Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

Sohaib, Muhammad; Haq, Sirajul; Mukhtar, Safyan; Khan, Ilyas

doi:10.1016/j.rinp.2018.01.065

Cited by 23 publications

(9 citation statements)

References 18 publications

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“…then equations (25) construct a system of algebraic equations. Particularly, linear or non-linear system of equations depend on formulation of HBVPs " (18) and ( 20)" of the problem. Hint, we shall be replacing some rows of D ij by the conditions (24).…”

Section: Proposed Methodsmentioning

confidence: 99%

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Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

Abdelhakem,

Ahmed,

El-kady

2021

Preprint

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This paper is focused on performing a new method for solving linear and nonlinear higher-order boundary value problems (HBVPs). This direct numerical method based on spectral method. The trial function of this method is the Monic Chebyshev polynomials (MCPs). This method was relying on derivative of MCPs which explicit in the series expansion. The advantage of this method is solved HBVPs without transforming it to a system of lower-order ordinary differential equations (ODEs). This method supported by examples of HBVPs in wide application. The mentioned examples showed that the proposed method is efficient and accurate.

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Section: Proposed Methodsmentioning

confidence: 99%

“…The fourth-order BVPs solved numerically in many papers by a different methods [15], [16] and [17]. The sixth-order is mentioned in [18] by Legendre wavelet collocation method, also solved in [19]. On the other hand, many authors solved high odd-order BVPs individuals [20].…”

Section: Introductionmentioning

confidence: 99%

Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

Abdelhakem,

Ahmed,

El-kady

2021

Preprint

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“…Spectral methods can be classified into three main categories. They are, Galerkin, 8–13 Tau, 14–18 and the collocation (pseudospectral) methods 19–24 …”

Section: Introductionmentioning

confidence: 99%

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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“…Even finite difference methods produce acceptable results for many BVPs, their local order refinement (p-refinement) is not easy task due to the direct disconnection of the higher order formulations. Another kind of direct method is the collocation methods which are based on direct substitutions of approximate solutions and getting algebraic equations at collocation points [11][12][13][14][15][16]. These types of methods are suitable for local order refinement and can also be used in either local form or global form.…”

Section: Introductionmentioning

confidence: 99%

A new implicit-explicit local differential method for boundary value problems

Tunc¹,

Sarı²

2021

Turk J Math

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In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

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Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method

Cited by 23 publications

References 18 publications

Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

Spectral Monic Chebyshev Approximation for Higher Order Differential Equations

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

A new implicit-explicit local differential method for boundary value problems

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