Orthonormal Polynomials Research Articles

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

MPSDynamics.jl: Tensor network simulations for finite-temperature (non-Markovian) open quantum system dynamics.

The MPSDynamics.jl package provides an easy-to-use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system-environment observables.

Orthogonal polynomial bases in the mixed virtual element method

AbstractThe use of orthonormal polynomial bases has been found to be efficient in preventing ill‐conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly‐shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill‐conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector‐polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high‐quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly‐shaped elements, is proposed to validate our procedures.

A numerical study for non-linear multi-term fractional order differential equations

In this article, a numerical method is presented for solving non-linear multi-order fractional differential equations (FDEs) using orthonormal Boubaker polynomials as basis functions. The operational matrix is constructed for the product of basis functions. The fractional derivative is considered in the Caputo sense. Using this technique, non-linear multi-order FDEs converted into a system of non-linear algebraic equations, which can then be solved numerically. The proposed method is implemented using MATLAB for numerical analysis. Using numerical results, validity, and accuracy of the proposed method is given and compared the results with existing literature

Parameter identification of fractional-order systems with time delays based on a hybrid of orthonormal Bernoulli polynomials and block pulse functions

Parameter identification of fractional-order systems with time delays based on a hybrid of orthonormal Bernoulli polynomials and block pulse functions

Gegenbauer Parameter Effect on Gegenbauer Wavelet Solutions of Lane-Emden Equations

In this study, we aim to solve Lane-Emden equations numerically by the Gegenbauer wavelet method. This method is mainly based on orthonormal Gegenbauer polynomials and takes advantage of orthonormality which reduces the computational cost. As a further advantage, Gegenbauer polynomials are associated with a real parameter allowing them to be defined as Legendre polynomials or Chebyshev polynomials for some values. Although this provides an opportunity to be able to analyze the problem under consideration from a wide point of view, the effect of the Gegenbauer parameter on the solution of Lane-Emden equations has not been studied so far. This study demonstrates the robustness of the Gegenbauer wavelet method on three problems of Lane-Emden equations considering different values of this parameter.

The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

In this paper, we study the large N behavior of the smallest eigenvalue λN of the (N+1)×(N+1) Hankel matrix, HN=(μj+k)0≤j,k≤N, generated by the γ dependent Jacobi weight w(z,γ)=e−γzzα(1−z)β,z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials PN(z), z∈C﹨[0,1], with the weight w(z,γ)=e−γzzα(1−z)β. Using the polynomials PN(z), we obtain the theoretical expression of λN, for large N. We also display the smallest eigenvalue λN for sufficiently large N, computed numerically.

Stable Spectral Methods for Time-Dependent Problems and the Preservation of Structure

AbstractThis paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any $$\text {C}^1(a,b)$$ C 1 ( a , b ) weight function such that $$w(a)=w(b)=0$$ w ( a ) = w ( b ) = 0 , we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $$a=-\infty $$ a = - ∞ , $$b=+\infty $$ b = + ∞ , only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function $$x^\alpha \textrm{e}^{-x}$$ x α e - x for $$x>0$$ x > 0 and $$\alpha >0$$ α > 0 and the ultraspherical weight function $$(1-x^2)^\alpha $$ ( 1 - x 2 ) α , $$x\in (-1,1)$$ x ∈ ( - 1 , 1 ) , $$\alpha >0$$ α > 0 , and establish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sensitive to the choice of $$\alpha $$ α , and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.

Numerical Methods Based on the Hybrid Shifted Orthonormal Polynomials and Block‐Pulse Functions for Solving a System of Fractional Differential Equations

This paper develops two numerical methods for solving a system of fractional differential equations based on hybrid shifted orthonormal Bernstein polynomials with generalized block‐pulse functions (HSOBBPFs) and hybrid shifted orthonormal Legendre polynomials with generalized block‐pulse functions (HSOLBPFs). Using these hybrid bases and the operational matrices method, the system of fractional differential equations is reduced to a system of algebraic equations. Error analysis is performed and some simulation examples are provided to demonstrate the efficacy of the proposed techniques. The numerical results of the proposed methods are compared to those of the existing numerical methods. These approaches are distinguished by their ability to work on the wide interval [0, a], as well as their high accuracy and rapid convergence, demonstrating the utility of the proposed approaches over other numerical methods.

Lagrange Interpolation Polynomials with Exponential Weights

Let Wρ be an exponential weight of the form Wρ (t):=|t|ρ exp(-Q(t)) on ℝ := (-∞,∞), and let Ln,ρ (F,t) for F ∈ C(ℝ) be the Lagrange interpolation polynomial at the zeros {tj,n,ρ}j=1)n of the orthonormal polynomial of degree n with respect to Wρ. A new Lp -norm convergence of Ln,ρ (F,t) is discussed.

Data Driven Approach to Determine Linear Stability of Delay Differential Equations Using Orthonormal History Functions

Abstract Delay differential equations (DDEs) appear in many applications, and determining their stability is a challenging task that has received considerable attention. Numerous methods for stability determination of a given DDE exist in the literature. However, in practical scenarios it may be beneficial to be able to determine the stability of a delayed system based solely on its response to given inputs, without the need to consider the underlying governing DDE. In this work, we propose such a data-driven method, assuming only three things about the underlying DDE: (i) it is linear, (ii) its coefficients are either constant or time-periodic with a known fundamental period, and (iii) the largest delay is known. Our approach involves giving the first few functions of an orthonormal polynomial basis as input, and measuring/computing the corresponding responses to generate a state transition matrix M, whose largest eigenvalue determines the stability. We demonstrate the correctness, efficacy and convergence of our method by studying four candidate DDEs with differing features. We show that our approach is robust to noise, thereby establishing its suitability for practical applications, wherein measurement errors are unavoidable.

Modeling Infectious Disease Trend using Sobolev Polynomials

Trend analysis plays an important role in infectious disease control. An analysis of the underlying trend in the number of cases or the mortality of a particular disease allows one to characterize its growth. Trend analysis may also be used to evaluate the effectiveness of an intervention to control the spread of an infectious disease. However, trends are often not readily observable because of noise in data that is commonly caused by random factors, short-term repeated patterns, or measurement error. In this paper, a smoothing technique that generalizes the Whittaker-Henderson method to infinite dimension and whose solution is represented by a polynomial is applied to extract the underlying trend in infectious disease data. The solution is obtained by projecting the problem to a finite-dimensional space using an orthonormal Sobolev polynomial basis obtained from Gram-Schmidt orthogonalization procedure and a smoothing parameter computed using the Philippine Eagle Optimization Algorithm, which is more efficient and consistent than a hybrid model used in earlier work. Because the trend is represented by the polynomial solution, extreme points, concavity, and periods when infectious disease cases are increasing or decreasing can be easily determined. Moreover, one can easily generate forecast of cases using the polynomial solution. This approach is applied in the analysis of trends, and in forecasting cases of different infectious diseases.

Bounds on Orthonormal Polynomials for Restricted Measures

Bounds on Orthonormal Polynomials for Restricted Measures

New adaptative numerical algorithm for solving partial integro-differential equations

The paper introduces an acurrate numerical appraoch based on orthonormal Bernoulli polynomials for solving parabolic partial integro- differential equations (PIDEs). This type of equations arises in physics and engineering. Some operational matrix are given for these polynomials and are also used to obtain the numerical solution. By this approach, the problem is transformed into a nonlinear algebraic system. Convergence analysis is given and some experiment tests are studied to examine the good accuracy of the numerical algorithm, the proposed technique is compared with some other well known methods.

On analyzing two dimensional fractional order brain tumor model based on orthonormal Bernoulli polynomials and Newton's method

Recently, modeling problems in various field of sciences and engineering with the help of fractional calculus has been welcomed by researchers. One of these interesting models is a brain tumor model. In this framework, a two dimensional expansion of the diffusion equation and glioma growth is considered. The analytical solution of this model is not an easy task, so in this study, a numerical approach based on the operational matrix of conventional orthonormal Bernoulli polynomials (OBPs) has been used to estimate the solution of this model. As an important advantage of the proposed method is to obtain the fractional derivative in matrix form, which makes calculations easier. Also, by using this technique, the problem under the study is converted into a system of nonlinear algebraic equations. This system is solved via Newton's method and the error analysis is presented. At the end to show the accuracy of the work, we have examined two examples and compared the numerical results with other works.

Orthonormal Chelyshkov polynomials for multi-term time fractional two-dimensional telegraph type equations

In this study, multi-term time fractional 2D telegraph type equations, as a new category of fractional differential equations, are introduced. The orthonormal Chelyshkov polynomials are used as basis functions to generate a collocation method for such equations. In this way, new results are obtained for the fractional and classical derivatives of these polynomials. In the proposed method, by representing the problem’s solution in terms of the orthonormal Chelyshkov polynomials (with unknown coefficients) and applying the extracted matrices, along with the collocation method, a solution is elicited for the problem under consideration. In fact, finding the solution of the initial problem turns into assigning the expansion coefficients through solving a system of algebraic equations. The accuracy of the proposed method is numerically checked with several examples.

Properties and Convergence Analysis of Orthogonal Polynomials, Reproducing Kernels, and Bases in Hilbert Spaces Associated with Norm-Attainable Operators

This research paper delves into the properties and convergence behaviors of various sequences of orthogonal polynomials, reproducing kernels, and bases within Hilbert spaces governed by norm-attainable operators. Through rigorous analysis, the study establishes the completeness of the sequences of monic orthogonal polynomials and orthonormal polynomials, highlighting their comprehensive representation and approximation capabilities in the Hilbert space. The paper also demonstrates the completeness and density attributes of the sequence of normalized reproducing kernels, showcasing its effective role in capturing the intrinsic structure of the space. Additionally, the research investigates the uniform convergence of these sequences, revealing their convergence to essential operators within the Hilbert space. Ultimately, these results contribute to both theoretical understanding and practical applications in various fields by providing insights into function approximation and representation within this mathematical framework.

Improving high-order VEM stability on badly-shaped elements

For the 2D and 3D Virtual Element Methods, a new approach to improve the conditioning of local and global matrices in the presence of badly-shaped polytopes is proposed. This new method defines the local projectors and the local degrees of freedom with respect to a set of scaled monomials recomputed on more well-shaped polytopes. This new approach is less computationally demanding than using the orthonormal polynomial basis. The effectiveness of our procedure is tested on different numerical examples characterized by challenging geometries of increasing complexity.

Data-driven sparse polynomial chaos expansion for models with dependent inputs

Polynomial chaos expansions (PCEs) have been used in many real-world engineering applications to quantify how the uncertainty of an output is propagated from inputs by decomposing the output in terms of polynomials of the inputs. PCEs for models with independent inputs have been extensively explored in the literature. Recently, different approaches have been proposed for models with dependent inputs to expand the use of PCEs to more real-world applications. Typical approaches include building PCEs based on the Gram–Schmidt algorithm or transforming the dependent inputs into independent inputs. However, the two approaches have their limitations regarding computational efficiency and additional assumptions about the input distributions, respectively. In this paper, we propose a data-driven approach to build sparse PCEs for models with dependent inputs without any distributional assumptions. The proposed algorithm recursively constructs orthonormal polynomials using a set of monomials based on their correlations with the output. The proposed algorithm on building sparse PCEs not only reduces the number of minimally required observations but also improves the numerical stability and computational efficiency. Four numerical examples are implemented to validate the proposed algorithm. The source code is made publicly available for reproducibility.

A new graph-theoretic approach for the study of the surface of a thin sheet of a viscous liquid model

In this study, we considered the model of the surface of a thin sheet of viscous liquid which is known as the Buckmaster equation (BME), and presented a new graph-theoretic polynomial collocation method named the Hosoya polynomial collocation method (HPCM) for the solution of nonlinear Buckmaster equation. In the literature, the majority of the developed numerical methods considered small time step sizes like 0.01s and 0.05s to obtain relatively accurate approximations for the nonlinear BME. This study focused on optimizing the time step sizes by adopting bigger time steps sizes like 1.0s,3.0s, and 5.0s, etc without adversely affecting accuracy. First, using the Gram- Schmidt process, we generated the orthonormal functions from the Hosoya polynomial of the path graph. Then developed the functional integration matrix using orthonormal Hosoya polynomials of path graphs. With this active matrix-involved method, the nonlinear BMEs are transformed into a system of nonlinear equations and solved the equations by Newton’s method through the Mathematica software for unknown coefficients. The exactness of the proposed strategy is tested with two numerical examples. The acquired results contrasted with the current analytical solutions to these problems. Also provided the convergence analysis, comparison of error norms, graphical plots of correlation of HPCM results, and the results of other numerical methods in the literature to validate the productivity and accuracy of the newly developed HPCM.