Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

/pdf/orthogonal-polynomials-of-several-variables-2cwkynqk1v.pdf

Orthogonal Polynomials of Several Variables

Orthogonal approximations in Sobolev spaces stability and convergence spectral methods and pseudospectral methods spectral methods for multi-dimensional and high order problems mixed spectral methods combined spectral methods spectral methods on the spherical surface.

Spectral Methods and Their Applications

Thank you for reading elements of mathematical ecology. As you may know, people have look numerous times for their favorite books like this elements of mathematical ecology, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some malicious bugs inside their computer. elements of mathematical ecology is available in our book collection an online access to it is set as public so you can download it instantly. Our books collection saves in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the elements of mathematical ecology is universally compatible with any devices to read.

Elements Of Mathematical Ecology

: Documentation is given for some subroutines which compute potentials and other functions. A set of subroutines uses rational approximations to compute Bessel functions of integral order. One subroutine uses the Debye approximation for the efficient computation of Bessel functions of complex argument and complex order. Empirical formulae have been developed to express the limiting boundaries of the modes of computation. (Author)

/pdf/computation-of-special-functions-m37di5noj9.pdf

Computation of Special Functions

On Sobolev orthogonal polynomials

Let ? be an open, simply connected, and bounded region in ? d , d???2, and assume its boundary ?? is smooth. Consider solving the elliptic partial differential equation ??Δu?+??u?=?f over ? with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. In the case the Neumann problem is uniquely solvable, and with sufficiently smooth problem parameters, the method is shown to have very rapid convergence. Numerical examples illustrate exponential convergence.

http://homepage.math.uiowa.edu/~atkinson/ftp/Spectral_Elliptic_Nmn.pdf

A spectral method for elliptic equations: the Neumann problem

We propose and analyze a numerical method for solving the nonlinear Poisson equation �Δu = f (·,u )o n the unit disk with zero Dirichlet boundary conditions. The problem is reformulated as a nonlinear integral equation. We use a Galerkin method with polynomials as approximations. The speed of convergence is shown to be very rapid; and ex- perimentally the maximum error is exponentially decreasing when it is regarded as a function of the degree of the approx- imating polynomial. 1. Introduction. In the earlier papers (2, 4) a Galerkin method was proposed, analyzed, and illustrated for the numerical solution of a Dirichlet problem for a semi-linear elliptic boundary value problem of the form

/pdf/solving-the-nonlinear-poisson-equation-on-the-unit-disk-3oomna9l5n.pdf

Solving the Nonlinear Poisson Equation on the Unit Disk

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in ℝ 2 . We show how hyperinterpolation can be used in connection with the Kumar―Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C 1,δ (D), δ > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n ―k ), for every k ∈ N if the solution of the nonlinear Poisson equation is in C°°(D).

/pdf/on-the-norm-of-the-hyperinterpolation-operator-on-the-unit-3phsjuylpi.pdf

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Let Ω be an open, simply connected, and bounded region in ℝd, d ≥ 2, and assume its boundary \(\partial\Omega\) is smooth. Consider solving an elliptic partial differential equation Lu = f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials un of degree ≤ n that is convergent to u. The transformation from Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty}( \overline{\Omega})\) and assuming \(\partial\Omega\) is a C ∞ boundary, the convergence of \(\left\Vert u-u_{n}\right\Vert _{H^{1}}\) to zero is faster than any power of 1/n. Numerical examples in ℝ2 and ℝ3 show experimentally an exponential rate of convergence.

/pdf/a-spectral-method-for-elliptic-equations-the-dirichlet-5d5sbc5vff.pdf

A spectral method for elliptic equations: the Dirichlet problem

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.

Olaf Hansen

Papers

A spectral method for elliptic equations: the Neumann problem

Solving the Nonlinear Poisson Equation on the Unit Disk

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

A spectral method for elliptic equations: the Dirichlet problem

A spectral method for elliptic equations: the Dirichlet problem