A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fractional order integro-differential equation into integral equation by fractional order integral,and transfer the integro equations into a system of linear equations by the Gausssian quadrature.We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞norm.Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and in- compressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangu- lar sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method （CSM） is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neu- mann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet bound- ary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison be- tween the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers Equations

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.

A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional derivatives.The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations.Then,a Legendre-based spectral collocation method is developed for solving the transformed system.Therefore,we can make good use of the advantages of the Gauss quadrature rule.We present the construction and analysis of the collocation method.These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations.Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid

This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid.The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid.With the proposed technique,the computation cost of collocation matrix entries is reduced from O(M2N4)to O(MN4),where N2 is the number of spherical harmonics(i.e.,size of the matrix)and M is the number of one-dimensional integration quadrature points.Numerical results demonstrate the spectral accuracy of the method.

Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel p（t, s） = （t - s）^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233：938 950], the error analysis for this approach is carried out for 0 〈 μ 〈 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-ype but also establish the error estimates under a more general regularity assumption on the exact solution.

RATIONAL SPECTRAL COLLOCATION METHOD FOR A COUPLED SYSTEM OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS

A, novel collocation method for a coupled system of singularly perturbed linear equations is presented. This method is based on rational spectral collocation method in barycentric form with sinh transform. By sinh transform, the original Chebyshev points are mapped into the transformed ones clustered near the singular points of the solution. The results from asymptotic analysis about the singularity solution are employed to determine the parameters in this sinh transform. Numerical experiments are carried out to demonstrate the high accuracy and efficiency of our method.

Korteweg-De Vries方程的Legendre时空谱配置方法

A Legendre-Legendre spectral collocation scheme is constructed for Korteweg-de Vries(KdV)equation on bounded domain by using the Legendre collocation method in both time and space,which is a nonlinear matrix equation that is changed to a nonlinear systems and can be solved by the usual fixed point iteration.Numerical results demonstrate the efficiency of the method and spectral accuracy.

On the rate of convergence of the Legendre spectral collocation method for multidimensional nonlinear Volterra-Fredholm integral equations

While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kermels are now well understood,no systematic studies of the numerical solutions of their multi-dimensional counterparts exist.In this paper,we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach.Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view.Consequently,rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors.The existence and uniqueness of the numerical solution are established.Numerical experiments are provided to support the theoretical convergence analysis.The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.

On the solution of the Human Immunodeficiency Virus(HIV)infection model using spectral collocation method

In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,Legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.

Legendre-Gauss Spectral Collocation Method for Second Order Nonlinear Delay Differential Equations

In this paper, we present and analyze a single interval Legendre-Gaussspectral collocation method for solving the second order nonlinear delay differentialequations with variable delays. We also propose a novel algorithm for the singleinterval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy ofthe proposed methods.

A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

In this paper,we firstly present a novel simple method based on a Picard integral type formulation for the nonlinear multi-dimensional variable coefficient fourthorder advection-dispersion equation with the time fractional derivative order a2(1,2).A new unknown function v(x,t)=■u(x,t)/■t is introduced and u(x,t)is recovered using the trapezoidal formula.As a result of the variable v(x,t)are introduced in each time step,the constraints of traditional plans considering the non-integer time situation of u(x,t)is no longer considered.The stability and solvability are proved with detailed proofs and the precise describe of error estimates is derived.Further,Chebyshev spectral collocation method supports accurate and efficient variable coefficient model with variable coefficients.Several numerical results are obtained and analyzed in multi-dimensional spatial domains and numerical convergence order are consistent with the theoretical value 3-a order for different a under infinite norm.

The Effects of an Inclined Magnetic Field, Brownian Motion, and Thermophoresis on the Flow of Electrically Conducting and Chemically Reacting Casson Nanofluids Using Soret-Dufour Mechanisms

This research explored the effects of an angled magnetic field, Brownian motion, and thermophoresis on the flow of an electrically conducting and chemically reacting Casson nanofluid under the influence of the Soret-Dufour mechanism. A set of partial differential equations is generated by the flow mode. The governing partial differential equations are solved numerically using the spectral collocation method after being transformed to self-similar forms. The effect of various fluid parameters on the velocity profile, temperature profile, and nanoparticle concentration is addressed. A quantitative agreement is observed when previous findings are compared to the current results. The skin friction, local Nusselt number, and local Sherwood number are also examined, and the results are presented in the table. This study discovered that the inclined magnetic field has a significant impact on the flow of the electrically conducting fluid by delaying its mobility within the boundary layer. The plastic dynamic viscosity, which acts as a barrier to fluid flow, is shown to degenerate the fluid velocity when the Casson parameter is increased. As a consequence, the findings may be used to improve thermal science instruments and increase industrial output.

Convergence analysis for delay Volterra integral equation

In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction.

A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods

The Klein-Cordon equation arises in many scientific areas of quantum mechanics and quantum field theory. In this paper a novel method based on spectral method and Jacobian free Newton method composed by generalized minimum residual （JFNGMRes） method with adaptive preconditioner will be introduced to solve nonlinear Klein-Gordon equation. In this work the nonlinear Klein-Gordon equation has been converted to a nonlinear system of algebraic equations using collocation method based on Bessel functions without any linearization, discretization and getting help of any other methods. Finally, by using JFNGMRes, solution of the nonlinear algebraic system will be achieved. To illustrate the reliability and efficiency of the proposed method, we solve some examples of the Klein-Gordon equation and compare our results with other methods.

A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

We study efficient spectral-collocation and continuation methods(SCCM)for rotating two-component Bose-Einstein condensates(BECs)and rotating two-component BECs in optical lattices,where the second kind Chebyshev polynomials are used as the basis functions for the trial function space.A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations(GPEs),where the classical tangent vector is split into two constraint conditions for the bordered linear systems.Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported.The results on the former are consistent with the published numerical results.

Numerical investigation of the flow of a micropolar fuid through a porous channel with expanding or contracting walls

In this paper,we study the flow of a micropolar fluid in a porous channel with expanding or contracting walls.First,we use spectral collocation method on the governing equations to obtain an initial approximation for the solution of equations.Then using the obtained initial approximation,we apply the homotopy analysis method to obtain a recursive fomula for the solution.