Computing Special Functions by Using Quadrature Rules

The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, differential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions.     

On nonoscillating integrals for computing inhomogeneous Airy functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation . The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of . In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

     

New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

In this paper, we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper, we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints.     

Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization

We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x ₁, . . . , x _N ) with certain separability and regularity properties. Assuming that f is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either f is pseudoconvex in every pair of coordinate blocks from among N-1 coordinate blocks or f has at most one minimum in each of N-2 coordinate blocks. If f is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity of f and compactness of the level set may be relaxed further. These results are applied to derive new (and old) convergence results for the proximal minimization algorithm, an algorithm of Arimoto and Blahut, and an algorithm of Han. They are applied also to a problem of blind source separation.     

A combination of potential reduction steps and steepest descent steps for solving convex programming problems

This paper studies the possibility of combining interior point strategy with a steepest descent method when solving convex programming problems, in such a way that the convergence property of the interior point method remains valid but many iterations do not request the solution of a system of equations. Motivated by this general idea, we propose a hybrid algorithm which combines a primal–dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. Our numerical experiments are also reported. Copyright © 2002 John Wiley & Sons, Ltd.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.