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.     

An extension of the Gegenbauer pseudospectral method for the time fractional Fokker‐Planck equation

The time fractional Fokker‐Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial‐boundary value problems of the time fractional Fokker‐Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.     

Stair Laguerre pseudospectral method for differential equations on the half line

A stair Laguerre pseudospectral method is proposed for numerical solutions of differential equations on the half line. Some approximation results are established. A stair Laguerre pseudospcetral scheme is constructed for a model problem. The convergence is proved. The numerical results show that this new method provides much more accurate numerical results than the standard Laguerre spectral method. Dedicated to Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 65N35, 41A10. Li-lian Wang: The work of this author is partially supported by The Shanghai Natural Science Foundation N. 00JC14057, The Shanghai Natural Science Foundation for Youth N. 01QN85 and The Special Funds for Major Specialities of Shanghai Education Committee. Ben-yu Guo: The work of this author is partially supported by The Special Funds for Major State Basic Research Projects of China G1999032804, The Shanghai Natural Science Foundation N. 00JC14057 and The Special Funds for Major Specialities of Shanghai Education Committee.     

Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions

In this article a numerical technique is presented for the solution of Fokker‐Planck equation. This method uses the cubic B‐spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B‐spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations

A Legendre–Gauss–Lobatto spectral collocation method is introduced for the numerical solutions of a class of nonlinear delay differential equations. An efficient algorithm is designed for the single‐step scheme and applied to the multiple‐domain case. As a theoretical result, we obtain a general convergence theorem for the single‐step case. Numerical results show that the suggested algorithm enjoys high‐order accuracy both in time and in the delayed argument and can be implemented in a robust and efficient manner. Copyright © 2013 John Wiley & Sons, Ltd.     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

A backward euler orthogonal spline collocation method for the time‐fractional Fokker–Planck equation

We formulate and analyze a novel numerical method for solving a time‐fractional Fokker–Planck equation which models an anomalous subdiffusion process. In this method, orthogonal spline collocation is used for the spatial discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1534–1550, 2015     

Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions

In this paper, we develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Some basic approximation results are established. The mixed spectral schemes are proposed for two model problems. Their convergences are proved. Numerical results demonstrate the efficiency of this new approach.     

A rational approximation and its applications to nonlinear partial differential equations on the whole line

An orthogonal system of rational functions is discussed. Some inverse inequalities, imbedding inequalities and approximation results are obtained. Two model problems are considered. The stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved. The techniques used in this paper are also applicable to other problems on the whole line. Numerical results show the efficiency of this approach.     

An h–p version of the Chebyshev spectral collocation method for nonlinear delay differential equations

We develop and analyze a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for nonlinear delay differential equations with vanishing delays. We derive an a priori error estimate in the H¹‐norm that is completely explicit with respect to the local time steps and the local polynomial degrees. Several numerical examples are provided to illustrate the theoretical results.     

Rational scaled generalized Laguerre function collocation method for solving the Blasius equation

In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate.     

Legendre-Laguerre coupled spectral element methods for second- and fourth-order equations on the half line

Some Legendre spectral element/Laguerre spectral coupled methods are proposed to numerically solve second- and fourth-order equations on the half line. The proposed methods are based on splitting the infinite domain into two parts, then using the Legendre spectral element method in the finite subdomain and Laguerre method in the infinite subdomain. C⁰ or C¹-continuity, according to the problem under consideration, is imposed to couple the two methods. Rigorous error analysis is carried out to establish the convergence of the method. More importantly, an efficient computational process is introduced to solve the discrete system. Several numerical examples are provided to confirm the theoretical results and the efficiency of the method.     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

Least‐squares spectral collocation method for the Stokes equations

First‐order system least‐squares spectral collocation methods are presented for the Stokes equations by adopting the first‐order system and modifying the least‐squares functionals in 2 . Then homogeneous Legendre and Chebyshev (continuous and discrete) functionals are shown to be elliptic and continuous with respect to appropriate product weighted norms. The spectral convergence is analyzed for the proposed methods with some numerical experiments. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 128–139, 2004     

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.     

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

In this paper, we study the numerical solution to time‐fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time‐fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

An optimization‐based domain decomposition method for the Boussinesq equations

We study an optimization based domain decomposition method for the Boussinesq equations governing natural convection problems. Domain decomposition is cast into a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the interface between solid and fluid subdomains. We showthat solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined. Finite element approximations of the solutions of the optimality system are examined. The domain decomposition method is also reformulated as a nonlinear least‐squares problem and the results of some numerical experiments are given. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 1–25, 2002     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

Jacobi rational approximation and spectral method for differential equations of degenerate type

We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach.