The barycentric rational interpolation collocation method for boundary value problems

Tian, Dan; He, Ji‐Huan

doi:10.2298/tsci1804773t

Cited by 6 publications

(5 citation statements)

References 9 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“… Also from Tables (2, 4 and 6) one can note that the BLP method has no results for n=8 and n=10 because of difficulty for finding the fractional derivatives which are so hard to compute exactly by hand or by using MATLAB within reason of shape of BLP appear in Eq. (18) and this result doses not agree with the other papers since the fractional derivative appears here only not in the other .  Due to the same reason, we suggest using numerical integration instead of exact value to avoid the difficulty of finding the integration in Eq.…”

Section: Discussioncontrasting

confidence: 72%

“…Pan and Huang (17) in 2017 presented a modified barycentric rational interpolation method for solving two-dimensional integral equations. Tian and He (18) in 2018 used barycentric rational interpolation collocation method to solve higherorder boundary value problems. Wu et al (19) in 2018 find numerical solution of a class of nonlinear partial differential equations using Barycentric interpolation collocation method.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

Salman

Mustfaf

2020

Baghdad Sci.J

View full text Add to dashboard Cite

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems.

show abstract

Section: Discussioncontrasting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

Salman

Mustfaf

2020

Baghdad Sci.J

View full text Add to dashboard Cite

show abstract

“…There are alternative methods for nonlinear oscillators; some famous ones include the variational iteration method, [74][75][76][77][78][79] the exp-function method, [80][81][82][83] the variational theory, [82][83][84] the G'/G-expansion method, 85 the Bayesian inference method, 86 the barycentric rational interpolation collocation method, 87 and others. 88 This section focuses itself on a simple method to find the frequency-amplitude relationship of a nonlinear oscillator using He's frequency formulation, [89][90][91] which represents a genius idea in converting a nonlinear equation into a linear equation.…”

Section: Quasi-exact Solution Based On He's Frequency Formulamentioning

confidence: 99%

A heuristic review on the homotopy perturbation method for non-conservative oscillators

El‐Dib

2021

Journal of Low Frequency Noise, Vibration and Active Control

View full text Add to dashboard Cite

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

show abstract

“…Boundary value problems arise everywhere in engineering (Khan, 2018; Bilige and Han, 2018; Yadav et al , 2019; Abu Arqub, 2018; Tian and He, 2018; Qiu, 2018; Karakas, 2018) and serve as “numerical building blocks” for many computational fluid dynamics software. More and more evidences have been reported that an unsmooth boundary or a porous medium can greatly affect the mass, energy and charge transfer through the boundary (Guo and Fu, 2019; Sivasankaran et al , 2019; Genbach et al , 2019; He et al , 2019).…”

Section: Introductionmentioning

confidence: 99%

A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives

2020

HFF

Self Cite

View full text Add to dashboard Cite

Purpose This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives. Design/methodology/approach Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution. Findings An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform. Originality/value This paper can be served as a paradigm for various practical applications.

show abstract

The barycentric rational interpolation collocation method for boundary value problems

Abstract: Higher-order boundary value problems have been widely studied in thermal science, though there are some analytical methods available for such problems, the barycentric rational interpolation collocation method is proved in this paper to be the most effective as shown in three examples.

Cited by 6 publications

References 9 publications

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

A heuristic review on the homotopy perturbation method for non-conservative oscillators

A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives

Contact Info

Product

Resources

About