Point Boundary Conditions Research Articles

Distribution of zeros of solutions of sixth order (2 ≤ n ≤ 5)-points boundary value problem in terms of semi-critical intervals

In this paper, the issue of distribution of zeros of the solutions of linear homogenous differential equations (LHDE) have been investigated in terms of semi-critical intervals. We shall follow a geometric approach to state and prove some properties of LHDEs of the sixth order with (2, 3, 4, and 5 points) boundary conditions and with measurable coefficients. Moreover, the relations between semi-critical intervals of the LHDEs have been explored. Also, the obtained results have been generalized for the 5th order differential equations.

Existence and uniqueness of solutions for system of Hilfer\u2013Hadamard sequential fractional differential equations with two point boundary conditions

In this paper, we study existence and uniqueness of solutions for a system of Hilfer–Hadamard sequential fractional differential equations via standard fixed point theorems. The existence is proved by using the Leray–Schauder alternative, while the existence and uniqueness by the Banach contraction mapping principle. Illustrative examples are also discussed.

Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

In this paper, we study the Riesz basis property and the problem of stabilization of two vibrating strings connected by a point mass with variable physical coefficients under a boundary feedback control acts at one extreme point and Dirichlet boundary condition on the other end. It is shown that the system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. By a detailed spectral analysis, it is proved that this hybrid system is asymptotically stable but not exponentially stable.

Ulam\u2013Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions

In this article, we discuss the sufficient conditions for the existence, uniqueness and stability of solutions to a class of nonlinear impulsive boundary value problem of fractional order differential equations. Using classical fixed point theorems, we develop the required conditions. Further, using the techniques of nonlinear functional analysis, we investigate Ulam–Hyers stability results to the proposed problem. For applications of our derived results, we present two numerical examples.

ON AN ITERATIVE METHOD FOR A CLASS OF 2 POINT & 3 POINT NONLINEAR SBVPS

In this article, we propose a novel modification to Quasi-Newton method, which is now a days popularly known as variation iteration method (VIM) and use it to solve the following class of nonlinear singular differential equations which arises in chemistry $ -y''(x)-\frac{\alpha}{x}y'(x) = f(x, y), \; x\in(0, 1), $ where $ \alpha\geq1 $, subject to certain two point and three point boundary conditions. We compute the relaxation parameter as a function of Bessel and the modified Bessel functions. Since rate of convergence of solutions to the iterative scheme depends on the relaxation parameter, thus we can have faster convergence. We validate our results for two point and three point boundary conditions. We allow $ \partial f/\partial y $ to take both positive and negative values.

Random coupled Caputo-Hadamard fractional differential systems with four- point boundary conditions in generalized banach spaces

This paper deals with some existence and uniqueness of random solutions for a coupled system of Caputo–Hadamard fractional differential equations with four-point boundary conditions and random effects in generalized Banach spaces. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces. An illustrative example is presented in the last section.

Framework for treating non-Linear multi-term fractional differential equations with reasonable spectrum of two-point boundary conditions

In this work, a novel bio-inspired meta-heuristic framework is presented for the assessment of linear and non-linear multi-term fractional differential equations (MFDEs) based on the idea of residual power series method (RPSM) in amalgamation with the bat algorithm (BATA) which, mimics the echolocation behavior of the foraging bats. The bat-inspired based methodology has been implemented to solve MFDEs with different possible variants of two point boundary conditions. The BATA is utilized in the recommended new residual optimization technique (NROT) for the minimization of the energy function attained by the spirit of RPSM. Moreover, to ratify the correctness and accuracy of the deliberated technique the results are evaluated by using two other meta-heuristic optimization techniques i.e., differential evolution algorithm (DEA) and the accelerated particle swarm optimization algorithm (APSOA) for the learning of unknown weights in the derived fitness function. The accuracy and competency of the NROT is validated by comparing the BATA computed results with the exact solution and the corresponding data acquired by the DEA and APSOA. Furthermore, detailed performance analysis is performed through statistical inference based on the large number of independent runs.

Source Attribution for Mercury Deposition with an Updated Atmospheric Mercury Emission Inventory in the Pearl River Delta Region, China.

We used CMAQ-Hg to simulate mercury pollution and identify main sources in the Pearl River Delta (PRD) with updated local emission inventory and latest regional and global emissions. The total anthropogenic mercury emissions in the PRD for 2014 were 11,939.6 kg. Power plants and industrial boilers were dominant sectors, responsible for 29.4 and 22.7%. We first compared model predictions and observations and the results showed a good performance. Then five scenarios with power plants (PP), municipal solid waste incineration (MSWI), industrial point sources (IP), natural sources (NAT), and boundary conditions (BCs) zeroed out separately were simulated and compared with the base case. BCs was responsible for over 30% of annual average mercury concentration and total deposition while NAT contributed around 15%. Among the anthropogenic sources, IP (22.9%) was dominant with a contribution over 20.0% and PP (18.9%) and MSWI (11.2%) ranked second and third. Results also showed that power plants were the most important emission sources in the central PRD, where the ultra-low emission for thermal power units need to be strengthened. In the northern and western PRD, cement and metal productions were priorities for mercury control. The fast growth of municipal solid waste incineration were also a key factor in the core areas. In addition, a coordinated regional mercury emission control was important for effectively controlling pollution. In the future, mercury emissions will decrease as control measures are strengthened, more attention should be paid to mercury deposition around the large point sources as high levels of pollution are observed.

Impulsive Fractional Differential Equations with p-Laplacian Operator in Banach Spaces

In this paper, we study a class of boundary value problem (BVP) with multiple point boundary conditions of impulsive p-Laplacian operator fractional differential equations. We establish the sufficient conditions for the existence of solutions in Banach spaces. Our analysis relies on the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to demonstrate the main results.

Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance

In this paper, by using Mawhin’s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results.

Existence of Solution, Filippov’s Theorem and Compactness of the Set of Solutions for a Third-Order Differential Inclusion with Three- Point Boundary Conditions

In this paper, we study a third-order differential inclusion with three-point boundary conditions. We prove the existence of a solution under convexity conditions on the multi-valued right-hand side; the proof is based on a nonlinear alternative of Leray-Schauder type. We also study the compactness of the set of solutions and establish some Filippov’s- type results for this problem.

Study of Fractional Boundary Value Problem Using Mittag-Leffler Function with Two Point Periodic Boundary Conditions

In this paper, two point periodic boundary value problem of fractional differential equation involving Caputo fractional derivative of order $$2<\alpha \le 3$$ are studied. Some properties of Mittag-Leffler function are used. Some important and useful results are investigated related to existence and uniqueness of fractional differential equations in terms of Mittag-Leffler by applying fixed point theorems. Three examples are given to illustrate the results.

Lyapunov-type inequalities for third order linear differential equations with two points boundary conditions

In this paper, by using Green's functions for second order differential equations, we establish new Lyapunov-type inequalities for third order linear differential equations with two points boundary conditions. By using such inequalities, we obtain sharp lower bounds for the eigenvalues of corresponding equations.

Shear Flows of an Ideal Fluid and Elliptic Equations in Unbounded Domains

We prove that, in a two‐dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half‐plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one‐dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n‐dimensional slabs in any dimension n.© 2016 Wiley Periodicals, Inc.

Existence of Positive Solution to a Class of Fractional Differential Equations with Three Point Boundary Conditions

Existence of Positive Solution to a Class of Fractional Differential Equations with Three Point Boundary Conditions

Existence of Solutions for a Coupled Fractional Differential Equations with Infinitely Many Points Boundary Conditions at Resonance on an Unbounded Domain

This paper investigates the existence of solutions for infinitely many points boundary value problems at resonance with \(\mathrm{dimker}\,L=2\) regarding fractional differential equations on an unbounded domain. By the well-known coincidence degree theory of Mawhin, it is shown that the considered system under certainly conditions admits at least one solution. An example to illustrate the applicability of the given conditions is also given.

The Numerical Solution of Third Order Boundary Value Problems Using Piecewise Bernstein polynomials by the Galerkin Weighted Residual Method

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Application of Fractional Order Legendre Polynomials: a New Procedure for Solution of Linear and Nonlinear Fractional Differential Equations under $m$-point Nonlocal Boundary Conditions

In this paper, we have proposed a new formulation for the solution of a general class of fractional differential equations (linear and nonlinear) under $\hat{m}$-point boundary conditions. We derive some new operational matrices and based on these operational matrices we develop scheme to approximate solution of the problem. The scheme convert the boundary value problem to a system of easily solvable algebraic equations. We show the applicability of the scheme by solving some test problems. The scheme is computer oriented.

Problem of Optimal Control for Parabolic-Hyperbolic Equations with Nonlocal Point Boundary Conditions and Semidefinite Quality Criterion

We consider a problem of optimal control for parabolic-hyperbolic equations with nonlocal boundary conditions and semidefinite quality criterion. The optimality conditions are constructed by reducing the problem to a sequence of one-dimensional problems, the optimal control is obtained in a closed form, and its convergence is proved. The form of the quality criterion is substantiated.

Constant sign solutions of two-point fourth order problems

In this paper we characterize the sign of the Green’s function related to the fourth order linear operator u(4) + M u coupled with the two point boundary conditions u(1) = u(0) = u′(0) = u′′(0) = 0. We obtain the exact values on the real parameter M for which the related Green’s function is negative in (0, 1) × (0, 1). Such property is equivalent to the fact that the operator satisfies a maximum principle in the space of functions that fulfil the homogeneous boundary conditions.When M > 0 the best estimate follows from spectral theory. When M < 0, we obtain an estimation by studying the disconjugacy properties of the solutions of the homogeneous equation u(4) + M u = 0. The optimal value is attained by studying the exact expression of the Green’s function. Such study allow us to ensure that there is no real parameter M for which the Green’s function is positive on (0, 1) × (0, 1).Moreover, we obtain maximum principles of this operator when the solutions verify suitable non-homogeneous boundary conditions.We apply the obtained results, by means of the method of lower and upper solutions, to nonlinear problems coupled with these boundary conditions.