Picard Iteration, Chebyshev Polynomials and Chebyshev-Picard Methods: Application in Astrodynamics

Junkins, John L.; Younes, Ahmad Bani; Woollands, Robyn; Bai, Xiaoli

doi:10.1007/s40295-015-0061-1

Cited by 31 publications

(20 citation statements)

References 14 publications

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“…(5) Analysis using Fourier bases. (6) Accuracy analysis of number of basis functions versus points distribution. (7) Extension to nonlinear DEs.…”

Section: Discussionmentioning

confidence: 99%

Least-Squares Solution of Linear Differential Equations

Mortari

2017

Mathematics

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This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g(t), and they satisfy the constraints, no matter what g(t) is. The second step consists of expressing g(t) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [−1, +1]. The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions' accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

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“…(5) Analysis using Fourier bases. (6) Accuracy analysis of number of basis functions versus points distribution. (7) Extension to nonlinear DEs.…”

Section: Discussionmentioning

confidence: 99%

Least-Squares Solution of Linear Differential Equations

Mortari

2017

Mathematics

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“…This section focuses on investigating the convergence characteristics of the JPI method. Similar to the CPI method and its modification, the convergence domain of the JPI method is different from the classical Picard iteration method due to the accumulation of round‐off error and approximation error based on the truncated Jacobi series of order N during the iterations. Here, based on the idea in Bai, we show that the convergence of the JPI method is not generally guaranteed, but we can provide a suitable condition to achieve it.…”

Section: Convergence Investigation Of the Jpi Methodsmentioning

confidence: 99%

“…Remark In the MCPI method and its applications, the original problem on the interval [ a , b ] is transformed to an equivalent problem on the interval [−1,1], and it may complicate the structure of the IVPs especially in higher order problems. To overcome this difficulty, we employed the shifted Jacobi polynomials on interval [ a , b ] instead of transforming the problem to the interval [−1,1].…”

Section: Vector‐matrix Formmentioning

confidence: 99%

“…Furthermore, she applied the MCPI method for numerical simulations of some important problems in aerospace engineering and optimized the approaches to utilize emerging parallel computing architectures . Also, a slight modification of CPM was introduced by Junkins et al . Thereafter, some researchers used the MCPI method for solving astronautical problems …”

Section: Introductionmentioning

confidence: 99%

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Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

Tafakkori-Bafghi

Loghmani

Heydari

et al. 2019

Math Methods in App Sciences

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In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi-Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi-Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation.Also, a vector-matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method. KEYWORDS initial value problems, Jacobi-Gauss quadrature formula, Picard iteration method, shifted Jacobi polynomials, vector-matrix form MSC CLASSIFICATION 65L05; 33C45; 33F05; 65D32 | INTRODUCTIONOne of the scientific methods for solving physics, chemistry, and engineering problems and analyzing natural phenomena is constructing mathematical models for these problems. Some of these models arise in the form of initial value problems (IVPs), namely, a differential equation with the initial condition(s). Various methods are presented to solve IVPs. These methods are classified as analytical, semianalytical, and basically numerical methods.The most commonly used semianalytical methods are the Adomian decomposition method (ADM) and its modifications, 1-4 the variational iteration method (VIM), 5-8 the hybrid spectral-variational iteration method (H-S-VIM), 9 the homotopy perturbation method (HPM), 10-12 homotopy analysis method (HAM), 13,14 and the differential transformation method (DTM) 15-17 ; on the other hand, collocation method using Chebyshev and Legendre as Jacobi polynomials, 18-29 radial basis function method, 30-34 cubic B-spline scaling function and Chebyshev cardinal function method, [35][36][37] Birkhoff-type interpolation method, 38,39 collocation method using Bernstein polynomials, 40-43 the Galerkin method, 44 the hybrid block-pulse and Chebyshev cardinal function method, 45 the linear barycentric rational interpolation method, 46 and also other numerical methods such as Runge-Kutta-Nyström method and multistep predictor-corrector method are proposed by some researchers such as Ahmad et al, 47 Jator and Lee, 48 Kosti et al, 49,50 Papadopoulos et al, 51 and Simos 52 for solving periodical oscillatory IVPs.

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“…where s = √ r 2 + z 2 , and γ r ,γ θ ,γ z are the acceleration components in cylindrical coordinate systems. These general trajectory equations of motion are vastly used in many spacecraft trajectory optimization problems [45], specifically for analyzing perturbed orbits [46] and low-thrust transfers [47]. Although the Cartesian and cylindrical forms are often used for typical spacecraft trajectory optimization problems [48], other forms based on the variation of parameters are sometimes used in spacecraft trajectory optimization.…”

Section: Inertial Coordinatesmentioning

confidence: 99%

Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions

Shirazi

Ceberio

Lozano

2018

Progress in Aerospace Sciences

118

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This article is a survey paper on solving spacecraft trajectory optimization problems. The solving process is decomposed into four key steps of mathematical modeling of the problem, defining the objective functions, development of an approach and obtaining the solution of the problem. Several subcategories for each step have been identified and described. Subsequently, important classifications and their characteristics have been discussed for solving the problems. Finally, a discussion on how to choose an element of each step for a given problem is provided.

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Picard Iteration, Chebyshev Polynomials and Chebyshev-Picard Methods: Application in Astrodynamics

Cited by 31 publications

References 14 publications

Least-Squares Solution of Linear Differential Equations

Least-Squares Solution of Linear Differential Equations

Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems

Spacecraft trajectory optimization: A review of models, objectives, approaches and solutions

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