Relevant bibliographies by topics / Gauss-Legendre collocation method

Academic literature on the topic 'Gauss-Legendre collocation method'

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Published: 10 March 2023

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Journal articles on the topic "Gauss-Legendre collocation method"

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Mai, Heng. "Convergence for the optimal control problems using collocation at Legendre-Gauss points." Transactions of the Institute of Measurement and Control 44, no. 6 (October 18, 2021): 1263–74. http://dx.doi.org/10.1177/01423312211043335.

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The convergence of the novel Legendre-Gauss method is established for solving a continuous optimal control problem using collocation at Legendre-Gauss points. The method allows for changes in the number of Legendre-Gauss points to meet the error tolerance. The continuous optimal control problem is first discretized into a nonlinear programming problem at Gauss collocations by the Legendre-Gauss method. Subsequently, we prove the convergence of the Legendre-Gauss algorithm under the assumption that the continuous optimal control problem has a smooth solution. Compared with those of the shooting method, the single step method, and the general pseudospectral method, the numerical example shows that the Legendre-Gauss method has higher computational efficiency and accuracy in solving the optimal control problem.
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Zhang, Ping, Te Li, and Yuan-Hao Zhang. "Space–time spectral collocation method for Klein–Gordon equation." Journal of Algorithms & Computational Technology 15 (January 2021): 174830262110653. http://dx.doi.org/10.1177/17483026211065385.

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By using the Legendre–Laguerre collocation method, we can construct a spectral collocation scheme to solve the Klein–Gordon equation on the half-line. The Laguerre function collocation method (based on the Lagrange interpolation) in space and the Legendre–Gauss–Lobatto collocation method in time are used. A Newton iterative algorithm is provided. The numerical results demonstrate the high efficiency and accuracy of suggested algorithms.
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Doha, E. H., D. Baleanu, A. H. Bhrawy, and R. M. Hafez. "A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/816473.

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A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.
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Abdelkawy, Mohamed A., Hijaz Ahmad, Mdi Begum Jeelani, and Abeer S. Alnahdi. "Fully Legendre spectral collocation technique for stochastic heat equations." Open Physics 19, no. 1 (January 1, 2021): 921–31. http://dx.doi.org/10.1515/phys-2021-0073.

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Abstract For the stochastic heat equation (SHE), a very accurate spectral method is considered. To solve the SHE, we suggest using a shifted Legendre Gauss–Lobatto collocation approach in combination with a shifted Legendre Gauss–Radau collocation technique. A comprehensive theoretical formulation is offered, together with numerical examples, to demonstrate the technique’s performance and competency. The scheme’s superiority in tackling the SHE is demonstrated.
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Algahtani, Obaid, M. A. Abdelkawy, and António M. Lopes. "A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations." AIMS Mathematics 7, no. 8 (2022): 15453–70. http://dx.doi.org/10.3934/math.2022846.

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<abstract><p>A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.</p></abstract>
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Bhrawy, A. H., and M. A. Alghamdi. "Approximate Solutions of Fisher's Type Equations with Variable Coefficients." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/176730.

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The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.
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Khan, Sami Ullah, and Ishtiaq Ali. "Numerical analysis of stochastic SIR model by Legendre spectral collocation method." Advances in Mechanical Engineering 11, no. 7 (July 2019): 168781401986291. http://dx.doi.org/10.1177/1687814019862918.

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This article represents Legendre spectral collocation method based on Legendre polynomials to solve a stochastic Susceptible, infected, Recovered (SIR) model. The Legendre polynomials on stochastic SIR model that convert it to a system of equations has been applied and then solved by the Legendre spectral method, which leads to excellent accuracy and convergence by implementing Legendre–Gauss–Lobatto collocation points permitting to generate coarser meshes. The numerical results for both the deterministic and stochastic models are presented. In case of probably small noise, the verge dynamics is analyzed. The large noise will show eradication of disease, which controls disease spreading. Various graphical results demonstrate the effectiveness of the proposed method to SIR model.
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Farzana, Humaira, and Md Shafiqul Islam. "Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method." GANIT: Journal of Bangladesh Mathematical Society 35 (June 28, 2016): 95–112. http://dx.doi.org/10.3329/ganit.v35i0.28574.

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We propose Chebyshev-Legendre spectral collocation method for solving second order linear and nonlinear eigenvalue problems exploiting Legendre derivative matrix. The Sturm-Liouville (SLP) problems are formulated utilizing Chebyshev-Gauss-Lobatto (CGL) nodes instead of Legendre Gauss-Lobatto (LGL) nodes and Legendre polynomials are taken as basis function. We discuss, in details, the formulations of the present method for the Sturm-Liouville problems (SLP) with Dirichlet and mixed type boundary conditions. The accuracy of this method is demonstrated by computing eigenvalues of three regular and two singular SLP's. Nonlinear Bratu type problem is also tested in this article. The numerical results are in good agreement with the other available relevant studies.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 95-112
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Liang, Hui. "hp-Legendre–Gauss collocation method for impulsive differential equations." International Journal of Computer Mathematics 94, no. 1 (October 21, 2015): 151–72. http://dx.doi.org/10.1080/00207160.2015.1099632.

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Zhang, Jian Ming, and Li Jun Yi. "A Legendre-Gauss Collocation Method for the Multi-Pantograph Delay Differential Equation." Applied Mechanics and Materials 444-445 (October 2013): 661–65. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.661.

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In this paper, we propose a single-interval Legendre-Gauss collocation method for multi-pantograph delay differential equations. Numerical experiments are carried out to illustrate the high order accuracy of the numerical scheme.
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Dissertations / Theses on the topic "Gauss-Legendre collocation method"

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FRASCA, CACCIA GIANLUCA. "A new efficient implementation for HBVMs and their application to the semilinear wave equation." Doctoral thesis, 2015. http://hdl.handle.net/2158/992629.

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In this thesis we have provided a detailed description of the low-rank Runge-Kutta family of Hamiltonian Boundary Value Methods (HBVMs) for the numerical solution of Hamiltonian problems. In particular, we have studied in detail their main property: the conservation of polynomial Hamiltonians, which results into a practical conservation for generic suitably regular Hamiltonians. This property turns out to play a fundamental role in some problems where the error on the Hamiltonian, usually obtained even when using a symplectic method, would be not negligible to the point of affecting the dynamics of the numerical solution. The research developed in this thesis has addressed two main topics. The first one is a new procedure, based on a particular splitting of the matrix defining the method, which turns out to be more effective of the well-known blended-implementation, as well as of a classical fixed-point iteration when the problem at hand is stiff. This procedure has been applied also to second order problems with separable Hamiltonian function, resulting in a cheaper computational cost. The second topic addressed is the application of HBVMs for the full discretization of a method of lines approach to numerically solve Hamiltonian PDEs. In particular, we have considered the semilinear wave equation coupled with either periodic, Dirichlet or Neumann boundary conditions, and the application of a (practically) energy conserving HBVM method to the semi-discrete problem obtained by means of a second order finite-difference approximation in space. When the problem is coupled with periodic boundary conditions we have also considered the case of higher-order finite-difference spatial discretizations and the case when a Fourier-Galerkin method is used for the spatial semi-discretization. The proposed methods are able to provide a numerical solution such that the energy (which can be conserved or not, depending on the assigned boundary conditions) practically satisfies its prescribed variation in time. A few numerical tests for the sine-Gordon equation have given evidence that, for some problems, there is an effective advantage in using an energy-conserving method for the time integration, with respect to the use of a symplectic one. Moreover, even though HBVMs are implicit method, their computational cost for the considered problem turns out to be competitive even with respect to that of explicit solvers of the same order, which, furthermore, may suffer from stepsize restrictions due to stability reasons, whereas HBVMs are A-stable.
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Conference papers on the topic "Gauss-Legendre collocation method"

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Moreno-Martin, Siro, Lluis Ros, and Enric Celaya. "A Legendre-Gauss Pseudospectral Collocation Method for Trajectory Optimization in Second Order Systems." In 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2022. http://dx.doi.org/10.1109/iros47612.2022.9981255.

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Kajani, M. Tavassoli, and I. Gholampoor. "A direct multi-step Legendre-Gauss collocation method for high-order Volterra integro-differential equation." In RECENT DEVELOPMENTS IN NONLINEAR ACOUSTICS: 20th International Symposium on Nonlinear Acoustics including the 2nd International Sonic Boom Forum. AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4934331.

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Hu, Xiaosong, Hector E. Perez, and Scott J. Moura. "Battery Charge Control With an Electro-Thermal-Aging Coupling." In ASME 2015 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/dscc2015-9705.

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Efficient and safe battery charge control is an important prerequisite for large-scale deployment of clean energy systems. This paper proposes an innovative approach to devising optimally health-conscious fast-safe charge protocols. A multi-objective optimal control problem is mathematically formulated via a coupled electro-thermal-aging battery model, where electrical and aging sub-models depend upon the core temperature captured by a two-state thermal sub-model. The Legendre-Gauss-Radau (LGR) pseudo-spectral method with adaptive multi-mesh-interval collocation is employed to solve the resulting highly nonlinear six-state optimal control problem. Charge time and health degradation are therefore optimally traded off, subject to both electrical and thermal constraints. Minimum-time, minimum-aging, and balanced charge scenarios are examined in detail. The implications of the upper voltage bound, ambient temperature, and cooling convection resistance to the optimization outcome are investigated as well.
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Bartali, Lorenzo, Marco Gabiccini, and Massimo Guiggiani. "A pnh-Adaptive Refinement Procedure for Numerical Optimal Control Problems." In ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/detc2022-89376.

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Abstract This paper presents an automatic procedure to enhance the accuracy of the numerical solution of an optimal control problem (OCP) discretized via direct collocation at Gauss-Legendre points. First, a numerical solution is obtained by solving a nonlinear program (NLP). Then, the method evaluates its accuracy and adaptively changes both the degree of the approximating polynomial within each mesh interval and the number of mesh intervals until a prescribed accuracy is met. The number of mesh intervals is increased for all state vector components alike, in a classical fashion. Instead, improving on state-of-the-art procedures, the degrees of the polynomials approximating the different components of the state vector are allowed to assume, in each finite element, distinct values. This explains the pnh definition, where n is the state dimension. Instead, in the literature, the degree is always raised to the highest order for all the state components, with a clear waste of resources. Numerical tests on three OCP problems highlight that, under the same maximum allowable error, by independently selecting the degree of the polynomial for each state, our method effectively picks lower degrees for some of the states, thus reducing the overall number of variables in the NLP. Accordingly, various advantages are brought about, the most remarkable being: (i) an increased computational efficiency for the final enhanced mesh with solution accuracy still within the specified tolerance, (ii) a reduced risk of being trapped by local minima due to the reduced NLP size.
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