Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Parand, Kourosh; Delkhosh, Mehdi

doi:10.1007/s11587-016-0291-y

Cited by 39 publications

(22 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There also exists a class of bases which is called semiorthogonal [15,16] , such as B-splines, which is considered as a non-orthogonal basis in the present paper's classification. Chebyshev, Laguerre, Hermite and Jacobi polynomials are some of orthogonal basis functions which are used in variety of problems such as astronomy, chemistry, physics and applied mathematic with a satisfactory result [17][18][19][20]. The stability and convergence analysis of the spectral methods are presented in [14].…”

Section: Spectral Methodsmentioning

confidence: 99%

A numerical approach based on B-spline basis functions to solve boundary layer flow model of a non-Newtonian fluid

Parand

Bajalan

2018

J Braz. Soc. Mech. Sci. Eng.

Self Cite

View full text Add to dashboard Cite

A spectral method based on B-spline functions is proposed for the numerical solution of the boundary layer flow of an Powell-Eyring fluid over a stretching sheet. First, the B-spline functions formula is represented, and the quasilinearization method is used to approximate the nonlinear equation of the problem with a sequence of linear equations. Then, the collocation method is used to approximate the answer of these equations and illustrate the relation between the shape parameters and the stream function using three various degrees of B-spline functions.

show abstract

Section: Spectral Methodsmentioning

confidence: 99%

A numerical approach based on B-spline basis functions to solve boundary layer flow model of a non-Newtonian fluid

Parand

Bajalan

2018

J Braz. Soc. Mech. Sci. Eng.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Some of Chebyshev polynomials properties are: orthogonality, recursive, real zeros, complete for the space of polynomials, etc. For these reasons, many researchers have employed these polynomials in their research [17,18,19,20,21,22,23].…”

Section: The Chebyshev Functionsmentioning

confidence: 99%

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Parand

Delkhosh

2018

B Soc Paran Mat

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…For example, the nonlinear oscillation of earthquake [8], the fractional optimal control problems for dynamic systems [9,10,11,12], and the fluid-dynamic models with fractional derivatives can eliminate the deficiency arising from the assumption of continuous traffic flow [13,14,15]. During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integro-differential equations, the initial and boundary value problems, and dynamic systems containing fractional derivatives, such as Adomian's decomposition method [16,17], fractional-order Legendre functions [18], fractional-order Chebyshev functions of the second kind [19], Homotopy analysis method [20], Bessel functions and spectral methods [21], Legendre and Bernstein polynomials [22], finite element methods [23], Legendre collocation [24], modified spline collocation [25], multiquadratic radial basis functions [26], and other methods [27,28,29,30,31,32,33]. …”

Section: Summary Of Fractional Calculus Historymentioning

confidence: 99%

Novel orthogonal functions for solving differential equations of arbitrary order

2017

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Cited by 39 publications

References 41 publications

A numerical approach based on B-spline basis functions to solve boundary layer flow model of a non-Newtonian fluid

A numerical approach based on B-spline basis functions to solve boundary layer flow model of a non-Newtonian fluid

Systems of nonlinear Volterra integro-differential equations of arbitrary order

Novel orthogonal functions for solving differential equations of arbitrary order

Contact Info

Product

Resources

About