High-order numerical solution of viscous Burgers' equation using an extended Cole–Hopf barycentric Gegenbauer integral pseudospectral method

Dahy, Sayed A.; Elgindy, Kareem T.

doi:10.1080/00207160.2021.1919302

Cited by 11 publications

(7 citation statements)

References 34 publications

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“…The presence of the parameter λ enriches their implementation in spectral and pseudospectral methods to solve many types of differential problems. Gegenbauer polynomials provide a convenient basis functions for polynomial approximations of many models involving differential, integral and integro-differential equations [14][15][16][17][18].…”

Section: Gegenbauer (Ultraspherical) Cmentioning

confidence: 99%

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Youssri

2021

Fractal Fract

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Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

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Section: Gegenbauer (Ultraspherical) Cmentioning

confidence: 99%

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Youssri

2021

Fractal Fract

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“…The current methods do not suffer from this limitation, as they generally remain stable for large mesh grids and certain parameter ranges, as we will demonstrate later in Sections 5 and 9. For a comprehensive review on IPS and IS methods, the reader may consult [7][8][9][10][11][12][13], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A direct integral pseudospectral method for solving a class of infinite-horizon optimal control problems using Gegenbauer polynomials and certain parametric maps

Elgindy

Refat

2023

MATH

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<abstract><p>We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Three illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS and IPS methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid, regardless of whether they are of Gauss/Gauss-Radau type or equally spaced.</p></abstract>

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“…For a comprehensive survey on the excellent virtues of integral reformulations and FIMs used in our work, we refer to [39][40][41][42][43][44] and the references therein. Clear expositions of the IPS methods can also be found in [45][46][47][48][49][50] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Optimal Periodic Control of Unmanned Aerial Vehicles Based on Fourier Integral Pseudospectral and Edge-Detection Methods

Elgindy¹

2023

Preprint

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This study describes the development of a novel numerical optimization framework to maximize the endurance of unmanned aerial vehicles (UAVs). We address the problem of numerically determining the optimal thrust and cruise angle of attack in a two-dimensional space for a UAV under certain initial, periodic, and bound constraints. The time horizon of the free final time optimal control problem (OCP) is first normalized, and the normalized OCP in integral form is discretized in physical space into a nonlinear programming problem (NLP) using Fourier collocation and quadrature based on equispaced points. Great attention in this work is placed on the accurate detection of jump discontinuities and resolving the thrust history effectively directly from the Fourier pseudospectral (FPS) data through a novel edge-detection method without any smoothing techniques. The numerical results demonstrate that the proposed method is simple, stable, and easy to implement.

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High-order numerical solution of viscous Burgers' equation using an extended Cole–Hopf barycentric Gegenbauer integral pseudospectral method

Cited by 11 publications

References 34 publications

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

A direct integral pseudospectral method for solving a class of infinite-horizon optimal control problems using Gegenbauer polynomials and certain parametric maps

Optimal Periodic Control of Unmanned Aerial Vehicles Based on Fourier Integral Pseudospectral and Edge-Detection Methods

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