Special Issue "New Trends in Applications of Orthogonal Polynomials and Special Functions"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2016)

Special Issue Editors

Guest Editor
Dr. Zhongqiang Zhang

Department of Mathematical Sciences - Worcester Polytechnic Institute, MA, USA
Website | E-Mail
Interests: numerical analysis, scientific computing, spectral methods
Guest Editor
Dr. Mohsen Zayernouri

Department of Computational Mathematics, Science, and Engineering, Department of Mechanical Engineering, Michigan State University, USA
Website | E-Mail
Interests: fractional sturm-liouville problems, high-order methods for fractional PDEs, scientific computing

Special Issue Information

Dear Colleagues,

Orthogonal polynomials and special functions play an important role in developing numerical and analytical methods in mathematics, physics, and engineering. Over the past decades, this area of research has received an ever-increasing attention and has gained a growing momentum in modern topics, such as computational probability, numerical analysis, computational fluid dynamics, data assimilation, statistics, image and signal processing etc.

Orthogonal polynomials are crucial to the stability of high-order numerical methods, such as hp/spectral-element methods for ordinary and partial differential equations and fast Fourier or wavelet transformations in signal processing. These high-order numerical methods, originally formulated for partial differential equations, have been extended to integral, integro-differential equations, stochastic differential equations, and yet, these methods have not been well understood in various fields. The study of orthogonal polynomials and corresponding numerical methods helps us deepen our understanding of  these more general mathematical models that can capture non-Gaussian, non-Markovian, and non-Newtonian phenomenon.

The purpose of this special issue is to report and review the recent developments in applications of orthogonal polynomials and special functions as numerical and analytical methods. This special issue of Mathematics will contain contributions from leading experts in areas ranging from mathematical modeling, high-order numerical methods for differential, integral and integro-differential equations, stochastic differential equations, statistics, information and communication sciences and beyond.

The guest editors aim that the papers in this special issue will help understand the state-of-art high-order methods for models of complex systems and boost in-depth insights and discussion in a wide research community of related topics.

Dr. Zhongqiang Zhang
Dr. Mohsen Zayernouri
Guest Editors

Manuscript Submission Information

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Keywords

  • orthogonal functions,
  • nonlocal problems,
  • integral transforms,
  • fractional differential equations,
  • high-order numerical methods,
  • stochastic dynamics

Published Papers (5 papers)

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Research

Open AccessFeature PaperArticle An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients
Mathematics 2016, 4(2), 29; https://doi.org/10.3390/math4020029
Received: 1 February 2016 / Revised: 20 April 2016 / Accepted: 21 April 2016 / Published: 3 May 2016
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Abstract
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the [...] Read more.
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme. Full article
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Open AccessArticle POD-Based Constrained Sensor Placement and Field Reconstruction from Noisy Wind Measurements: A Perturbation Study
Mathematics 2016, 4(2), 26; https://doi.org/10.3390/math4020026
Received: 12 January 2016 / Revised: 15 March 2016 / Accepted: 21 March 2016 / Published: 14 April 2016
Cited by 2 | PDF Full-text (11129 KB) | HTML Full-text | XML Full-text
Abstract
It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in [...] Read more.
It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in the oceanography study. In this paper, we extend this approach of sensor placement and take into account measurement errors and detect possible malfunctioning sensors. We use the 24 hourly spatial wind field simulation data sets simulated using the Weather Research and Forecasting (WRF) model applied to the Maine Bay to evaluate the performances of our methods. Specifically, we use an exclusion disk strategy to distribute sensors when the extrema of POD modes are close. We demonstrate that this strategy can improve the accuracy of the reconstruction of the velocity field. It is also capable of reducing the standard deviation of the reconstruction from noisy measurements. Moreover, by a cross-validation technique, we successfully locate the malfunctioning sensors. Full article
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Open AccessArticle Recurrence Relations for Orthogonal Polynomials on Triangular Domains
Mathematics 2016, 4(2), 25; https://doi.org/10.3390/math4020025
Received: 26 December 2015 / Revised: 15 March 2016 / Accepted: 7 April 2016 / Published: 12 April 2016
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Abstract
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials Pn,r(u,v,w),r=0,1,,n,n0 on the triangular domain T={(u [...] Read more.
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , , n , n 0 on the triangular domain T = { ( u , v , w ) : u , v , w 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation. Full article
Open AccessArticle Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure
Mathematics 2015, 3(4), 1045-1068; https://doi.org/10.3390/math3041045
Received: 15 July 2015 / Revised: 15 October 2015 / Accepted: 26 October 2015 / Published: 5 November 2015
Cited by 2 | PDF Full-text (592 KB) | HTML Full-text | XML Full-text
Abstract
Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L2 convergent, non-orthonormal, [...] Read more.
Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L2 convergent, non-orthonormal, basis expansion with sufficient pointwise convergent behaviors, enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed The quantization approach pursued here defines an alternative strategy emphasizing the relevance of OPPQ to the reconstruction of the local structure of the physical states. Full article
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Open AccessArticle The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials
Mathematics 2015, 3(2), 382-397; https://doi.org/10.3390/math3020382
Received: 20 March 2015 / Accepted: 6 May 2015 / Published: 14 May 2015
Cited by 2 | PDF Full-text (216 KB) | HTML Full-text | XML Full-text
Abstract
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many [...] Read more.
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014). Full article
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