Generation of higher order pseudospectral integration matrices
ABSTRACT A new explicit expression of the higher order pseudospectral integration matrices is presented using an explicit formula for computing iterated integrals of Chebyshev polynomials. Applications to initial value problems, boundary value problems, linear integral and integro-differential equations are presented. The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods.
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- "Subsequently, Elbarbary  presented a new approach for computing the same HPIM based on the exact relation between the Chebyshev polynomials and their derivatives. Recently, Elgindy  has derived the above HPIM using an explicit formula for the iterated integrals of Chebyshev polynomials. The motivation of this paper is to provide two new HPIMs for the Chebyshev–Gauss (CG) and flipped Chebyshev–Gauss–Radau (FCGR) points, respectively, and present an exact, efficient, and stable approach for computing the HPIMs. "
ABSTRACT: The main purpose of this work is to provide new higher-order pseudospectral integration matrices (HPIMs) for the Chebyshev-type points, and present an exact, efficient, and stable approach for computing the HPIMs. The essential idea is to reduce the computation of HPIMs to that of higher-order Chebyshev integration matrices (HCIMs), and take a very simple and recursive way to compute the HCIMs efficiently and stably. Extensive numerical results show that the new approach for computing the HPIMs has better stability than that of the recently derived Elgindy’s approach for large number of Chebyshev-type points.Applied Mathematics and Computation 06/2015; 261. DOI:10.1016/j.amc.2015.03.090 · 1.60 Impact Factor
- "Elbarbary (2007) presented a modification of the El- Gendi successive integration matrix in (El-Gendi et al., 1992) which yields more accurate results than those computed by El-Gendi matrix in solving problems. Finally, Elgindy (2009) developed a new explicit expression of the higher order pseudo-spectral integration matrices. Applications to initial value problems, boundary value problems and linear integral and integro-differential equations are presented. "
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- "(iii) Greengard and Rokhlin  showed that the integral equation formulation, when applied to twopoint boundary value problems (TPBVPs) for instance, is insensitive to boundary layers, insensitive to end-point singularities , and leads to small condition numbers while achieving high computational efficiency. (iv) The use of integration for constructing the spectral approximations improves the rate of convergence of the spectral interpolants, and allows the multiple boundary conditions to be incorporated more efficiently –, . These useful features in addition to the promising results obtained by Elgindy and Smith-Miles ,  motivate us to apply the Gegenbauer integration scheme for approximating the underlying dynamical system of the CTOCP. "
ABSTRACT: In this paper we describe a novel direct optimization method using Gegenbauer-Gauss (GG) collocation for solving continuous-time optimal control (OC) problems (CTOCPs) with nonlinear dynamics, state and control constraints. The time domain is mapped onto the interval [0; 1], and the dynamical system formulated as a system of ordinary differential equations is transformed into its integral formulation through direct integration. The state and the control variables are fully parameterized using Gegenbauer expansion series with some unknown Gegenbauer spectral coefficients. The proposed Gegenbauer transcription method (GTM) then recasts the performance index, the reduced dynamical system, and the constraints into systems of algebraic equations using optimal Gegenbauer quadratures. Finally, the GTM transcribes the infinite-dimensional OC problem into a parameter nonlinear programming (NLP) problem which can be solved in the spectral space; thus approximating the state and the control variables along the entire time horizon. The high precision and the spectral convergence of the discrete solutions are verified through two OC test problems with nonlinear dynamics and some inequality constraints. The present GTM offers many useful properties and a viable alternative over the available direct optimization methods.