Articles de revues : « 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, n^o 6 (18 octobre 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 et Yuan-Hao Zhang. « Space–time spectral collocation method for Klein–Gordon equation ». Journal of Algorithms & ; Computational Technology 15 (janvier 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 et 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 et Abeer S. Alnahdi. « Fully Legendre spectral collocation technique for stochastic heat equations ». Open Physics 19, n^o 1 (1 janvier 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 et António M. Lopes. « A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations ». AIMS Mathematics 7, n^o 8 (2022) : 15453–70. http://dx.doi.org/10.3934/math.2022846.

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<abstract>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.</abstract>

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Bhrawy, A. H., et 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, et Ishtiaq Ali. « Numerical analysis of stochastic SIR model by Legendre spectral collocation method ». Advances in Mechanical Engineering 11, n^o 7 (juillet 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, et Md Shafiqul Islam. « Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method ». GANIT : Journal of Bangladesh Mathematical Society 35 (28 juin 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, n^o 1 (21 octobre 2015) : 151–72. http://dx.doi.org/10.1080/00207160.2015.1099632.

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Zhang, Jian Ming, et Li Jun Yi. « A Legendre-Gauss Collocation Method for the Multi-Pantograph Delay Differential Equation ». Applied Mechanics and Materials 444-445 (octobre 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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Wu, Chuanhua, et Ziqiang Wang. « The spectral collocation method for solving a fractional integro-differential equation ». AIMS Mathematics 7, n^o 6 (2022) : 9577–87. http://dx.doi.org/10.3934/math.2022532.

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<abstract>In this paper, we propose a high-precision numerical algorithm for a fractional integro-differential equation based on the shifted Legendre polynomials and the idea of Gauss-Legendre quadrature rule and spectral collocation method. The error analysis of this method is also given in detail. Some numerical examples are give to illustrate the exponential convergence of our method.</abstract>

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Bhrawy, A. H., et M. A. Abdelkawy. « Efficient Spectral Collocation Algorithm for Solving Parabolic Inverse Problems ». International Journal of Computational Methods 13, n^o 06 (2 novembre 2016) : 1650036. http://dx.doi.org/10.1142/s0219876216500365.

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This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to solve nonclassical parabolic initial-boundary value problems. Accordingly, the inverse problem is reduced into a system of ordinary differential equations (ODEs) and afterwards, such system can be solved numerically using implicit Runge–Kutta (IRK) method of order four. Four examples are introduced to demonstrate the applicability, validity, effectiveness and stable approximations of the present method. Numerical results show the exponential convergence property and error characteristics of presented method.

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Mashali-Firouzi, Mahmoud, et Mohammad Maleki. « A multiple-step adaptive pseudospectral method for solving multi-order fractional differential equations ». Nonlinear Engineering 8, n^o 1 (28 janvier 2019) : 702–18. http://dx.doi.org/10.1515/nleng-2018-0079.

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Abstract The nonlocal nature of the fractional derivative makes the numerical treatment of fractional differential equations expensive in terms of computational accuracy in large domains. This paper presents a new multiple-step adaptive pseudospectral method for solving nonlinear multi-order fractional initial value problems (FIVPs), based on piecewise Legendre–Gauss interpolation. The fractional derivatives are described in the Caputo sense. We derive an adaptive pseudospectral scheme for approximating the fractional derivatives at the shifted Legendre–Gauss collocation points. By choosing a step-size, the original FIVP is replaced with a sequence of FIVPs in subintervals. Then the obtained FIVPs are consecutively reduced to systems of algebraic equations using collocation. Some error estimates are investigated. It is shown that in the present multiple-step pseudospectral method the accuracy of the solution can be improved either by decreasing the step-size or by increasing the number of collocation points within subintervals. The main advantage of the present method is its superior accuracy and suitability for large-domain calculations. Numerical examples are given to demonstrate the validity and high accuracy of the proposed technique.

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Wang, Zhong-Qing, et Li-Lian Wang. « A Legendre-Gauss collocation method for nonlinear delay differential equations ». Discrete & ; Continuous Dynamical Systems - B 13, n^o 3 (2010) : 685–708. http://dx.doi.org/10.3934/dcdsb.2010.13.685.

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Zhao, Ting Gang, Zi Lang Zhan, Jin Xia Huo et Zi Guang Yang. « Legendre Collocation Solution to Fractional Ordinary Differential Equations ». Applied Mechanics and Materials 687-691 (novembre 2014) : 601–5. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.601.

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In this paper, we propose an efficient numerical method for ordinary differential equation with fractional order, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We apply the proposed method to multi-order fractional ordinary differential equation. Numerical results demonstrate the effectiveness of the approach.

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Li, Mingwu, Haijun Peng et Zhigang Wu. « Symplectic Irregular Interpolation Algorithms for Optimal Control Problems ». International Journal of Computational Methods 12, n^o 06 (décembre 2015) : 1550040. http://dx.doi.org/10.1142/s0219876215500401.

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Symplectic numerical methods for optimal control problems with irregular interpolation schemes are developed and the comparisons between irregular interpolation schemes and equidistant scheme are made in this paper. The irregular interpolation points, which are the collocation points usually adopted by pseudospectral (PS) methods, such as Legendre–Gauss, Legendre–Gauss–Radau, Legendre–Gauss–Lobatto and Chebyshev–Gauss–Lobatto points, are taken into consideration in this study. The symplectic numerical method with irregular points is proposed firstly. Then, several examples with different complexities highlight the differences in performance between different kinds of interpolation schemes. The numerical results show that the convergence of the present symplectic numerical methods can be obtained by increasing the number of sub-intervals or the number of interpolation points. Moreover, the comparison results show that the convergence of the symplectic numerical methods are generally independent on the type of interpolation points and the computational efficiency is not sensitive to the choice of interpolation points in general. Thus, the symplectic numerical methods with different interpolation schemes have obvious difference with the PS methods.

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Laouar, Zineb, Nouria Arar et Abdellatif Ben Makhlouf. « Spectral Collocation Method for Handling Integral and Integrodifferential Equations of n-th Order via Certain Combinations of Shifted Legendre Polynomials ». Mathematical Problems in Engineering 2022 (8 octobre 2022) : 1–10. http://dx.doi.org/10.1155/2022/9043428.

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In this study, an accurate and efficient numerical method based on spectral collocation is presented to solve integral equations and integrodifferential equations of n -th order. The method is developed using compact combinations of shifted Legendre polynomials as a spectral basis and shifted Legendre–Gauss–Lobatto nodes as collocation points to construct the appropriate algorithm that makes simple systems easy to solve. The technique treats both types of equations: linear and nonlinear equations. The study aims to provide the relevant spectral basis by the use of compact combinations, which allows us to take advantage of shifted Legendre polynomials and to reduce the dimension of the space of approximation. The reliability of the proposed algorithms is proven via different examples of several cases and the results are discussed to confirm the effectiveness of the spectral approach.

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Tavassoli Kajani, Majid, Mohammad Maleki et Adem Kılıçman. « A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra’s Population Growth Model ». Mathematical Problems in Engineering 2013 (2013) : 1–6. http://dx.doi.org/10.1155/2013/783069.

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A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra’s model for population growth of a species in a closed system. Volterra’s model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.

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Wu, Hua, Jiajia Pan et Haichuan Zheng. « Chebyshev-Legendre Spectral Domain Decomposition Method for Two-Dimensional Vorticity Equations ». Communications in Computational Physics 19, n^o 5 (mai 2016) : 1221–41. http://dx.doi.org/10.4208/cicp.scpde14.18s.

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AbstractWe extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkin method while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse. The algorithm can be implemented efficiently and in parallel way. The numerical analysis results in the case of one-dimensional multi-domain are generalized to two-dimensional case. The stability and convergence of the method are proved. Numerical results are given.

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Razzaghi, M., J. Nazarzadeh et K. Y. Nikravesh. « A collocation method for optimal control of linear systems with inequality constraints ». Mathematical Problems in Engineering 3, n^o 6 (1998) : 503–15. http://dx.doi.org/10.1155/s1024123x97000653.

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A numerical method for solving linear quadratic optimal control problems with control inequality constraints is presented in this paper. The method is based upon hybrid function approximations. The properties of hybrid functions which are the combinations of block-pulse functions and Legendre polynomials are first presented. The operational matrix of integration is then utilized to reduce the optimal control problem to a set of simultaneous nonlinear equations. The inequality constraints are first converted to a system of algebraic equalities, these equalities are then collocated at Legendre–Gauss–Lobatto nodes. An illustrative example is included to demonstrate the validity and applicability of the technique.

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Bhrawy, A. H., M. A. Alghamdi et D. Baleanu. « Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method ». Abstract and Applied Analysis 2013 (2013) : 1–9. http://dx.doi.org/10.1155/2013/513808.

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The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-legθ-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

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Li-jun, Yi, Liang Zi-qiang et Wang Zhong-qing. « Legendre-Gauss-Lobatto spectral collocation method for nonlinear delay differential equations ». Mathematical Methods in the Applied Sciences 36, n^o 18 (21 mars 2013) : 2476–91. http://dx.doi.org/10.1002/mma.2769.

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Wang, Lijun Yi and Zhongqing. « Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations ». Numerical Mathematics : Theory, Methods and Applications 7, n^o 2 (juin 2014) : 149–78. http://dx.doi.org/10.4208/nmtma.2014.1309nm.

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Sheng, Chang-Tao, Zhong-Qing Wang et Ben-Yu Guo. « A Multistep Legendre--Gauss Spectral Collocation Method for Nonlinear Volterra Integral Equations ». SIAM Journal on Numerical Analysis 52, n^o 4 (janvier 2014) : 1953–80. http://dx.doi.org/10.1137/130915200.

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Bhrawy, A. H., et M. A. Alghamdi. « A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems ». Abstract and Applied Analysis 2013 (2013) : 1–10. http://dx.doi.org/10.1155/2013/306746.

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We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

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Assari, Pouria. « The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions ». Filomat 33, n^o 3 (2019) : 667–82. http://dx.doi.org/10.2298/fil1903667a.

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Hammerstein integral equations have been arisen from mathematical models in various branches of applied sciences and engineering. This article investigates an approximate scheme to solve Fredholm-Hammerstein integral equations of the second kind. The new method uses the discrete collocation method together with radial basis functions (RBFs) constructed on scattered points as a basis. The discrete collocation method results from the numerical integration of all integrals appeared in the approach. We employ the composite Gauss-Legendre integration rule to estimate the integrals appeared in the method. Since the scheme does not need any background meshes, it can be identified as a meshless method. The algorithm of the presented scheme is interesting and easy to implement on computers. We also provide the error bound and the convergence rate of the presented method. The results of numerical experiments confirm the accuracy and efficiency of the new scheme presented in this paper and are compared with the Legendre wavelet technique.

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Bhrawy, A. H. « A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation ». Abstract and Applied Analysis 2014 (2014) : 1–10. http://dx.doi.org/10.1155/2014/636191.

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A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomialsPL,n(x)PL,m(y), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

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Urevc, Janez, et Miroslav Halilovič. « Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs ». Mathematics 9, n^o 2 (16 janvier 2021) : 174. http://dx.doi.org/10.3390/math9020174.

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In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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Zhao, Ting Jing. « Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation Method ». Applied Mechanics and Materials 687-691 (novembre 2014) : 1522–27. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1522.

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The purpose of this paper is to propose an efficient numerical method for solving Volterra-type integro-differential equation of the second kinds. This method based on Legendre-Gauss-Radau collocation, which is easy to be implemented especially for nonlinear and possesses high accuracy. Also, the method can be done by proceeding in time step by step. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

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Steppeler, J., J. Li, F. Fang, J. Zhu et P. A. Ullrich. « o3o3 : A Variant of Spectral Elements with a Regular Collocation Grid ». Monthly Weather Review 147, n^o 6 (15 mai 2019) : 2067–82. http://dx.doi.org/10.1175/mwr-d-18-0288.1.

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Abstract In this study, an alternative local Galerkin method (LGM), the o3o3 scheme, is proposed. o3o3 is a variant or generalization of the third-order spectral element method (SEM3). It uses third-order piecewise polynomials for the representation of a field and piecewise third-degree polynomials for fluxes. For the discretization, SEM3 uses the irregular Legendre–Gauss–Lobatto grid while o3o3 uses a regular collocation grid. o3o3 can be regarded as an inhomogeneous finite-difference scheme on a uniform grid, which means that the finite-difference equations are different for each group with three points. A particular version of o3o3 is set as an example of many possibilities to construct LGM schemes on piecewise polynomial spaces in which the basis functions used are continuous at corner points and function spaces having continuous derivatives are shortly discussed. We propose a standard o3o3 scheme and a spectral o3o3 scheme as alternatives to the standard method of using the quadrature approximation. These two particular schemes selected were chosen for ease of implementation rather than optimal performance. In one dimension, compared to standard SEM3, o3o3 has a larger CFL condition benefiting from the use of a regular collocation grid. While SEM3 uses the irregular Legendre–Gauss–Lobatto collocation grid, o3o3 uses a regular grid. This is considered an advantage for physical parameterizations. The shortest resolved wave is marginally smaller than that with SEM3. In two dimensions, o3o3 is implemented on a sparse grid where only a part of the points on the underlying regular grid are used for forecasting.

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Ezz-Eldien, SS, EH Doha, D. Baleanu et AH Bhrawy. « A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems ». Journal of Vibration and Control 23, n^o 1 (9 août 2016) : 16–30. http://dx.doi.org/10.1177/1077546315573916.

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The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method.

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Guo, Ben-yu, et Jian-ping Yan. « Legendre–Gauss collocation method for initial value problems of second order ordinary differential equations ». Applied Numerical Mathematics 59, n^o 6 (juin 2009) : 1386–408. http://dx.doi.org/10.1016/j.apnum.2008.08.007.

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Shan, Yingying, Wenjie Liu et Boying Wu. « Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized sine-Gordon equation ». Applied Numerical Mathematics 122 (décembre 2017) : 92–107. http://dx.doi.org/10.1016/j.apnum.2017.08.003.

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Yi, Lijun, et Zhongqing Wang. « A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations ». Calcolo 53, n^o 4 (11 décembre 2015) : 691–721. http://dx.doi.org/10.1007/s10092-015-0169-5.

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Magagula, V. M., S. S. Motsa et P. Sibanda. « A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations ». International Journal of Computational Methods 14, n^o 04 (18 avril 2017) : 1750041. http://dx.doi.org/10.1142/s0219876217500414.

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In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

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Hedayati, Mehrnoosh, Hojjat Ahsani Tehrani, Alireza Fakharzadeh Jahromi, Mohammad Hadi Noori Skandari et Dumitru Baleanu. « A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables ». AIMS Mathematics 7, n^o 6 (2022) : 9789–808. http://dx.doi.org/10.3934/math.2022545.

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<abstract>In this study, we intend to present a numerical method with highly accurate to solve the time-delay optimal control problems with delay on both the state and control variables. These problems can be seen in many sciences such as medicine, biology, chemistry, engineering, etc. Most of the methods used to work out time delay optimal control problems have high complexity and cost of computing. We extend a direct Legendre-Gauss-Lobatto spectral collocation method for numerically solving the issues mentioned above, which have some difficulties with other methods. The simple structure, convergence, and high accuracy of our approach are the advantages that distinguish it from different processes. At first, by replacing the delay functions of state and control variables in the dynamical method, we propose an equivalent system. Then discretizing the problem at the collocation points, we achieve a nonlinear programming problem. We can solve this discrete problem to obtain the approximate solutions for the main problem. Moreover, we prove the gained approximate solutions convergent to the exact optimal solutions when the number of collocation points increases. Finally, we show the capability and the superiority of the presented method by solving some numeral examples and comparing the results with those of others.</abstract>

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Rakhshan, Seyed Ali, et Sohrab Effati. « A generalized Legendre–Gauss collocation method for solving nonlinear fractional differential equations with time varying delays ». Applied Numerical Mathematics 146 (décembre 2019) : 342–60. http://dx.doi.org/10.1016/j.apnum.2019.07.016.

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Wang, Zhong-qing, et Ben-yu Guo. « Legendre-Gauss-Radau Collocation Method for Solving Initial Value Problems of First Order Ordinary Differential Equations ». Journal of Scientific Computing 52, n^o 1 (22 septembre 2011) : 226–55. http://dx.doi.org/10.1007/s10915-011-9538-7.

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Khaksari, Neda, Mahmoud Paripour et Nasrin Karamikabir. « An Effective Local Radial Basis Function Method for Solving the Delay Volterra Integral Equation of Nonvanishing and Vanishing Types ». Journal of Mathematics 2022 (22 juin 2022) : 1–11. http://dx.doi.org/10.1155/2022/1527399.

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This paper presents a numerical method for solving a class of the delay Volterra integral equation of nonvanishing and vanishing types by applying the local radial basis function method. This method converts these types of integral equations into an easily solvable system of algebraic equations. To prove the method, we use the discrete collocation method and the local radial basis function method to approximate the delay Volterra integral equation. Also, we use the nonuniform Gauss–Legendre integration method to calculate the integral part appearing in the method. In addition, the existence, uniqueness, and convergence of the solution are investigated in this paper. Finally, some numerical examples are shown to observe the accuracy and effectiveness of the numerical method. Some problems have been plotted and compared with other methods. Obtained numerical results and their comparison with other methods show the reliability and accuracy of this method.

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Rakhshan, Seyed Ali, Sohrab Effati et Ali Vahidian Kamyad. « Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation ». Journal of Vibration and Control 24, n^o 9 (14 septembre 2016) : 1741–56. http://dx.doi.org/10.1177/1077546316668467.

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The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final Time have been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The method is based upon finding the numerical solution of the Hamilton–Jacobi–Bellman equation, corresponding to this problem, by the Legendre–Gauss collocation method. The main reason for using this technique is its efficiency and simple application. Also, in this work, we use the fractional derivative in the Riemann–Liouville sense and explain our method for a fractional derivative of order of [Formula: see text]. Numerical examples are provided to show the effectiveness of the formulation and solution scheme.

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Zanger, Benjamin, Christian B. Mendl, Martin Schulz et Martin Schreiber. « Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods ». Quantum 5 (13 juillet 2021) : 502. http://dx.doi.org/10.22331/q-2021-07-13-502.

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Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6th order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

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Bhrawy, A. H., et D. Baleanu. « A Spectral Legendre–Gauss–Lobatto Collocation Method for a Space-Fractional Advection Diffusion Equations with Variable Coefficients ». Reports on Mathematical Physics 72, n^o 2 (octobre 2013) : 219–33. http://dx.doi.org/10.1016/s0034-4877(14)60015-x.

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Baus, Franziska, Axel Klar, Nicole Marheineke et Raimund Wegener. « Low-Mach-number and slenderness limit for elastic Cosserat rods and its numerical investigation ». Asymptotic Analysis 120, n^o 1-2 (6 octobre 2020) : 103–21. http://dx.doi.org/10.3233/asy-191581.

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This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e., low-Mach-number–slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a Gauss–Legendre collocation in space and an α-method in time.

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Nemati, S., P. M. Lima et S. Sedaghat. « Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations ». Applied Numerical Mathematics 149 (mars 2020) : 99–112. http://dx.doi.org/10.1016/j.apnum.2019.05.024.

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Maleki, Mohammad, Ishak Hashim, Majid Tavassoli Kajani et Saeid Abbasbandy. « An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems ». Abstract and Applied Analysis 2012 (2012) : 1–19. http://dx.doi.org/10.1155/2012/381708.

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An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.

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Ali, Ishtiaq. « Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method ». AIMS Mathematics 7, n^o 4 (2022) : 4946–59. http://dx.doi.org/10.3934/math.2022275.

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<abstract> Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L∞ norm is provided. Several numerical experiments were performed to confirm the theoretical results. </abstract>

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Magagula, Vusi Mpendulo. « On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations ». International Journal of Mathematics and Mathematical Sciences 2019 (10 juin 2019) : 1–18. http://dx.doi.org/10.1155/2019/6423294.

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In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

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Zhao, Gang, Ran Zhang, Wei Wang et Xiaoxiao Du. « Two-dimensional frictionless large deformation contact problems using isogeometric analysis and Nitsche’s method ». Journal of Computational Design and Engineering 9, n^o 1 (30 décembre 2021) : 82–99. http://dx.doi.org/10.1093/jcde/qwab070.

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ABSTRACT The simulation of large deformation contact problems has been a tough subject due to the existence of multiple nonlinearities, including geometric nonlinearity and contact interface nonlinearity. In this paper, we develop a novel method to compute the large deformation of 2D frictionless contact by employing Nitsche-based isogeometric analysis. The enforcement of contact constraints as one of the main issues in contact simulation is implemented by using Nitsche’s method, and the node-to-segment scheme is applied to the contact interface discretization. We detailedly derive the discrete formulations for 2D large deformation frictionless contact where NURBS is used for geometrical modeling and the Neo-Hookean hyperelastic materials are applied to describe the deformation of the model. The collocation method with Greville points is employed to integrate the contact interface and the classical Legendre–Gauss quadrature rule is used for contact bodies’ integration. The Lagrange multiplier method and penalty method are also implemented for comparison with Nitsche’s method. Several examples are investigated, and the obtained results are compared with that from commercial software ABAQUS to verify the effectiveness and accuracy of the Nitsche-based isogeometric analysis.

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Chen, C., X. Li, X. Shen et F. Xiao. « A high-order conservative collocation scheme and its application to global shallow-water equations ». Geoscientific Model Development 8, n^o 2 (10 février 2015) : 221–33. http://dx.doi.org/10.5194/gmd-8-221-2015.

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Abstract. In this paper, an efficient and conservative collocation method is proposed and used to develop a global shallow-water model. Being a nodal type high-order scheme, the present method solves the pointwise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve high-order accuracy. The time evolution equations to update the unknowns are derived under the flux reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the pointwise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the computational element, which are obtained as Riemann solutions between the adjacent elements. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid, where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

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Chen, C., X. Li, X. Shen et F. Xiao. « A high-order conservative collocation scheme and its application to global shallow water equations ». Geoscientific Model Development Discussions 7, n^o 4 (10 juillet 2014) : 4251–90. http://dx.doi.org/10.5194/gmdd-7-4251-2014.

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Abstract. An efficient and conservative collocation method is proposed and used to develop a global shallow water model in this paper. Being a nodal type high-order scheme, the present method solves the point-wise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve the high-order accuracy. The time evolution equations to update the unknowns are derived under the flux-reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the point-wise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the control volume which are obtained as the Riemann solutions between the adjacent cells. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

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