The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers

Abstract

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initialboundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

Orthogonal polynomials on a class of planar algebraic curves

“…14) and(5.15) satisfied by the OPs and the structures of the matrices given above imply that for 0 ≤ 𝑛 ≤ 𝑑 − 2 and 𝑘 = 1, … , 𝑛 + recurrences satisfied by the OPs for the cases 𝑛 = 𝑑 − 1, 𝑘 = 1, … , 𝑑 and 𝑛 ≥ 𝑑, 𝑘 = 1, … , 𝑑 can be found similarly using (5.43)-(5…”

“…4 These OPs were used in applications, such as the statistics of determinantal point processes, the computation of Stieltjes transforms, multivariate function approximation and orthogonal series, [5][6][7][8][9][10] the approximation of singular or nearly singular functions of one variable (including those that arise as solutions to differential equations), the explicit solution to wave equations, 11 spectral methods for partial differential equations on trapeziums, disk slices and spherical caps, 12,13 and a spectral method for fractional integral equations. 14 In this paper, we generalize both the types of curves on which OPs are constructed as well as their method of construction, in particular, to OPs defined on curves of the form 𝑦 𝑚 = 𝜙(𝑥) in ℝ 2 where 𝑚 = 1, 2 and 𝜙 is a polynomial of arbitrary degree 𝑑. Unlike the previously mentioned studies, our constructions of OPs are computational, not explicit.…”

Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

“…Consequently, the inner product spaces are the natural way to give sense to infinite sums of vectors. Now our question of finding the basis set for infinite dimensional vector spaces turns into finding an orthonormal set u n fg such that it spans whole space V. If it happens then the set u n fg is called the orthonormal basis for V. This idea is very useful in many applications of mathematics, particularly in approximation theory, see [5][6][7][8][9][10][11][12].…”

Orthogonal Polynomials on Planar Cubic Curves

“…The OPs were constructed explicitly on the wedge and the square [11], planar quadratic curves [14], cubic curves [6], as well as on and inside quadratic surfaces of revolution [12]. These OPs were used in applications, such as the statistics of determinantal point processes, the computation of Stieltjes transforms, multivariate function approximation and orthogonal series [21,22,23,24,25,26], the approximation of singular or nearly singular functions of one variable (including those that arise as solutions to differential equations), the explicit solution to wave equations [15], spectral methods for partial differential equations on trapeziums, disk slices and spherical caps [17,18] and a spectral method for fractional integral equations [16].…”