Point Boundary Value Problem Research Articles

Stability Analysis of Hermite Collocation Method for Pulp Washing Models

Pulp washing is concerned with detaching cellulose fibres from black liquor with the use of a minimal amount of wash liquor. An efficient numerical technique of hermite collocation method is used for the solution of mathematical models related to pulp washing. The linear and non-linear models are solved using Quintic Hermite collocation method with Dirichlet’s and mixed Robin’s boundary conditions. Numerical solution of the models are derived using MATLAB ode15s. This study deals with the justification of accuracy of the method with stability analysis. The present method is more convenient, simple and elegant for solving the two point boundary value problems and the results found are very much stable from numerical point of view

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Existence of multiple unbounded solutions for a three-point boundary value problems on an infinite time scales

Abstract In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

Three Point Boundary Value Problems for Generalized Fractional Integro Differential Equations

This article deals with some existence results for a class of boundary value problem with three-point boundary conditions involving a nonlinear θ-Caputo fractional proportional integro differential equation. By means of some standard fixed point theorems, sufficient conditions for the existence of solutions are presented. Additionally, some applications of the main results are demonstrated.

GPINN with Neural Tangent Kernel Technique for Nonlinear Two Point Boundary Value Problems

Neural networks as differential equation solvers are a good choice of numerical technique because of their fast solutions and their nature in tackling some classical problems which traditional numerical solvers faced. In this article, we look at the famous gradient descent optimization technique, which trains the network by updating parameters which minimizes the loss function. We look at the theoretical part of gradient descent to understand why the network works great for some terms of the loss function and not so much for other terms. The loss function considered here is built in such a way that it incorporates the differential equation as well as the derivative of the differential equation. The fully connected feed-forward network is designed in such a way that, without training at boundary points, it automatically satisfies the boundary conditions. The neural tangent kernel for gradient enhanced physics informed neural networks is examined in this work, and we demonstrate how it may be used to generate a closed-form expression for the kernel function. We also provide numerical experiments demonstrating the effectiveness of the new approach for several two point boundary value problems. Our results suggest that the neural tangent kernel based approach can significantly improve the computational accuracy of the gradient enhanced physics informed neural network while reducing the computational cost of training these models.

Tangential velocity constraint for orbital maneuvers with Theory of Functional Connections

Maneuvering a spacecraft in the cislunar space is a complex problem, since it is highly perturbed by the gravitational influence of both the Earth and the Moon, and possibly also the Sun. Trajectories minimizing the needed fuel are generally preferred in order to decrease the mass of the payload. A classical method to constrain maneuvers is mathematically modeling them using the Two Point Boundary Value Problem (TPBVP), defining spacecraft positions at the start and end of the trajectory. Solutions to this problem can then be obtained with optimization techniques like the nonlinear least squares conjugated with the Theory of Functional Connections (TFC) to embed the constraints, which recently became an effective method for deducing orbit transfers. In this paper, we propose a tangential velocity (TV) type of constraints to design orbital maneuvers. We show that the technique presented in this paper can be used to transfer a spacecraft (e.g. from the Earth to the Moon) and perform gravity assist maneuvers (e.g. a swing-by with the Moon). In comparison with the TPBVP, solving the TV constraints via TFC offers several advantages, leading to a significant reduction in computational time. Hence, it proves to be an efficient technique to design these maneuvers.

Numerical Solution of Eighth Order Two Point Boundary Value Problems by Taylor Series Method

A Taylor series method which is taught at undergraduate levels and which has been hitherto used for the solution of initial value problem is successfully used in this work for the solutions of eighth-order two point boundary-value problems. The method is based on successive differentiation of the governing equation to obtain high order derivatives and then evaluated at the boundary point x=a. The solution is expressed inform a Taylor series with the unknown coefficients at a point x=a. By applying boundary conditions at x=b in the Taylor series solution, the system of unknown coefficient is obtained. After solving the system, then unknown coefficient are determined. The procedure is applied on both linear and nonlinear boundary-value problems. A comparison of the results obtained by the present method with results obtained by other methods reveals that the present method is simple, effective and also is in good agreement with the previous result and exact solution as showing in the tables and figures.

Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four-body problem

We develop constructive, computer assisted arguments for proving the existence of transverse homoclinic connections for hyperbolic periodic orbits in Hamiltonian systems. These arguments are applied at a number of non-perturbative parameter and energy values in the spatial equilateral circular restricted four-body problem: a six dimensional vector field. The idea is to formulate the desired connecting orbits as solutions of certain two point boundary value problems for orbit segments which originate and terminate on the local stable/unstable manifolds attached to a periodic orbit. These boundary value problems are studied via a Newton-Kantorovich argument in an appropriate Cartesian product of Banach algebras of rapidly decaying sequences of Chebyshev coefficients. Perhaps the most delicate part of the problem is controlling the boundary conditions, which must lie on the local stable/unstable manifolds of the periodic orbit. For this portion of the problem we use a parameterization method to develop Fourier-Taylor approximations equipped with mathematically rigorous a-posteriori error bounds. This requires validated computation of a finite number of Fourier-Taylor coefficients via Newton-Kantorovich arguments in appropriate Cartesian product of rapidly decaying sequences of Fourier coefficients, followed by a fixed point argument to bound the tail terms of the Taylor expansion. After some geometric considerations in the energy manifold, transversality of the connection follows as a direct consequence of the Newton-Kantorovich argument.

A Note on Single Impulse Approach for Trajectory Correction

Trajectory correction in the terminal phase of artillery projectile is studied using single impulse. The point of implementation of impulse on the trajectory and magnitude of impulse is obtained for desired correction. The flight path angle at point of implementation of impulse for given target position is evaluated and analysed by two point boundary value problem.

Powered lunar flyby for transfers between non-coplanar orbits around Earth

This work studies transfer between non-coplanar circular orbits around Earth with the space vehicle performing a powered lunar flyby maneuver. The complete transfer trajectory is accomplished by an application of two or three impulsive velocity increments. First and final velocity increments are applied tangentially, respectively, to the departing and the arrival orbits around Earth. An optional second velocity increment is applied at the perilune in order to increase the effects of the flyby maneuver. Despite many works consider the powered lunar flyby instead of a natural lunar flyby, it is important to compare both maneuvers in the context of the complete trajectory. In this direction, the present work formulates and solves multiple point boundary value problems that determine the transfer trajectories considering three models: a three-dimensional patched-conic approximation, a model based on the spatial restricted three-body problem, and, a model based on the spatial bi-circular restricted four-body in which the influence of the Sun is included. The transfer trajectory solutions are compared with classical maneuvers and with transfers that perform a natural flyby maneuver. An interesting result shows that a decelerating propulsion during the flyby maneuver can provide a transfer trajectory with a fuel consumption smaller than the one of bi-parabolic maneuver even if the Sun’s attraction is considered. Moreover, the influence of the Sun can decrease the time of flight and the apogee of the trajectory and it can save fuel consumption if the Sun’s initial phase angle is properly chosen.

Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Strömgren's asymptotic periodic orbits (transverse homoclinics for L4,5), families of periodic orbits passing through collision, and orbits connecting L4 to ejection or collision.

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

It is our main objective to find the critical load for three beams with cross sectional heterogeneity. Each beam has three supports, of which the intermediate one is a spring support. Determination of the critical load for these beams leads to three point boundary value problems (BVPs) associated with homogeneous boundary conditions—the mentioned BVPs constitute three eigenvalue problems. They are solved by using a novel solution strategy based on the Green functions that belong to these BVPs: the eigenvalue problems established for the critical load are transformed into eigenvalue problems governed by homogeneous Fredholm integral equations with kernels that can be given in closed forms provided that the Green function of each BVP is known. Then the eigenvalue problems governed by the Fredholm integral equations can be manipulated into algebraic eigenvalue problems solved numerically by using effective algorithms. It is an advantage of the way we attack these problems that the formalism established and the results obtained remain valid for homogeneous beams as well. The numerical results for the critical forces can be applied to solve some stability problems in the engineering practice.

Wavelet Approach to Estimate The solution for Two Points Boundary Value Problems

In this paper, a new algorithm based on using Chebyshev waveletoperational matrix of integration instead of operational matrix of derivativeto find the approximate solution of two points boundary value problem.Wavelet first Chebyshev with aid of collocation method employed totransform the differential equation with its boundary condition to a systemof linear algebraic equations. A very high level of accuracy explicitlyreflected by the proposed examples.

Consistency of an Infinite system of third order three-point boundary value problem in the $bv_0$ space by the theory of measure of noncompactness

Several authors have examined the solvability conditions for an infinite system of differential equations in different Banach spaces using the concept of measure of noncompactness. In all these studies, they have considered differential equations where the boundary conditions are defined on two points. In this paper, we have studied the solvability conditions for an infinite system of third-order three point boundary value problem in the sequence space of bounded variation $bv_0$ with the help of the theory of measure of noncompactness and have given a suitable example to illustrate the result.

A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz’s formula, Rolle and Cauchy’s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Approximation of optimal control problems for the Navier-Stokes equation via multilinear HJB-POD

We consider the approximation of some optimal control problems for the Navier-Stokes equation via a Dynamic Programming approach. These control problems arise in many industrial applications and are very challenging from the numerical point of view since the semi-discretization of the dynamics corresponds to an evolutive system of ordinary differential equations in very high-dimension. The typical approach is based on the Pontryagin maximum principle and leads to a two point boundary value problem. Here we present a different approach based on the value function and the solution of a Bellman equation, a challenging problem in high-dimension. We mitigate the curse of dimensionality via a recent multilinear approximation of the dynamics coupled with a dynamic programming scheme on a tree structure. We discuss several aspects related to the implementation of this new approach and we present some numerical examples to illustrate the results on classical control problems studied in the literature.

Spacecraft orbital pursuit–evasion games with [formula omitted] perturbations and direction-constrained thrust

In this paper, we study spacecraft orbital pursuit–evasion games under J2 perturbations. We consider that situation where the thrust of each player is constrained and the control direction cannot deviate from the velocity more than a given angle. After characterizing the optimal control of players under direction constraints, we transfer the pursuit–evasion game into a two point boundary value problem, which is solved by the shooting method. Different with the classical research, the initial guess for the unknown initial adjoint variables and the game ending time are not generated by heuristic approaches, but from the solution of the orbital one-side interception problem with a non-maneuverable evader. We solve the one-side interception problem also by shooting, starting iteratively from multiple initial guesses. In each initial guess, the initial adjoint vector is chosen randomly small while the interception termination time is selected from contiguous feasible time subintervals. The simulations show that the shooting method for the one-side interception problem can converge even if the feasible interval of the termination time is partitioned into a small number of subintervals, and the whole method for solving the pursuit–evasion game can quickly find the saddle-point solution. The effect of the control direction constraint on the game ending time is also discussed.

Finding extremals of Lagrangian actions

Given a smooth m-manifold M, a smooth Lagrangian L:TM→R and endpoints x0,xT∈M, we look for an extremal x:[0,T]→M of the action ∫0TL(x(t),ẋ(t))dt satisfying x(0)=x0 and x(T)=xT. When interpolating between endpoints, this amounts to a 2-point boundary value problem for the Euler–Lagrange equation. Single or multiple shooting is one of the most popular methods to solve boundary value problems, but the efficiency of shooting and the quality of solutions depends heavily on initial guesses. In the present paper, by dividing the interval [0,T] into several sub-intervals, on which extremals can be found efficiently by shooting when good initial guesses are available from the geometry of a variational problem, we then adjust all junctions by finding zeros of vector fields associated with the velocities at junctions with Newton’s method. We discuss the cases where L is the difference between kinetic energy and potential, M is a hypersurface in Euclidean space, or M is a Lie group. We make some comparisons in numerical experiments for a double pendulum, for obstacle avoidance by a moving particle on the 2-sphere, and for obstacle avoidance by a planar rigid body.

Existence of positive solutions for boundary value problems with p-Laplacian operator

Abstract This paper is concerned with the existence of positive solutions for three point boundary value problems of Riemann-Liouville fractional differential equations with p-Laplacian operator. By means of the properties of the Green’s function and Avery-Peterson fixed point theorem, we establish a condition ensuring the existence of at least three positive solutions for the problem.

Near-Optimal Control for Offshore Structures with Nonlinear Energy Sink Mechanisms

To improve the safety and reliability of offshore structures subject to wave loading, the active vibration control problem is always one of significant issues in the field of ocean engineering. This paper deals with the near–optimal control problem of offshore structures with a nonlinear energy sink (NES) mechanism. By taking the dominant vibration mode of the offshore structure with the NES into account, a nonlinear dynamic model of the steel–jacket structure subject to wave loading is presented first. Then, using the parameter perturbation approach to solve a nonlinear two–point boundary value problem, an NES–based optimal controller with the form of infinite series sum is presented to suppress the vibration of the offshore structure. Third, an iteration algorithm is provided to obtain the near–optimal controller. Simulation results demonstrate that the NES–based near–optimal controller can mitigate the oscillation amplitude of offshore structures significantly. Moreover, the NES–based optimal controller outperforms the one based on active tuned mass damper.