Composite generalized Laguerre–Legendre pseudospectral method for Fokker–Planck equation in an infinite channel

In this paper, we propose a composite generalized Laguerre–Legendre pseudospectral method for the Fokker–Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre–Legendre–Gauss–Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types.     

A pseudospectral method for nonlinear Duffing equation involving both integral and non‐integral forcing terms

The Legendre pseudospectral method is developed for the numerical solution of nonlinear Duffing equation involving both integral and non‐integral forcing terms. By using differentiation matrix, the problem is reduced to the solution of a system of algebraic equations. The method is general, easy to implement, and yields very accurate results. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. Copyright © 2014 John Wiley & Sons, Ltd.     

Determination of a control function in three‐dimensional parabolic equations by Legendre pseudospectral method

A Legendre pseudospectral method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x, y, z, t) in a diffusion equation in a three‐dimensional region. The diffusion equation is to be solved subject to suitably prescribed initial‐boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This extra condition considered as the integral overspecification over the spacial domain. For discretizing the problem, after homogenization of the boundary conditions, we apply the Legendre pseudospectral method in a matrix based manner. As a results a system of nonlinear differential algebraic equations is generated. Then by using suitable transformation, the problem will be converted to a homogeneous time varying system of linear ordinary differential equations. Also a pseudospectral method for efficient solving of the resulted system of ordinary differential equations is proposed. The solution of this system gives the approximation to values of u and p. The matrix based structure of the present method makes it easy to implement. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 74‐93, 2012     

Optimal control computation for integro-differential aerodynamic equations

In order to maintain spectrally accurate solutions, the grids on which a non-linear physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a pseudospectral computational method of solving integro-differential systems with quadratic performance index. The proposed method is based on the idea of relating grid points to the structure of orthogonal interpolating polynomials. The optimal control and the trajectory are approximated by the m th degree interpolating polynomial. This interpolating polynomial is spectrally constructed using Legendre–Gauss–Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. The integrals involved in the formulation of the problem are calculated by Gauss–Lobatto integration rule, thereby reducing the problem to a mathematical programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. An illustrative example is included to confirm the convergence of the pseudospectral Legendre method, and a comparison is made with an existing result in the literature. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

Quadrature discretization method in tethered satellite control

A pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented. Traditional pseudospectral methods expand the state and control trajectories using global Lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the Jacobi family, such as Legendre or Chebyshev polynomials, which are orthogonal with respect to a specific weight function over a fixed interval. Although these methods have many advantages, the location of the collocation points are more or less fixed. The method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval. The trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method.     

Numerical solution of two‐dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral Legendre method

In this research, the problem of solving the two‐dimensional parabolic equation subject to a given initial condition and nonlocal boundary specifications is considered. A technique based on the pseudospectral Legendre method is proposed for the numerical solution of the studied problem. Several examples are given and the numerical results are shown to demonstrate the efficiently of the newly proposed method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Recovering a time‐dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method

The aim of this article is to discuss the problem of finding the unknown function u(x,t) and the time‐dependent coefficient a(t) in a parabolic partial differential equation. The pseudospectral Legendre method is employed to solve this problem. The results of numerical experiments are given. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

This article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt) . Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system

In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.     

A LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVINGNONLINEAR KLEIN-GORDON EQUATION

1.IntroductionAsweknow,theKlein-Gordonequationisanimportantmathematicalmodelinquantummechanics.Itisoftheformwherefi=(--1,1)",x=(xl,x25'jx.),bisarealnumber.AssumethatU000=UI(x)=0onOnandAsin[1],itcanbeshownthatifUOEHI(n)nH(n),UIEL'(fl)andfEL'(o,T;L'(n)),then(1.1)hasuniquesolutionUELoo(o,T;Hi(n)nH(fl)).oftheotherhand,somefinitedifferenceschemeswereproposedin12,31withstrictproo]ofgeneralizedstabilityandconvergence.TheirnumericalsolutionskeepthediscretEconservations.Oneofspecialcases(…     

Uniform convergence of the legendre spectral method for the Zakharov equations

The Legendre spectral and pseudospectral approximations are proposed for the standard Zakharov equations with initial boundary conditions. Optimal H¹ error estimate of the method is given for both semidiscrete and fully discrete schemes. The uniform convergence for the parameter ε relative to the acoustic speed is proved. Moreover, the multidomain Legendre spectral scheme is also constructed, which can be implemented in parallel. Finally, numerical results in single domain and multidomain verify the high accuracy of the Legendre spectral method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Modified Laguerre pseudospectral method refined by multidomain Legendre pseudospectral approximation

A modified Laguerre pseudospectral method is proposed for differential equations on the half-line. The numerical solutions are refined by multidomain Legendre pseudospectral approximation. Numerical results show the spectral accuracy of this approach. Some approximation results on the modified Laguerre and Legendre interpolations are established. The convergence of proposed method is proved.     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

A hybrid approximation method for solving Hutchinson’s equation

The hybrid function approximation method for solving Hutchinson’s equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre-Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson’s equation.