Point Boundary Value Problem Research Articles

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control

A representation of a fundamental solution group for a class of wave equations is constructed by exploiting connections between stationary action and optimal control. By using a Yosida approximation of the associated generator, an approximation of the group of interest is represented for sufficiently short time horizons via an idempotent convolution kernel that describes all possible solutions of a corresponding short time horizon optimal control problem. It is shown that this representation of the approximate group can be extended to longer horizons via a concatenation of such short horizon optimal control problems, provided that the associated initial and terminal conditions employed in concatenating trajectories are determined via a stationarity rather than an optimality based condition. The construction is illustrated by its application to the approximate solution of a two point boundary value problem.

Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

Two-point boundary value problems for ordinary differential equations, uniqueness implies existence

We consider a family of three point n − 2, 1, 1 conjugate boundary value problems for nth order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of n−point boundary value problems. Here, we replace that standard hypothesis with one in which we assume uniqueness of solutions of large families of two and three point boundary value problems. We then close the paper with verifiable conditions on the nonlinear term that in fact imply global uniqueness of solutions of the large family of three point boundary value problems.

Dichotomy and well conditioning of two-point boundary value problems on time scale dynamical systems

In this paper, we establish close relationships between the stability constants on one hand and the global behaviour of fundamental matrices on the other hand to the two-point boundary value problems on time-scale dynamical systems. We introduce the concept of conditioning number k and show that co nditioning number is the right criteria in estimating the global error due to small perturbations of two point boundary value problems on time scale dynamical systems. Further, the moderate stability constants imply a dichotomy with moderate k-bound will be developed. Further, the exponential behaviour of solutions of the Green’s matrix will be investigated. We also investigate the conditions under which strong dichotomy exists for two-point boundary value problems when the boundary conditions are separable.

Relation-Theoretic Nonlinear Contractions in $$an~ {{\mathcal {F}}}$$-Metric Space and Applications

We introduce relation-theoretic contractions to establish a fixed point in $${\mathcal {F}}$$ -metric spaces. Numerical examples are also given to illustrate the theoretical findings. As applications of our main result, we solve two point boundary value problems.

Implementation of Differential Transform Method (DTM) for Large Deformation Analysis of Cantilever Beam

Large deformation of a cantilever beam is an eminent structural analysis and engineering problem. A non-linear differential equation governs the problem of large deformation of beams. It has many important applications scientific and engineering fields. It is quite intricate and complex to find a closed-form or an exact solution for such a problem. It is relatively easier to use analytic expressions for calculations than numerical or experimental analysis. Large deformation analysis has always been an active area of research for researchers. The main focus of this research is to implement a new method namely Differential Transformation Method (DTM) for non-linear beam problem analysis. DTM is quite reliable and efficient in giving an approximate analytical solution to beam problems. Moreover, it is a new application for DTM. Hence, the primary purpose of this work is to evaluate the utility of the method for solution of beam problem. DTM yields a 2-point boundary value problem. It is based Taylor’s series expansion. DTM helps to calculate the deflection angle and horizontal as well as vertical displacements of a cantilever beam, experiencing large deformation, in an explicit analytical form. Mathematical formulation code will be developed using MATLAB. Variation of rotation angle at free end will be analysed using different number of terms used and values of non-dimensional load “β”. Results of DTM prove the method to be quite effective and suitable for predicting solutions to non-linear beam problems. Therefore, DTM can be used for widespread applications for new, emerging and complex engineering problems.

A Fitted Finite Difference Scheme for solving Singularly Perturbed Two Point Boundary Value Problems

A Fitted Finite Difference Scheme for solving Singularly Perturbed Two Point Boundary Value Problems

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.

A Novel Computational Method for Coupled Two Point Boundary Value Problem

Abstract The paper deals with Finite difference Method of solving a boundary value problem involving a coupled pair of system of Ordinary Differential Equations. A novel iterative scheme is given for solving the Finite Difference Equations. Quasi-linearization used to convert a nonlinear problem into a series of linear problems. A problem from a flow of a nanofluid is presented as an example.

Retraction Note to: Existence of weak solutions for two point boundary value problems of Schr\xf6dingerean predator-prey system and their applications

The Editors-in-Chief have retracted this article [1] because it significantly overlaps with an article by different authors that was simultaneously under consideration with another journal [2]. In addition, the identity of the corresponding author could not be verified: Universidad Austral confirmed that Tanriver Ülker was never affiliated to this institution. The authors have not responded to any correspondence regarding this retraction.

Existence of Solutions for $${\mathrm {m}}$$-Point Boundary Value Problems on an Infinite Time Scale

In this study, we present the existence result for the the second order $${\mathrm {m}}$$-point boundary value problems on infinite time scales. Nagumo condition, lower and upper solutions play an important role in the arguments.

Uncertain Lambert Problem: A Probabilistic Approach

A complete solution to the uncertain Lambert problem is considered in this paper. While the deterministic Lambert problem considers the two point boundary value problem of the two body problem with a fixed transfer time, its uncertain counterpart is shown to be related to the propagation of uncertainty from a set of initial conditions. In contrast to the linearized solutions of the uncertain Lambert problem associated with a particular solution, our paper outlines a numerical process to characterize the non-Gaussian uncertainty associated with all solutions. Applications of solutions to the uncertain Lambert problem are detailed using representative examples.

A new high accuracy method in exponential form based on off-step discretization for non-linear two point boundary value problems

ABSTRACTThis paper is intended to describe a graded mesh implicit method based upon half step discretization of order three for the solution of non-linear two point boundary value problem of the form , . The devised method is compact and uses three nodal points for the unknown function u(x) and two nodal points for the known variable ‘x’ in spatial axis. A detailed convergence analysis has also been included. This method, though meant for scalar equations was further extended to compute the vector equations of two point nonlinear boundary value problems. The method is directly applicable to singular problems. Computational results are provided to assess the performance and reliability of the method.

Fixed point results for single valued and set valued P-contractions and application to second order boundary value problems

In this paper, by considering the concept of set-valued nonlinear P-contraction which is newly introduced, we present some new fixed point theorems for set-valued mappings on complete metric space. Then by considering a single-valued case we provide an existence and uniqueness result for a kind of second order two point boundary value problem.

A NEW THIRD ORDER EXPONENTIALLY FITTED DISCRETIZATION FOR THE SOLUTION OF NON-LINEAR TWO POINT BOUNDARY VALUE PROBLEMS ON A GRADED MESH

This paper puts forward a novel graded mesh implicit scheme resting upon full step discretization of order three for computation of non-linear two point boundary value problems. The suggested method is compact and employs three nodal points for the unknown function $ u(x) $ in spatial axis. We have also performed error analysis of the cited method. The given method was tried (implemented) upon multiple problems in Cartesian and Polar coordinates with extremely favorable outcomes. This method, though meant for scalar equations, was further extended to compute the vector equations of two point nonlinear boundary value problems. To check the validity of the proposed scheme, we applied it to multiple problems and obtained supporting numerical computations.

Computational Solution of Two–Point Boundary Value Problems with Derivative Boundary Conditions using Cubic-Spline Process

Computational Solution of Two–Point Boundary Value Problems with Derivative Boundary Conditions using Cubic-Spline Process

Three point boundary value problems for ordinary differential equations, uniqueness implies existence

Three point boundary value problems for ordinary differential equations, uniqueness implies existence

Comparative Study of Pontryagin Maximum Principle and Direct Method in Aircraft Optimal Control Design

This paper deals with the problem of open-loop control design for a dynamic object which is described by a nonlinear system of differential equations. To solve this problem a direct method is used, in which the target functional is minimized by population-based algorithm. The method is applied to a test problem which consists of finding optimal control for spacial motion of maneuverable aircraft. Proposed technique is compared with two classical solutions to considered problem. One of them is based on equating to zero the partial control derivatives of the Hamilton function, wherease the other on the Hamilton function maximum over controls (Pontryagin maximum principle). The paper confirms high degree of similarity between solutions obtained by all considered methods of selecting target functional. However, classical algotithms show slightly worse accuracy and higher sensitivity to the quality of initial approximation. In addition, proposed direct method permits to evade the necessity to solve a two–point boundary value problem required in classical algorithms.

Higher Order Fitted Operator Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems

In this paper, we consider singularly perturbed parabolic convection-diffusion initial boundary value problems with two small positive parameters to construct higher order fitted operator finite difference method. At the beginning, we discretize the solution domain in time direction to approximate the derivative with respect to time and considering average levels for other terms that yields two point boundary value problems which covers two time level. Then, full discretization of the solution domain followed by the derivatives in two point boundary value problem are replaced by central finite difference approximation, introducing and determining the value of fitting parameter ended at system of equations that can be solved by tri-diagonal solver. To improve accuracy of the solution with corresponding higher orders of convergence, we applying Richardson extrapolation method that accelerates second order to fourth order convergent. Stability and consistency of the proposed method have been established very well to assure the convergence of the method. Finally, validate by considering test examples and then produce numerical results to care the theoretical results and to establish its effectiveness. Generally, the formulated method is stable, consistent and gives more accurate numerical solution than some methods existing in the literature for solving singularly perturbed parabolic convection- diffusion initial boundary value problems with two small positive parameters.