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Journal articles on the topic 'Hamiltonian Boundary Value Method'

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Published: 10 March 2023

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1

Song, Mingzhan, Xu Qian, Hong Zhang, and Songhe Song. "Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation." Advances in Applied Mathematics and Mechanics 9, no. 4 (January 18, 2017): 868–86. http://dx.doi.org/10.4208/aamm.2015.m1356.

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AbstractIn this paper, we introduce the Hamiltonian boundary value method (HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectral method in spatial direction, which leads to the finite-dimensional Hamiltonian system. The HBVM, which can preserve the Hamiltonian effectively, is applied in time direction. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation are taken as examples to show the validity of the proposed method. Numerical results confirm that the proposed method can simulate the propagation and collision of different solitons well. Meanwhile the corresponding errors in Hamiltonian and other intrinsic invariants are presented to show the good preservation property of the proposed method during long-time numerical calculation.

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2

Brugnano, L., and D. Trigiante. "Block Boundary Value Methods for linear Hamiltonian systems." Applied Mathematics and Computation 81, no. 1 (January 1997): 49–68. http://dx.doi.org/10.1016/0096-3003(95)00308-8.

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3

Brugnano, Luigi, Gianluca Frasca Caccia, and Felice Iavernaro. "Efficient implementation of Gauss collocation and Hamiltonian boundary value methods." Numerical Algorithms 65, no. 3 (January 17, 2014): 633–50. http://dx.doi.org/10.1007/s11075-014-9825-0.

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4

Erbe, L. H., Xinzhi Liu, and Jianhong Wu. "Solvability of boundary value problems for vector differential systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 114, no. 3-4 (1990): 181–93. http://dx.doi.org/10.1017/s0308210500024379.

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SynopsisBrouwer topological degree theory, the shooting type method, the disconjugacy theory of Hamiltonian systems and the Liapunov-Razumikhin technique of Volterra integrodifferential equations are employed to establish some solvability results for the 2n-dimensional differential systemsubject to one of the following boundary conditions:(i) x(0) = Qx(l), Qg(l x(1), qy(1)= g(0, x(0), y(0)),(ii) Blx(O) = B2g(O, x(O), y(O)), C1x(l) = −C2g(l, x(l), y(l)),where Q, Bi, Ci, i = 1, 2, are n x n real matrices. An application is given to the second order equation xn = h(t, x, x') subject to certain nonlinear boundary conditions.

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Amodio, Pierluigi, Luigi Brugnano, and Felice Iavernaro. "Energy-conserving methods for Hamiltonian boundary value problems and applications in astrodynamics." Advances in Computational Mathematics 41, no. 4 (November 14, 2014): 881–905. http://dx.doi.org/10.1007/s10444-014-9390-z.

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6

Al-Hawasy, Jamil A. Ali. "The Continuous Classical Boundary Optimal Control of Couple Nonlinear Hyperbolic Boundary Value Problem with Equality and Inequality Constraints." Baghdad Science Journal 16, no. 4(Suppl.) (December 18, 2019): 1064. http://dx.doi.org/10.21123/bsj.2019.16.4(suppl.).1064.

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The paper is concerned with the state and proof of the existence theorem of a unique solution (state vector) of couple nonlinear hyperbolic equations (CNLHEQS) via the Galerkin method (GM) with the Aubin theorem. When the continuous classical boundary control vector (CCBCV) is known, the theorem of existence a CCBOCV with equality and inequality state vector constraints (EIESVC) is stated and proved, the existence theorem of a unique solution of the adjoint couple equations (ADCEQS) associated with the state equations is studied. The Frcéhet derivative derivation of the "Hamiltonian" is obtained. Finally the necessary theorem (necessary conditions "NCs") and the sufficient theorem (sufficient conditions" SCs") for optimality of the state constrained problem are stated and proved.

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Al-Hawasy, Jamil A. Ali, and Lamyaa H. Ali. "Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem." Journal of Applied Mathematics 2020 (April 10, 2020): 1–14. http://dx.doi.org/10.1155/2020/8021635.

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The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.

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Song, Mingliang, and Shuyuan Mei. "Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations." Journal of Function Spaces 2021 (January 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/6668037.

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The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

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Brugnano, Luigi, Gianluca Frasca-Caccia, and Felice Iavernaro. "Line Integral Solution of Hamiltonian PDEs." Mathematics 7, no. 3 (March 18, 2019): 275. http://dx.doi.org/10.3390/math7030275.

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In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach.

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Amodio, Pierluigi, Luigi Brugnano, and Felice Iavernaro. "Continuous-Stage Runge–Kutta Approximation to Differential Problems." Axioms 11, no. 5 (April 21, 2022): 192. http://dx.doi.org/10.3390/axioms11050192.

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In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge–Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge–Kutta methods. In this review paper, we recall this aspect and extend it to higher-order differential problems.

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WELLS, J. C., V. E. OBERACKER, M. R. STRAYER, and A. S. UMAR. "SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD." International Journal of Modern Physics C 06, no. 01 (February 1995): 143–67. http://dx.doi.org/10.1142/s0129183195000125.

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We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

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Abarbanel, Henry D. I., Paul J. Rozdeba, and Sasha Shirman. "Machine Learning: Deepest Learning as Statistical Data Assimilation Problems." Neural Computation 30, no. 8 (August 2018): 2025–55. http://dx.doi.org/10.1162/neco_a_01094.

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We formulate an equivalence between machine learning and the formulation of statistical data assimilation as used widely in physical and biological sciences. The correspondence is that layer number in a feedforward artificial network setting is the analog of time in the data assimilation setting. This connection has been noted in the machine learning literature. We add a perspective that expands on how methods from statistical physics and aspects of Lagrangian and Hamiltonian dynamics play a role in how networks can be trained and designed. Within the discussion of this equivalence, we show that adding more layers (making the network deeper) is analogous to adding temporal resolution in a data assimilation framework. Extending this equivalence to recurrent networks is also discussed. We explore how one can find a candidate for the global minimum of the cost functions in the machine learning context using a method from data assimilation. Calculations on simple models from both sides of the equivalence are reported. Also discussed is a framework in which the time or layer label is taken to be continuous, providing a differential equation, the Euler-Lagrange equation and its boundary conditions, as a necessary condition for a minimum of the cost function. This shows that the problem being solved is a two-point boundary value problem familiar in the discussion of variational methods. The use of continuous layers is denoted “deepest learning.” These problems respect a symplectic symmetry in continuous layer phase space. Both Lagrangian versions and Hamiltonian versions of these problems are presented. Their well-studied implementation in a discrete time/layer, while respecting the symplectic structure, is addressed. The Hamiltonian version provides a direct rationale for backpropagation as a solution method for a certain two-point boundary value problem.

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13

Mehrpouya, M. A., M. Shamsi, and V. Azhmyakov. "An efficient solution of hamiltonian boundary value problems by combined gauss pseudospectral method with differential continuation approach." Journal of the Franklin Institute 351, no. 10 (October 2014): 4765–85. http://dx.doi.org/10.1016/j.jfranklin.2014.07.005.

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Iavernaro, Felice, and Francesca Mazzia. "A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods." Mathematics 9, no. 10 (May 13, 2021): 1103. http://dx.doi.org/10.3390/math9101103.

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The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

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15

Hayes, Michael, Tasso J. Kaper, Nancy Kopell, and Kinya Ono. "On the Application of Geometric Singular Perturbation Theory to Some Classical Two Point Boundary Value Problems." International Journal of Bifurcation and Chaos 08, no. 02 (February 1998): 189–209. http://dx.doi.org/10.1142/s0218127498000140.

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In this tutorial, we illustrate how geometric singular perturbation theory provides a complementary dynamical systems-based approach to the method of matched asymptotic expansions for some classical singularly-perturbed boundary value problems. The central theme is that the criterion of matching corresponds to the criterion of transverse intersection of manifolds of solutions. This theme is studied in three classes of problems, linear: ∊y″+αy′+βy=0, semilinear: ∊y″+αy′+f(y)=0, and quasilinear: ∊y″+g(y) y′+f(y)=0, on the interval [0,1], where t∈[0,1], ′=d/dt, 0<∊≪1, and general boundary conditions y(0)=A, y(1)=B hold. Chosen for their relatively simple structure, these problems provide a useful introduction to the methods of geometric singular perturbation theory that are now widely used in dynamical systems, from reaction-diffusion equations with traveling waves to perturbed N-degree-of-freedom Hamiltonian systems, and in applications to a variety of fields.

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Amodio, Pierluigi, Luigi Brugnano, and Felice Iavernaro. "Analysis of spectral Hamiltonian boundary value methods (SHBVMs) for the numerical solution of ODE problems." Numerical Algorithms 83, no. 4 (May 25, 2019): 1489–508. http://dx.doi.org/10.1007/s11075-019-00733-7.

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17

Brugnano, Luigi, Felice Iavernaro, and Donato Trigiante. "Analysis of Hamiltonian Boundary Value Methods (HBVMs): A class of energy-preserving Runge–Kutta methods for the numerical solution of polynomial Hamiltonian systems." Communications in Nonlinear Science and Numerical Simulation 20, no. 3 (March 2015): 650–67. http://dx.doi.org/10.1016/j.cnsns.2014.05.030.

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18

Zheng, Qiang. "Homoclinic Solutions for a Second-Order Nonperiodic Asymptotically Linear Hamiltonian Systems." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/417020.

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We establish a new existence result on homoclinic solutions for a second-order nonperiodic Hamiltonian systems. This homoclinic solution is obtained as a limit of solutions of a certain sequence of nil-boundary value problems which are obtained by the minimax methods. Some recent results in the literature are generalized and extended.

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Amodio, Pierluigi, Luigi Brugnano, and Felice Iavernaro. "A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)." Applied Mathematics and Computation 363 (December 2019): 124634. http://dx.doi.org/10.1016/j.amc.2019.124634.

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20

Brugnano, Luigi, Chengjian Zhang, and Dongfang Li. "A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schrödinger equation with wave operator." Communications in Nonlinear Science and Numerical Simulation 60 (July 2018): 33–49. http://dx.doi.org/10.1016/j.cnsns.2017.12.018.

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21

Brugnano, Luigi, Gianmarco Gurioli, and Yajuan Sun. "Energy-conserving Hamiltonian Boundary Value Methods for the numerical solution of the Korteweg–de Vries equation." Journal of Computational and Applied Mathematics 351 (May 2019): 117–35. http://dx.doi.org/10.1016/j.cam.2018.10.014.

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22

Mazzia, Francesca, and Giuseppina Settanni. "BVPs Codes for Solving Optimal Control Problems." Mathematics 9, no. 20 (October 17, 2021): 2618. http://dx.doi.org/10.3390/math9202618.

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Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems.

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23

Uspenskii, Alexandr Alexandrovich, and Pavel Dmitrievich Lebedev. "EUCLIDEAN DISTANCE TO A CLOSED SET AS A MINIMAX SOLUTION OF THE DIRICHLET PROBLEM FOR THE HAMILTON–JACOBI EQUATION." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 797–804. http://dx.doi.org/10.20310/1810-0198-2018-23-124-797-804.

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A combined (jointing analytical methods and computational procedures) approach to the construction of solutions in a class of boundary-value problems for a Hamiltonian-type equation is proposed. In the class of problems under consideration, the minimax (generalized) solution coincides with the Euclidean distance to the boundary set. The properties of this function are studied depending on the geometry of the boundary set and the differential properties of its boundary. Methods are developed for detecting pseudo-vertices of a boundary set and for constructing singular solution sets with their help. The methods are based on the properties of local diffeomorphisms and use partial one-sided limits. The effectiveness of the research approaches developed is illustrated by the example of solving a planar timecontrol problem for the case of a nonconvex target set with boundary of variable smoothness.

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Brugnano, Luigi, Felice Iavernaro, and Donato Trigiante. "Reprint of Analysis of Hamiltonian Boundary Value Methods (HBVMs): A class of energy-preserving Runge–Kutta methods for the numerical solution of polynomial Hamiltonian systems." Communications in Nonlinear Science and Numerical Simulation 21, no. 1-3 (April 2015): 34–51. http://dx.doi.org/10.1016/j.cnsns.2014.10.015.

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25

Mehrpouya, Mohammad A. "A modified pseudospectral method for indirect solving a class of switching optimal control problems." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 234, no. 9 (April 3, 2020): 1531–42. http://dx.doi.org/10.1177/0954410020916303.

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In the present paper, an efficient pseudospectral method for solving the Hamiltonian boundary value problems arising from a class of switching optimal control problems is presented. For this purpose, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality are derived. Then, by partitioning the time interval related to the problem under study into some subintervals, the states (and costates) and control functions are approximated on each subintervals with piecewise interpolating polynomials based on Legendre–Gauss–Radau points and a piecewise constant function, respectively. As a result, solution of the problem is turned into solution of a number of algebraic equations, in which the values of the states (and costates) and control functions at Legendre–Gauss–Radau points as well as switching and terminal points are allowed to be unknown. Numerical examples are presented at the end to show the efficiency of the proposed method.

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KUMAR, PRAVEEN, SITANSH SHARMA, and HARJINDER SINGH. "OPTIMALLY CONTROLLED VIBRATIONAL POPULATION TRANSFER IN A DIATOMIC QUANTUM SYSTEM." Journal of Theoretical and Computational Chemistry 08, no. 01 (February 2009): 157–80. http://dx.doi.org/10.1142/s0219633609004605.

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A time-dependent formulation of quantum control is employed to investigate optimally controlled vibrational population transfer in a diatomic quantum system. The problem of finding the optimal laser field needed to achieve a specific quantum transition from an initial state to the desired target goal is formulated using an iterative method and the conjugate gradient method (CGM). The time-dependent Schrödinger equation is solved with interaction of laser radiation with matter included within a dipole approximation in the Hamiltonian. Appropriate boundary conditions are chosen for the evolution problem. The control objective is chosen as the value of transition probability from an initial state to a target state. A comparison is made between the results obtained using the iterative method and the CGM for optimization. Finally, quantum bits are encoded using the vibrational states of the diatomic in the regime of low-vibrational excitation.

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27

Athanassoulis, G. A., and Ch E. Papoutsellis. "Exact semi-separation of variables in waveguides with non-planar boundaries." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2201 (May 2017): 20170017. http://dx.doi.org/10.1098/rspa.2017.0017.

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Series expansions of unknown fields Φ = ∑ φ n Z n in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions Z n are determined by solving local Sturm–Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to Z n cannot be compatible with the physical boundary conditions of Φ , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (Athanassoulis & Belibassakis 1999 J. Fluid Mech . 389 , 275–301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field Φ , valid for any smooth, non-planar boundaries and any smooth enough Φ . This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.

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Sands, Timothy. "Virtual Sensoring of Motion Using Pontryagin’s Treatment of Hamiltonian Systems." Sensors 21, no. 13 (July 5, 2021): 4603. http://dx.doi.org/10.3390/s21134603.

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To aid the development of future unmanned naval vessels, this manuscript investigates algorithm options for combining physical (noisy) sensors and computational models to provide additional information about system states, inputs, and parameters emphasizing deterministic options rather than stochastic ones. The computational model is formulated using Pontryagin’s treatment of Hamiltonian systems resulting in optimal and near-optimal results dependent upon the algorithm option chosen. Feedback is proposed to re-initialize the initial values of a reformulated two-point boundary value problem rather than using state feedback to form errors that are corrected by tuned estimators. Four algorithm options are proposed with two optional branches, and all of these are compared to three manifestations of classical estimation methods including linear-quadratic optimal. Over ten-thousand simulations were run to evaluate each proposed method’s vulnerability to variations in plant parameters amidst typically noisy state and rate sensors. The proposed methods achieved 69–72% improved state estimation, 29–33% improved rate improvement, while simultaneously achieving mathematically minimal costs of utilization in guidance, navigation, and control decision criteria. The next stage of research is indicated throughout the manuscript: investigation of the proposed methods’ efficacy amidst unknown wave disturbances.

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da Silva Fernandes, Sandro. "Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities." Mathematical Problems in Engineering 2009 (2009): 1–35. http://dx.doi.org/10.1155/2009/503168.

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Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonicJ2in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First-order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

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Yan, Jinliang, and Zhiyue Zhang. "New energy-preserving schemes using Hamiltonian Boundary Value and Fourier pseudospectral methods for the numerical solution of the “good” Boussinesq equation." Computer Physics Communications 201 (April 2016): 33–42. http://dx.doi.org/10.1016/j.cpc.2015.12.013.

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Kwon, Younghak, and Georg Menz. "Uniform LSI for the canonical ensemble on the 1D-lattice with strong, finite-range interaction." ESAIM: Probability and Statistics 24 (2020): 341–73. http://dx.doi.org/10.1051/ps/2020001.

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We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show that the canonical ensemble (ce) satisfies a uniform logarithmic Sobolev inequality (LSI). The LSI constant is uniform in the boundary data, the external field and scales optimally in the system size. This extends a classical result of H.T. Yau from discrete to unbounded, real-valued spins. It also extends prior results of Landim et al. or Menz for unbounded, real-valued spins from absent- or weak- to strong-interaction. We deduce the LSI by combining two competing methods, the two-scale approach and the Zegarlinski method. Main ingredients are the strict convexity of the coarse-grained Hamiltonian, the equivalence of ensembles and the decay of correlations in the ce.

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Phornphatcharaphong, Wutthichai, and Nawapak Eua-Anant. "Edge-Based Color Image Segmentation Using Particle Motion in a Vector Image Field Derived from Local Color Distance Images." Journal of Imaging 6, no. 7 (July 16, 2020): 72. http://dx.doi.org/10.3390/jimaging6070072.

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This paper presents an edge-based color image segmentation approach, derived from the method of particle motion in a vector image field, which could previously be applied only to monochrome images. Rather than using an edge vector field derived from a gradient vector field and a normal compressive vector field derived from a Laplacian-gradient vector field, two novel orthogonal vector fields were directly computed from a color image, one parallel and another orthogonal to the edges. These were then used in the model to force a particle to move along the object edges. The normal compressive vector field is created from the collection of the center-to-centroid vectors of local color distance images. The edge vector field is later derived from the normal compressive vector field so as to obtain a vector field analogous to a Hamiltonian gradient vector field. Using the PASCAL Visual Object Classes Challenge 2012 (VOC2012), the Berkeley Segmentation Data Set, and Benchmarks 500 (BSDS500), the benchmark score of the proposed method is provided in comparison to those of the traditional particle motion in a vector image field (PMVIF), Watershed, simple linear iterative clustering (SLIC), K-means, mean shift, and J-value segmentation (JSEG). The proposed method yields better Rand index (RI), global consistency error (GCE), normalized variation of information (NVI), boundary displacement error (BDE), Dice coefficients, faster computation time, and noise resistance.

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BARRY, J. H., and N. S. SULLIVAN. "EXACT PHASE DIAGRAMS FOR THE CONDENSATION OF A KAGOMÉ LATTICE GAS WITH THREE-PARTICLE INTERACTIONS." International Journal of Modern Physics B 07, no. 15 (July 10, 1993): 2831–57. http://dx.doi.org/10.1142/s0217979293003061.

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The condensation of a two-dimensional kagomé lattice gas having purely three-particle interactions is first theoretically investigated. The Hamiltonian Hℓg=−∈3Σ<i, j, k> ninjnk, where ∈3>0 is the strength. parameter of the short-range attractive triplet interaction, the sum is taken over all elementary triangles of the kagomé lattice, and nℓ=0, 1 is an idempotent site-occupation number. The method initially involves transforming the lattice-gas model into a generalized kagomé Ising model having both pair and triplet interactions as well as a magnetic field. Since the canonical partition function of a generalized kagomé Ising model is equivalent (aside from known prefactors) to the canonical partition function of a standard honeycomb Ising model in a magnetic field, one can deduce the exact liquid-vapor phase diagrams of the triplet-interaction kagomé lattice gas from its grand canonical partition function. As results, the liquid-vapor phase boundary (reduced chemical potential μ/∈3 vs reduced temperature T/Tc) is found to be curvilinear with a positive slope, originating at zero temperature with μ/∈3=−2/3 and analytic at its terminating critical point whose coordinates are T/Tc=1, μc/∈3=−0.64469…, where ∈3/kBTc=3.96992…. The companion coexistence curve (particle number density ρ vs. reduced temperature T/Tc) exhibits an asymmetric rounded shape with a positive-slope curvilinear diameter, and the value of the critical density ρc=0.58931…. At criticality, the expression for the coexistence curve superposes a pair of branch point singularities resulting in an infinite (vertical) slope at the critical point (T/Tc=ρ/ρc=1). The case of a kagomé lattice gas having mixed attractive pair interactions and very weak repulsive triplet interactions (Axilrod-Teller) is next considered. Perturbation analyses upon exact expressions relating to the phase diagrams reveal, over chosen ranges of reduced temperatures, that the phase boundary and the diameter of the two-phase coexistence region each have a negative slope due to the repulsive three-particle interactions.

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McLachlan, R. I., and C. Offen. "Bifurcation of solutions to Hamiltonian boundary value problems." Nonlinearity 31, no. 6 (May 9, 2018): 2895–927. http://dx.doi.org/10.1088/1361-6544/aab630.

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35

Guseinov, Gusein Sh. "Boundary value problems for nonlinear impulsive Hamiltonian systems." Journal of Computational and Applied Mathematics 259 (March 2014): 780–89. http://dx.doi.org/10.1016/j.cam.2013.06.034.

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Kayar, Zeynep, and Ağacık Zafer. "Impulsive Boundary Value Problems for Planar Hamiltonian Systems." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/892475.

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We give an existence and uniqueness theorem for solutions of inhomogeneous impulsive boundary value problem (BVP) for planar Hamiltonian systems. Green's function that is needed for representing the solutions is obtained and its properties are listed. The uniqueness of solutions is connected to a Lyapunov type inequality for the corresponding homogeneous BVP.

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37

Kirichuka, A. "Multiple Solutions of Boundary-Value Problems for Hamiltonian Systems." Journal of Mathematical Sciences 231, no. 6 (May 19, 2018): 730–44. http://dx.doi.org/10.1007/s10958-018-3847-4.

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McLachlan, Robert I., and Christian Offen. "Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation." Foundations of Computational Mathematics 20, no. 6 (March 10, 2020): 1363–400. http://dx.doi.org/10.1007/s10208-020-09454-z.

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39

Guseinov, Gusein Sh. "On the impulsive boundary value problems for nonlinear Hamiltonian systems." Mathematical Methods in the Applied Sciences 39, no. 15 (April 7, 2016): 4496–503. http://dx.doi.org/10.1002/mma.3877.

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40

Kostyukova, O. I. "An Investigation of a Linear Hamiltonian Singularly Perturbed Boundary Value Problem." Differential Equations 40, no. 5 (May 2004): 652–61. http://dx.doi.org/10.1023/b:dieq.0000043523.61257.6c.

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41

FAN, HONG-YI, HAO WU, and XUE-FEN XU. "VIBRATIONAL SPECTRUM FOR THE LINEAR LATTICE CHAIN GAINED BY VIRTUE OF THE "INVARIANT EIGEN-OPERATOR" METHOD." International Journal of Modern Physics B 19, no. 27 (October 30, 2005): 4073–80. http://dx.doi.org/10.1142/s0217979205032590.

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We propose an operator Hamiltonian (a ring of identically coupled harmonic oscillators) to describe the linear lattice chain with Born–von Karmann boundary condition. We apply the method of "invariant eigen-operator" to study this Hamiltonian and derive its invariant eigen-operator. The vibrational spectrum is thus obtained. This approach seems concise and direct and can be extended to tackle other Hamiltonian models.

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42

Modebei, Mark I., and Raphael B. Adeniyi. "Boundary Value Method for Numerically Solving Fifth-order Boundary Value Problems." Asian Research Journal of Mathematics 12, no. 2 (January 25, 2019): 1–14. http://dx.doi.org/10.9734/arjom/2019/46477.

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Cai, Hao, and Nian-Ning Huang. "The Hamiltonian formalism of the DNLS equation with a nonvanished boundary value." Journal of Physics A: Mathematical and General 39, no. 18 (April 19, 2006): 5007–14. http://dx.doi.org/10.1088/0305-4470/39/18/016.

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Elyseeva, Julia. "The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems." EPJ Web of Conferences 248 (2021): 01002. http://dx.doi.org/10.1051/epjconf/202124801002.

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In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.

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Sun, Shurong, Martin Bohner, and Shaozhu Chen. "Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems." Abstract and Applied Analysis 2010 (2010): 1–18. http://dx.doi.org/10.1155/2010/514760.

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We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for𝕋=ℝand𝕋=ℤwithin one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i)M(λ)theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

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Brahim, Benali, Mohammed Tayeb Meftah, and Rai Vandana. "Green's Function for A Piecewise Continous Potential via Integral Equations Method." Journal of the Indonesian Mathematical Society 24, no. 2 (May 18, 2018): 20–35. http://dx.doi.org/10.22342/jims.24.2.387.20-35.

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The aim of this work is to provide Green's function for the Schrodingerequation. The potential part in the Hamiltonian is piecewise continuous operator.It is a zero operator on a disk of radius "a" and a constant V0 outside this disk (intwo dimensions). We have used, to construct the Green's function, the technique ofthe integral equations. We have respected the boundary conditions of the problem.The discrete spectra of the Hamiltonian operator have been also derived.

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Watanabe, Haruki. "A Proof of the Bloch Theorem for Lattice Models." Journal of Statistical Physics 177, no. 4 (September 17, 2019): 717–26. http://dx.doi.org/10.1007/s10955-019-02386-1.

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Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.

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GogoiKonwar, Madhumita, and Arun Kumar Baruah. "Power Method on Boundary Value Problems." International Journal of Computer Applications 77, no. 11 (September 18, 2013): 46–50. http://dx.doi.org/10.5120/13442-1309.

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Akinlabi, G. O., A. A. Busari, O. G. Abatan, and O. A. Odunlami. "Numerical approximation of second-order boundary value problems via hybrid boundary value method." Journal of Physics: Conference Series 1734 (January 2021): 012022. http://dx.doi.org/10.1088/1742-6596/1734/1/012022.

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Zhang, W. X., Y. Bai, and F. Yuan. "The Hamiltonian System Method for the Stress Analysis in Axisymmetric Problems of Viscoelastic Solids." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/945238.

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With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.

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