Two-point Boundary Value Problem Research Articles

Post-Buckling Behavior of Variable-Arc-Length Elastica Pipe Conveying Fluid Including Effects of Self-weight and Pressure Variation

This paper investigated the post-buckling behaviors of a variable-arc-length (VAL) elastica pipe caused by internal transporting fluid motion, accounting for the effects of the self-weight and variation of internal fluid pressure. Both weak-form and strong-form formulations of the VAL elastica pipe were developed. The weak-form formulation was derived by the virtual work principle, and later solved using the nonlinear finite element method (FEM). The strong-form formulation was derived using equilibrium of the force and moment equations, the geometrical relationship of the differential pipe segment, and the energy conservation of the transported fluid. Because of the large displacement analysis, the variation in the internal pressure inside the pipe was considered, which was taken into account by energy conservation based on the Bernoulli principle. Then, the set of nonlinear governing first-order differential equations, corresponding to the two-point boundary value problem, was conveniently solved using the shooting optimization method (SOM). The numerical results obtained from the presented FEM and SOM analyses independently verified each other, with good agreement between these two methods. The parametric study considered the effects of gravity load (pipe and fluid self-weights) and internal fluid pressure on the post-buckling characteristics of the VAL pipe. The numerical results showed that without the pipe and internal fluid weights, the transporting fluid exerted an axial compressive force on the pipe, causing post-buckling phenomena. Beyond the critical state, the support rotation and the length of the VAL pipe increased as the internal fluid velocity decreased. Finally, by including the effect of self-weight, the mode exchange of the odd buckling modes (1st and 3rd modes) was found from the load-displacement path.

A Theory of Functional Connections-Based hp-Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems

This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians.

A collocation method for an RLC fractional derivative two-point boundary value problem with a singular solution

A collocation method for an RLC fractional derivative two-point boundary value problem with a singular solution

Computational Approach to Third-Order Nonlinear Boundary Value Problems via Efficient Decomposition Shooting Method

The present manuscript proposes a computational approach to efficiently tackle a class of two-point boundary value problems that features third-order nonlinear ordinary differential equations. Specifically, this approach is based upon a combination of the shooting method with a modification of the renowned Adomian decomposition method. The approach starts by transforming the governing BVP into two appropriate initial-value problems, and thereafter, solves the resulting IVPs recurrently. In addition, the application of this method to varied test models remains feasible—of course, this is supported by the competing Runge–Kutta method, among others, and reported through comparison plots and tables.

Heat and Mass Transfer in Unsteady Radiating MHD Flow of a Maxwell Fluid with a Porous Vertically Stretching Sheet in the Presence of Activation Energy and Thermal Diffusion Effects

The aim of this study is to analyze the effects of activation energy and thermal diffusion an unsteady MHD reactive Maxwell fluid flow past a porous stretching sheet in the presence of Brownian motion, Thermophoresis and nonlinear thermal. The non-linear partial differential equations that govern the fluid flow have been transformed into a two-point boundary value problem using similarity variables and then solved numerically by fourth order Runge–Kutta method with shooting technique. Graphical results are discussed for non-dimensional velocity, temperature and concentration profiles while numerical values of the skin friction, Nusselt number and Sherwood number are presented in tabular form for various values of parameters controlling the flow system. The present study is compared with the previous literature and found to be in good agreement.

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.

Infinitely Many Nodal Solutions of Superlinear Third Order Two-Point Boundary Value Problems

Infinitely Many Nodal Solutions of Superlinear Third Order Two-Point Boundary Value Problems

Boundary value problems associated with Hamiltonian systems coupled with positively-(p, q)-homogeneous systems

We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated.

Learning Fuel-Optimal Trajectories for Space Applications via Pontryagin Neural Networks

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem’s Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method.

Fast initialization of the indirect optimization problem in the solar sail circle-to-circle orbit transfer

This Short Communication presents a procedure to approximate, in an analytical form, the solution of the optimization problem that gives the minimum-time trajectory of a low-performance solar sail, in a classical circle-to-circle heliocentric transfer. More precisely, the proposed procedure gives an accurate approximation of the solution of the two-point boundary value problem associated to the optimization process, when an indirect approach is exploited to obtain the optimal transfer trajectory and an ideal force model is used to describe the solar sail thrust vector. In particular, the analytical results illustrated in this work extend the model discussed in the recent literature, obtained with an ideal electric thruster, to the case of a flat, perfectly reflecting, solar sail-based scenario.

Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

Low-Thrust Trajectory Optimization with Averaged Dynamics Using Analytical Switching Detection

This paper presents an efficient indirect optimization method for the time- and fuel-optimal low-thrust many-revolution transfers with averaged dynamics. Compared with the previous indirect method, the proposed method involves a new set of variables and several near-optimal control laws to formulate the two-point boundary value problem based on three practical assumptions. Then, the costate transformations between two reference frames are first derived with respect to the the corresponding Eulerian angles, such that the switching function can be expressed with a six-degree polynomial function and the number of parameters used to determine the energy-optimal solutions is reduced to 6. Finally, a neural network is trained to generated the initial guesses, and the solution algorithms for solving the time- and fuel-optimal problem are formulated. Numerical simulations demonstrate that the convergence ratio of initial guesses is higher than 98%, the proposed assumptions are reasonable for guaranteeing the near optimality, and the presented method can obtain convergent solutions within tenths of a second for the Earth- and moon-centered external/interior transfers considering the shadow J2 and third-body perturbations.

A highly efficient and accurate numerical method for the electromagnetic scattering problem with rectangular cavities

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted.

Second-Order Averaging of Low-Thrust Transfers in Fixed Time

An algorithm for the second-order approximation of fixed-time, low-thrust orbital transfers and generation of Bacon plots is proposed. After averaging the extremal flow of the optimal-control Hamiltonian, a one-parameter family of solutions of a reduced-order, two-point boundary value problems is computed by means of a differential-continuation scheme. Sensitivities of the shooting function are then used in conjunction with an ad hoc near-identity transformation between averaged and osculating variables to achieve an accurate solution for all longitudes of the departure and arrival orbits. Hence, a single simplified shooting problem has to be solved to approximate the solution for any combination of departure and arrival dates and to draw a Bacon plot for an arbitrary launch window. Both the averaged flow and the near-identity transformation are efficiently evaluated via the fast-Fourier transform algorithm, yielding a fully numerical procedure.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

Bioconvection in a Porous Medium Saturated with a Casson Nanofluid

The bio-convective heat transfer of an incompressible, viscous, electrically conducting Casson nanofluid past a wedge has been analyzed. Furthermore, using a similarity variable, the governing flow equations are transformed to non-linear coupled differential equations corresponding to a two-point boundary value problem, which is solved numerically. A comparison of the solution technique is carried out with previous work and the results are found to be in good agreement. Numerical results for the coefficient of skin friction, Nusselt number, Sherwood number, micro-organism flux as well as the velocity, temperature, nanoparticles and micro-organisms concentration profiles are presented for different physical parameters. The analysis of the obtained results shows that the field of flow is significantly influenced by these parameters.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

Well-posedness criteria for one family of boundary value problems

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

Coincidence and common fixed points for F-Contractive mappings

The purpose of this article is to establish the existence and uniqueness of coincidence and common fixed point of discontinuous non-compatible faintly compatible pair of self maps in non-complete metric space without using containment requirement of range space of involved maps satisfying Ciric type F-contraction and Hardy-Roger type F-contraction. Some illustrative examples associated with pictographic validations are provided to demonstrate the main results and to show the genuineness of our results. We consider the application of our results to the study of a two-point boundary value problem related to second order differential equation, solve the two-point boundary value problem of the second-order differential equation arising in electric circuit equation, and also apply our results to Volterra type integral equation using Ciric type F-contraction as well as Hardy Roger type F-contraction. Mathematics subject classification: 47H10; 54H25; 54E50

Fourth order computational spline method for two-parameter singularly perturbed boundary value problem

The current research work considers a two-parameter singularly perturbed two-point boundary value problem. Here, we suggest a computational scheme derived by using an exponential spline for the numerical solution of the problem on a uniform mesh. The proposed numerical scheme is analyzed for convergence and an accuracy of O(h4) is achieved. Numerical experiments are considered to validate the efficiency of the spline method, and compared comparison with the existing method to prove the superiority of the proposed scheme.