Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

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Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

Mathematics ◽

10.3390/math8020210 ◽

2020 ◽

Vol 8 (2) ◽

pp. 210

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Hyunseok Lee ◽

Jongkyum Kwon

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Linear Combinations ◽

The Third ◽

Linearization Problem ◽

Finite Products ◽

Fourth Kind ◽

Classical Linearization

In this paper, we consider three sums of finite products of Chebyshev polynomials of two different kinds, namely sums of finite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of finite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefficients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .

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Exponentially Accurate Rayleigh–Ritz Method for Fractional Variational Problems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4028581 ◽

2015 ◽

Vol 10 (5) ◽

Cited By ~ 2

Author(s):

Zhi Mao ◽

Aiguo Xiao ◽

Dongling Wang ◽

Zuguo Yu ◽

Long Shi

Keyword(s):

Exponential Decay ◽

Numerical Approximation ◽

Exponential Convergence ◽

Ritz Method ◽

Variational Problems ◽

Basis Functions ◽

Algebraic Equations ◽

Numerical Examples ◽

Fractional Variational Problems ◽

Sturm Liouville

A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

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Deriving the explicit formula of Chebyshev polynomials of the third kind and the fourth kind

10.1063/1.5064199 ◽

2018 ◽

Author(s):

I. P. Dewi ◽

S. Utama ◽

S. Aminah

Keyword(s):

Explicit Formula ◽

Chebyshev Polynomials ◽

The Third ◽

Fourth Kind

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Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

Advances in Difference Equations ◽

10.1186/s13662-021-03588-2 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

H. Jafari ◽

S. Nemati ◽

R. M. Ganji

Keyword(s):

Differential Equations ◽

Chebyshev Polynomials ◽

Original Problem ◽

High Accuracy ◽

Operational Matrices ◽

Variable Order ◽

Algebraic Equations ◽

Numerical Tests ◽

Collocation Points ◽

Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

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Dyson’s “Favorite” Identity and Chebyshev Polynomials of the Third and Fourth Kind

Trends in Mathematics - George E. Andrews 80 Years of Combinatory Analysis ◽

10.1007/978-3-030-57050-7_6 ◽

2019 ◽

pp. 89-110

Author(s):

George E. Andrews

Keyword(s):

Chebyshev Polynomials ◽

The Third ◽

Fourth Kind

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Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

Abstract and Applied Analysis ◽

10.1155/2015/672703 ◽

2015 ◽

Vol 2015 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

W. M. Abd-Elhameed

Keyword(s):

Boundary Value Problems ◽

Jacobi Polynomials ◽

Boundary Value ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Third Order ◽

Generalized Jacobi Polynomials ◽

Nonlinear Algebraic Equations ◽

Derivatives Of

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.

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Efficient numerical treatments for a fractional optimal control nonlinear Tuberculosis model

International Journal of Biomathematics ◽

10.1142/s1793524518501152 ◽

2018 ◽

Vol 11 (08) ◽

pp. 1850115 ◽

Cited By ~ 11

Author(s):

N. H. Sweilam ◽

S. M. AL-Mekhlafi ◽

D. Baleanu

Keyword(s):

Jacobi Polynomials ◽

Euler Method ◽

Algebraic Equations ◽

Order Model ◽

Newton’S Iterative Method ◽

Generalized Euler Method ◽

Tuberculosis Model ◽

Nonlinear Algebraic Equations ◽

Numerical Treatments ◽

Objective Functional

In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.

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The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree

Symmetry ◽

10.3390/sym11020274 ◽

2019 ◽

Vol 11 (2) ◽

pp. 274 ◽

Cited By ~ 3

Author(s):

Clemente Cesarano ◽

Sandra Pinelas ◽

Paolo Ricci

Keyword(s):

Chebyshev Polynomials ◽

Orthogonal Functions ◽

The Third ◽

Half Integer ◽

Fourth Kind

New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown.

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Connection Problem for Sums of Finite Products of Chebyshev Polynomials of the Third and Fourth Kinds

Symmetry ◽

10.3390/sym10110617 ◽

2018 ◽

Vol 10 (11) ◽

pp. 617 ◽

Cited By ~ 5

Author(s):

Dmitry Dolgy ◽

Dae Kim ◽

Taekyun Kim ◽

Jongkyum Kwon

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Classical Orthogonal Polynomials ◽

Linear Combinations ◽

The Third ◽

Connection Problem ◽

Finite Products

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions F 0 2 , F 1 2 , and F 2 3 .

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