Multi-point Boundary Value Problems Research Articles

New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative

<abstract><p>In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.</p></abstract>

The continuation method for calculation of continuous beams of arbitrary bending rigidity

This article presents an effective mathematical continuation method for the numerical implementation of the multipoint boundary value problem, to which the calculation of a beam of arbitrary rigidity at any of its supports is reduced. The problem can be treated as a direct one in the matter of constructing an optimal design based on beam systems. A test example of the calculation is given.

Limit Theorems for the Solutions of Multipoint Boundary-Value Problems with Parameter in Sobolev Spaces

We consider the most general class of multipoint boundary-value problems for systems of linear ordinary differential equations of any order whose solutions belong to a given Sobolev space $$ {W}_p^{n+r} $$ , with n ≥ 0, r ≥ 1, and 1 ≤ p ≤ ∞. We establish constructive sufficient conditions under which the solutions of the analyzed problems are continuous with respect to the parameter e for e = 0 in the space $$ {W}_p^{n+r} $$ .

Multipoint boundary value problem of optimal control for parabolic equations with degeneration

Multipoint boundary value problem of optimal control for parabolic equations with degeneration

EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS FOR A SYSTEM OF NONLINEAR FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEMS WITH <i>P</i> -LAPLACIAN OPERATOR

<abstract> In this paper, we deal with a coupled system of nonlinear fractional multi-point boundary value problems with $ p $-Laplacian operator. The existence and multiplicity of positive solutions are obtained by employing Leray-Schauder alternative theory, Leggett-Williams fixed point theorem and Avery-Henderson fixed point theorem. As an application, two examples are given to illustrate the effectiveness of our main results. </abstract>

Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

In this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

Solving complex optimal control problems with nonlinear controls using trigonometric functions

AbstractThis study investigates the use of trigonometric functions to resolve two major issues encountered when solving practical optimal control problems (OCPs) that are characterized by nonlinear controls. First, OCPs with constraints on nonlinear controls require the solution to a multipoint boundary value problem, which poses additional computational difficulties. Second, in certain unconstrained OCPs with nonlinear controls, the extremum found from the necessary conditions can be opposite than expected (e.g., a maximum instead of a minimum) due to the absence of control options. The aforementioned issues and their effective resolution by using trigonometric functions are explained through two examples including a second‐order system problem and an aerocapture problem of a slender entry vehicle in the Martian atmosphere.

Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation

In this paper, the Sinc-derivative collocation method is used to solve linear and nonlinear multi-point boundary value problems. This is done by interpolating the first derivative of the unknown variable via Sinc numerical methods and obtaining the desired solution through numerical integration of the interpolation and all higher order derivatives through successive differentiation of the interpolation. Non-homogeneous boundary conditions are reduced to homogeneous using suitable transformations. The efficiency and the accuracy of the method are tested using illustrative examples previously considered by other researchers who used different approaches. The results show the excellent performance of the Sinc-derivative collocation method.

Direct integration of the third-order two point and multipoint Robin type boundary value problems

This numerical study exclusively focused on the direct two point diagonally multistep block method of order four (2DDM4) in the form of Adams-type formulas. The proposed predictor–corrector scheme was applied in this study to compute two equally spaced numerical solutions for the third-order two point and multipoint boundary value problems (BVPs) subject to Robin boundary conditions concurrently at each step. The optimization of the computational costs was taken into consideration by not resolving the equation into a set of first-order differential equations. Instead, its implementation involved the use of shooting technique, which included the Newton divided difference formula employed for the iterative part, for the estimation of the initial guess. Apart from studying the local truncation error, the study also included the method analysis, including the order, stability, and convergence. The results of eight numerical problems demonstrated and highlighted competitive computational cost attained by the scheme, as compared to the existing method.

EXISTENCE AND UNIQUENESS RESULTS FOR A COUPLED SYSTEM OF HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTI-POINT BOUNDARY CONDITIONS

In this paper, we have studied existence and uniqueness of solutions for a coupled system of multi-point boundary value problems for Hadamard fractional differential equations. By applying principle contraction and Shaefer's fixed point theorem new existence results have been obtained.

A composite framework for co-optimization of spacecraft trajectory and propulsion system

Indirect optimization methods convert optimal control problems (OCPs) into two- or multi-point boundary-value problems. A highly desirable feature of indirect methods, specifically for space applications, is that high-resolution trajectories can be generated, which satisfy the first-order necessary conditions of optimality. A recently developed Composite Smoothing Control (CSC) framework is utilized to formulate and solve the problem of simultaneous trajectory optimization and propulsion sub-system design of spacecraft. A reasonable parameterized breakdown of the spacecraft mass is adopted, which captures the impact of power produced by the solar arrays and its contribution to the total spacecraft mass. Thus, the implicit trade-offs can be considered in the indirect optimization approach. The function space co-optimization problem of spacecraft power subsystem parameters along with the main trajectory is solved with the objective to maximize the payload delivered. The proposed framework amounts to an invariant embedding that reduces the original, difficult-to-solve, multi-point boundary-value problem into a two-point boundary-value problem with continuous, differentiable control inputs. Utility of the proposed construct is demonstrated through a low-thrust, multi-revolution, multi-year rendezvous maneuver to asteroid Dionysus with a variable-specific-impulse, variable-thrust modeled engine. This is the first time that indirect optimization methods have tackled such a complex co-optimization problem using the CSC framework.

Eigenvalue Intervals of Multivalued Operator and its Application for a Multipoint Boundary Value Problem

In this paper, we establish a general existence result on positive eigenvalue interval for the multivalued operators in cone. By using the main result, we investigate the existence of positive solutions for a multi-point boundary value problem. We study this problem via a parametric multivalued inclusion.

A Modification of the Parameterization Method for a Linear Boundary Value Problem for a Fredholm Integro-Differential Equation

A modification of the parameterization method is proposed to solve a linear two-point boundary value problem for a Fredholm integro-differential equation. The domain of the problem is partitioned and additional parameters are set as the values of the solution at interior points of the partition subintervals. Definition of a regular pair consisting of a partition and chosen interior points is given. The original problem is transformed into a multipoint boundary value problem with parameters. For fixed values of parameters, we get a special Cauchy problem for a system of integro-differential equations on the subintervals. Using the solution to this problem, the boundary condition and continuity conditions of solutions at the interior mesh points of the partition, we construct a system of linear algebraic equations in parameters. It is established that the solvability of the problem under consideration is equivalent to that of the constructed system.

Граничні теореми для розв’язків багатоточкових крайових задач із параметром у просторах Соболєва

УДК 517.927 Розглянуто найбільш загальний клас багатоточкових крайових задач для систем лінійних звичайних диференціальних рівнянь довільного порядку, розв'язки яких належать заданому простору Соболєва , де , і . Встановлено конструктивні достатні умови, за яких розв'язки цих задач неперервні за параметром при у просторі .

Optimal two-impulse space interception with multiple constraints

We consider optimal two-impulse space interception problems with multiple constraints. The multiple constraints are imposed on the terminal position of a space interceptor, impulse and impact instants, and the component-wise magnitudes of velocity impulses. These optimization problems are formulated as multi-point boundary value problems and solved by the calculus of variations. Slackness variable methods are used to convert all inequality constraints into equality constraints so that the Lagrange multiplier method can be used. A new dynamic slackness variable method is presented. As a result, an indirect optimization method is developed. Subsequently, our method is used to solve the two-impulse space interception problems of free-flight ballistic missiles. A number of conclusions for local optimal solutions have been drawn based on highly accurate numerical solutions. Specifically, by numerical examples, we show that when time and velocity impulse constraints are imposed, optimal two-impulse solutions may occur; if two-impulse instants are free, then a two-impulse space interception problem with velocity impulse constraints may degenerate to a one-impulse case.

Approximation properties of multipoint boundary-value problems

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable functions $y:[a,b]\to\mathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $\mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.

Existence Results for Nonlocal Multi-Point and Multi-Term Fractional Order Boundary Value Problems

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and integral boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples.

Existence and Uniqueness Results for a Class of Singular Fractional Boundary Value Problems with the p-Laplacian Operator via the Upper and Lower Solutions Approach

In this paper, we study the existence and uniqueness of positive solutions to a class of multipoint boundary value problems for singular fractional differential equations with the p-Laplacian operator. Here, the nonlinear source term f permits singularity with respect to its time variable t. Some fixed-point theorems such as the Leray-Schauder nonlinear alternative, the Schauder fixed-point theorem, and the Banach contraction mapping principle and the properties of the Gauss hypergeometric function are used to prove our main results. And by employing the upper and lower solutions technique, we derive a new approach to obtain the maximal and minimal solutions to the given problem. Finally, we present some examples to demonstrate our existence and uniqueness results.

Solvability for p‐Laplacian generalized fractional coupled systems with two‐sided memory effects

This paper determines the solvability of multipoint boundary value problems for p‐Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two‐sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum‐type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.

Existence of multiple solutions for nonlinear multi-point boundary value problems

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.