Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

Abdelhakem, M.; Ahmed, Abd El-Latif S.; El-Kady, M.

doi:10.18576/msl/080201

Cited by 25 publications

(2 citation statements)

References 2 publications

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“…The unique system of MCPs is defined by: Using relation ( 8 ), we have: The recursive formula of MCPs is: The the recursive relation for MCPs in terms of its derivatives is Abdelhakem et al. 2019b : The MCPs constitute an orthogonal basis w.r.t. (the same weight of CPs): …”

Section: Preliminaries and Notationsmentioning

confidence: 99%

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Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

et al. 2022

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We introduce new differentiation matrices based on the pseudospectral collocation method. Monic Chebyshev polynomials (MCPs) were used as trial functions in differentiation matrices (D-matrices). Those matrices have been used to approximate the solutions of higher-order ordinary differential equations (H-ODEs). Two techniques will be used in this work. The first technique is a direct approximation of the H-ODE. While the second technique depends on transforming the H-ODE into a system of lower order ODEs. We discuss the error analysis of these D-matrices in-depth. Also, the approximation and truncation error convergence have been presented to improve the error analysis. Some numerical test functions and examples are illustrated to show the constructed D-matrices’ efficiency and accuracy.

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Section: Preliminaries and Notationsmentioning

confidence: 99%

“…Spectral methods solved ODEs by expressing these equations in terms of a series of known functions (Abdelhakem et al. 2019b ). The basic concept of any spectral method is to use trial functions, called basis or expansion approximating functions.…”

Section: Introductionmentioning

confidence: 99%

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

et al. 2022

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A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

Youssri

Abd-Elhameed

Abdelhakem

2021

Math Methods in App Sciences

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New modified shifted Chebyshev polynomials (MSCPs) have been constructed over the interval [α, β]. These polynomials are utilized as basis functions with the application of the spectral collocation method. The operational matrix of integer order derivatives of these polynomials is introduced. The elements of this matrix are explicitly given. The introduced operational matrix along with the collocation method is used to find a direct solver of linear/nonlinear class of IVPs. Furthermore, the convergence and error analysis of the modified Chebyshev expansion are discussed. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithm. The obtained results from the tested numerical examples are convincing. Also, the introduced approximate solutions are very close to the analytical ones.

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Numerical Study of a Nonlinear High Order Boundary Value Problems Using Genocchi Collocation Technique

El-Gamel

Mohamed

Adel

2022

Int. J. Appl. Comput. Math

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Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

Cited by 25 publications

References 2 publications

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials

Numerical Study of a Nonlinear High Order Boundary Value Problems Using Genocchi Collocation Technique

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