Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: A comparison

Parand, Kourosh; Rezaei, A.; Taghavi, A.

doi:10.1002/mma.1318

Cited by 41 publications

(33 citation statements)

References 42 publications

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“…The number of researchers by using some transformations extended Chebyshev polynomials to semi-infinite or infinite domain for example by using x = t−L t+L , L > 0 the rational Chebyshev functions are introduced [24,25,26,27,28,29,30].…”

Section: The Chebyshev Functionsmentioning

confidence: 99%

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Parand

Delkhosh

2018

B Soc Paran Mat

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In this paper, a new approximate method for solving the system of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order α in the Caputo's definition for GFCFs. This method reduces a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

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Section: The Chebyshev Functionsmentioning

confidence: 99%

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Parand

Delkhosh

2018

B Soc Paran Mat

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“…For these reasons, many researchers have employed these polynomials in their research [37,38,39,40,41]. Using some transformations, some researchers extended Chebyshev polynomials to semi-infinite or infinite domain, for example by using x = t−L t+L , L > 0 the rational Chebyshev functions on semi-infinite domain [42,43,44,45,46,47,48], by using…”

Section: The Chebyshev Functionsmentioning

confidence: 99%

Novel orthogonal functions for solving differential equations of arbitrary order

2017

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Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

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“…There are many approximate and numerical solutions for Volterra's population model, we name a few [18,20,21,22,23,24,25,26,27,28,29,30,31,32]. We can solve Eq.…”

Section: The Model Of Volterra Populationmentioning

confidence: 99%

A multiple-scale power series method for solving nonlinear ordinary differential equations

Liu¹,

Kuo²,

Jhao³

2016

Communications in Numerical Analysis

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The power series solution is a cheap and effective method to solve nonlinear problems, like the Duffing-van der Pol oscillator, the Volterra population model and the nonlinear boundary value problems. A novel power series method by considering the multiple scales R k in the power term (t/R k ) k is developed, which are derived explicitly to reduce the ill-conditioned behavior in the data interpolation. In the method a huge value times a tiny value is avoided, such that we can decrease the numerical instability and which is the main reason to cause the failure of the conventional power series method. The multiple scales derived from an integral can be used in the power series expansion, which provide very accurate numerical solutions of the problems considered in this paper.

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Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: A comparison

Cited by 41 publications

References 42 publications

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Novel orthogonal functions for solving differential equations of arbitrary order

A multiple-scale power series method for solving nonlinear ordinary differential equations

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