scholarly journals A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernels

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Timon S. Gutleb
Keyword(s):  
Open Source ◽  
Spectral Method ◽  
Jacobi Polynomial ◽  
Numerical Experiments ◽  
Exponential Convergence ◽  
Analytic Solutions ◽  
Convolution Type ◽  
Volterra Equations ◽  
Polynomial Bases ◽  
Sparsity Structure

AbstractWe present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.

scholarly journals Some dual series and triple integral equations

1969 ◽  
Vol 16 (4) ◽  
pp. 273-280 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Jacobi Polynomial ◽  
Triple Integral Equations ◽  
Dual Series Equations ◽  
Image Position

In this paper we first of all solve the dual series equationswhere ƒ(ρ) and g(ρ) are prescribed functions,is the Jacobi polynomial (2).

Solution of the Pellet Equation by use of the Orthogonal Collocation and Least Squares Methods: Effects of Different Orthogonal Jacobi Polynomials

2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Jannike Solsvik ◽  
Hugo Jakobsen
Keyword(s):  
Numerical Methods ◽  
Least Squares ◽  
Collocation Method ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Least Squares Method ◽  
Condition Numbers ◽  
Orthogonal Collocation ◽  
Point Distribution ◽  
Boundary Points

Two numerical methods in the family of weighted residual methods; the orthogonal collocation and least squares methods, are used within the spectral framework to solve a linear reaction-diffusion pellet problem with slab and spherical geometries. The node points are in this work taken as the roots of orthogonal polynomials in the Jacobi family. Two Jacobi polynomial parameters, alpha and beta, can be used to tune the distribution of the roots within the domain. Further, the internal points and the boundary points of the boundary-value problem can be given according to: i) Gauss-Lobatto-Jacobi points, or ii) Gauss-Jacobi points plus the boundary points. The objective of this paper is thus to investigate the influence of the distribution of the node points within the domain adopting the orthogonal collocation and least squares methods. Moreover, the results of the two numerical methods are compared to examine whether the methods show the same sensitivity and accuracy to the node point distribution. The notifying findings are as follows: i) The Legendre polynomial, i.e., alpha=beta=0, is a very robust Jacobi polynomial giving the better condition number of the coefficient matrix and the polynomial also give good behavior of the error as a function of polynomial order. This polynomial gives good results for small and large gradients within both slab and spherical pellet geometries. This trend is observed for both of the weighted residual methods applied. ii) Applying the least squares method the error decreases faster with increasing polynomial order than observed with the orthogonal collocation method. However, the orthogonal collocation method is not so sensitive to the choice of Jacobi polynomial and the method also obtains lower error values than the least squares method due to favorable lower condition numbers of the coefficient matrices. Thus, for this particular problem, the orthogonal collocation method is recommended above the least squares method. iii) The orthogonal collocation method show minor differences between Gauss-Lobatto-Jacobi points and Gauss-Jacobi plus boundary points.

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

Linearization of the Product of Jacobi Polynomials. II

1970 ◽  
Vol 22 (3) ◽  
pp. 582-593 ◽  
Author(s):  
George Gasper
Keyword(s):  
Harmonic Analysis ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position

Let [3, p. 170, (16)](1.1)denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by(1.2)where Rn(α, β)(x) = Pn(α, β)(x)/Pn(α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.Question 1. For which (α, β) do we have(1.3)Question 2. For which (α, β) do we have(1.4)where G depends only on (α, β)?

scholarly journals On Hermite-Fejér type interpolation

1983 ◽  
Vol 28 (1) ◽  
pp. 39-51 ◽  
Author(s):  
H.-B. Knoop ◽  
B. Stockenberg
Keyword(s):  
Jacobi Polynomial ◽  
Higher Order ◽  
Interpolation Operator ◽  
Type Interpolation

For the Hermite-Fejér interpolation operator of higher order constructed on the roots , 1 ≤ k ≤ m, of the Jacobi-polynomial it is shown that is positive for all m ∈ N, if (α, β) ∈ [−¾, −¼]2. Further there is given an bound, which implies for arbitrary f ∈ C(I) and (α, β) ∈ [−¾, −¼]2.

scholarly journals Some triple series equations involving Jacobi polynomials

1968 ◽  
Vol 16 (2) ◽  
pp. 101-108 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Jacobi Polynomial ◽  
Present Author ◽  
Jacobi Polynomials ◽  
Dual Series Equations ◽  
Image Position

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kindand the triple series equations of the second kindwhere f, f1, g, g1h and h1 are all known functions,is the Jacobi polynomial (3).

scholarly journals Positive Jacobi polynomial sums

1972 ◽  
Vol 24 (2) ◽  
pp. 109-119 ◽  
Author(s):  
Richard Askey
Keyword(s):  
Jacobi Polynomial
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