On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations

An analytical formula expressing the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is stated in a more compact form and proved in a simpler way than the formula suggested by Phillips and Karageorghis (27 (1990) 823). A new formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is given. The tensor product of ultraspherical polynomials is used to approximate a function of more than one variable. Formulae expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are described.     

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

In the spectral Petrov‐Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein‐spectral Petrov‐Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.     

Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations

A method is presented for obtaining recurrence relations for the coefficients in ultraspherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zx), -1\leq x\leq 1,z\in C,|z|\leq 1$, where y is a solution of a linear differential equation. In particular, rational approximations of $y(z)$ result if $x$ is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Some Results on the Coefficients of Integrated Expansions of Ultraspherical Polynomials and their Integrals

The formula of expressing the coefficients of an expansion of ultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

Some Results on the Coefficients of Integrated Expansions of Ultraspherical Polynomials and their Integrals

The formula of expressing the coefficients of an expansion of ultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

Polynomial particular solutions for certain partial differential operators

In this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo‐spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill‐conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz‐type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary‐type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in R² and R³ . Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 112–133, 2003     

Numerical algorithm for solving multi-pantograph delay equations on the half-line using Jacobi rational functions with convergence analysis

A new spectral Jacobi rational-Gauss collocation (JRC) method is proposed for solving the multi-pantograph delay differential equations on the half-line. The method is based on Jacobi rational functions and Gauss quadrature integration formula. The main idea for obtaining a semi-analytical solution for these equations is essentially developed by reducing the pantograph equations with their initial conditions to systems of algebraic equations in the unknown expansion coefficients. The convergence analysis of the method is analyzed. The method possesses the spectral accuracy. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Indeed, the present method is compared favorably with other methods.     

DJKM algebras and non-classical orthogonal polynomials

We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa (1983) [6] in their work on the Landau–Lifshitz equations. We show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials.     

三阶微分方程的多区域Legendre-Petrov-Galerkin谱方法

提出三阶微分方程初边值问题的多区域Legendre-Petrov-Galerkin谱方法.对于三阶线性微分方程,证明该方法全离散格式的稳定性,并给出L~2-误差估计.进而将该方法和Legendre配置方法相结合,应用于某些非线性问题.数值算例对单区域和多区域方法的结果进行比较.     

A Jacobi spectral Galerkin method for the integrated forms of fourth‐order elliptic differential equations

This article analyzes the solution of the integrated forms of fourth‐order elliptic differential equations on a rectilinear domain using a spectral Galerkin method. The spatial approximation is based on Jacobi polynomials P (x), with α, β ∈ (?1, ∞) and n the polynomial degree. For α = β, one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for α = β = ?½, α = β = 0, the Chebyshev of the first and second kinds and Legendre polynomials respectively; and for the nonsymmetric Jacobi polynomials, the two important special cases α = ?β = ±½ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two‐dimensional version of the approximations is obtained by tensor products of the one‐dimensional bases. The various matrix systems resulting from these discretizations are carefully investigated, especially their condition number. An algebraic preconditioning yields a condition number of O(N), N being the polynomial degree of approximation, which is an improvement with respect to the well‐known condition number O(N⁸) of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to N^d+1 for a d‐dimensional problem. This operational count is the best one can achieve with a spectral method. The numerical results illustrate the theory and constitute a convincing argument for the feasibility of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Adomian polynomials: A powerful tool for iterative methods of series solution of nonlinear equations

In this article, we illustrate how the Adomian polynomials can be utilized with different types of iterative series solution methods for nonlinear equations. Two methods are considered here: the differential transform method that transforms a problem into a recurrence algebraic equation and the homotopy analysis method as a generalization of the methods that use inverse integral operator. The advantage of the proposed techniques is that equations with any analytic nonlinearity can be solved with less computational work due to the properties and available algorithms of the Adomian polynomials. Numerical examples of initial and boundary value problems for differential and integro-differential equations with different types of nonlinearities show good results.     

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials \({P_{L,n}(x), \ x\in[0,L]}\) is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.     

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.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.