation are well-posed, we should note that the method presented here results in an ill-conditioned system and this is a feature of the numerical method employed in the present approach. The ill-posedness of this system is given by the potential function. In order to overcome the ill-posedness of the system, the Tikhonov regularization method, as well as Morozov’s discrepancy principle for selecting an appropriate regularization parameter, are used to increase the stability of this method. Then three kinds of boundary value problems are presented to show the effectiveness of this method with some examples, whilst the comparisons with the MFS is presented. The numerical convergence and stability of this method are also analyzed.

Recovery of the temperature and the heat flux by a novel meshless method from the measured noisy data

In this paper, we give an invariant method of fundamental solutions (MFS) for recovering the temperature and the heat flux. The invariant MFS is to keep a very basic natural property, which is called the invariance property under trivial coordinate changes in the problem description. The optimal regularization parameter is chosen by Morozov discrepancy principle. Then the reason for introducing the regularization is explained clearly by using the potential function. Three kinds of boundary value problems are investigated to show the effectiveness of this method with some examples. In especial, when the classical MFS does not give accurate results for some problems, it is shown that the proposed method is effective and stable. For each example, the numerical convergence, accuracy, and stability with respect to the number of source points, the distance between the pseudo and real boundary, and decreasing the amount of noise added into the input data, respectively, are also analyzed.

Similar Constructing Method for Solving the Boundary Value Problem of the Compound Kummer Equation

This article is devoted to analyze the compound Kummer equation. Through the structure of left region and right region similarity kernel function, have similar structure of solution for this boundary value problems. The left field solution of this kind of boundary value problems can be obtained by combing the coefficients of boundary value problems on the left and the left area similar kernel function. The right field solution of this kind of boundary value problems can be expressed by assembling the left, the right area similar kernel function, the left lead solution functions and the coefficients of the convergence conditions. Thus put forward the simple method which was named similar constructive method to solve the boundary value problems of the compound Kummer equation. The put forward of this method provides the convenience for solving this kind of differential equation boundary value problems.

Fractional-order multivalued problems with non-separated integral-flux boundary conditions

In this paper, we study the existence of solutions for a new kind of boundary value problem of Caputo type fractional differential inclusions with non-separated lo- cal and nonlocal integral-flux boundary conditions. We apply appropriate fixed point theorems for multivalued maps to obtain the existence results for the given problems covering convex as well as non-convex cases for multivalued maps. We also include Riemann-Stieltjes integral conditions in our discussion. Some illustrative examples are also presented. The paper concludes with some interesting observations.