Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations

Rahimkhani, Parisa; Ordokhani, Yadollah; Babolian, E.

doi:10.1016/j.apnum.2017.08.002

Cited by 57 publications

(33 citation statements)

References 35 publications

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“…Definition The Riemann‐Liouville fractional integral operator of order α of a function f ∈ C μ , μ > 1, is given by

I^{α} f false(x false) = ({, \begin{matrix} array \frac{1}{Γ (α)} \int_{0}^{x} \frac{f (s)}{{(x - s)}^{1 - α}} d s, & array α > 0, x > 0, \\ array f (x), & array α = 0 . \end{matrix})

…”

Section: Some Preliminaries In Fractional Calculusmentioning

confidence: 99%

“…Definition Caputo's fractional derivative of order α is given by

D^{α} f false(x false) = \frac{1}{normalΓ false(n - α false)} \int_{0}^{x} \frac{f^{false(n false)} false(s false)}{{false(x - s false)}^{α + 1 - n}} normald s, n - 1 < α \leq n, n \in double-struckN, f \in C_{- 1}^{m} .

…”

Section: Some Preliminaries In Fractional Calculusmentioning

confidence: 99%

“…Nowadays, fractional calculus have a wide range of applications in fields of fluid‐dynamic traffic, frequency dependent damping behavior of various viscoelastic materials, colored noise, solid mechanics, economics, signal processing, and control theory . Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, finite difference method, wavelet method, least square approximation method, finite element method, spline collocation method, fractional wavelet method, shifted fractional‐order Jacobi orthogonal functions, Jacobi Tau method, and Legendre Tau method …”

Section: Introductionmentioning

confidence: 99%

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Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

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In this paper, a method for finding an approximate solution of a class of 2D fractional optimal control problems with fractional-order dynamical system is discussed. In the proposed method, the fractional derivative is expressed in the Caputo sense. The method consists of expanding unknown functions as the elements of two-dimensional (2D) Müntz-Legendre wavelets. The 2DMüntz-Legendre wavelets are constructed and their properties are presented.The operational matrix of fractional-order integration for these wavelets is utilized to reduce the solution of 2D fractional optimal control problem to an optimization problem, which can then be solved easily. Some results concerning the error analysis are obtained. Finally, two illustrative test problems are included to demonstrate the validity and applicability of the technique. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. KEYWORDSCaputo derivative, fractional optimal control problem, numerical solution, Riemann-Liouville fractional integral operator, wavelet method INTRODUCTIONNowadays, fractional calculus have a wide range of applications in fields of fluid-dynamic traffic, 1 frequency dependent damping behavior of various viscoelastic materials, 2 colored noise, 3 solid mechanics, 4 economics, 5 signal processing, 6 and control theory. 7 Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, 8 finite difference method, 9 wavelet method, 10 least square approximation method, 11 finite element method, 12 spline collocation method, 13 fractional wavelet method, 14 shifted fractional-order Jacobi orthogonal functions, 15 Jacobi Tau method, 16,17 and Legendre Tau method. 18 An optimal control problem (OCP) is defined to minimize the functional over an admissible set of control functions subject to dynamic constraints on the state and control variables. A fractional OCP (FOCP) is an optimal control problem in which the performance index or the differential equations governing the dynamics of the system or both contains at least one fractional-order derivative term. In recent the years, several numerical methods have been devoted to solve of one-dimensional FOCPs. We list here some of these numerical methods, as follows: • Eigen functions method (Agrawal 19 ); • Quadratic numerical scheme (Agrawal 20 ); • Rational approximation method (Tricaud and Chen 21 ); • Legendre multiwavelet collocation method (Yousefi et al 22 ); 1916

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“…Definition The Riemann‐Liouville fractional integral operator of order α of a function f ∈ C μ , μ > 1, is given by

I^{α} f false(x false) = ({, \begin{matrix} array \frac{1}{Γ (α)} \int_{0}^{x} \frac{f (s)}{{(x - s)}^{1 - α}} d s, & array α > 0, x > 0, \\ array f (x), & array α = 0 . \end{matrix})

…”

Section: Some Preliminaries In Fractional Calculusmentioning

confidence: 99%

“…Definition Caputo's fractional derivative of order α is given by

D^{α} f false(x false) = \frac{1}{normalΓ false(n - α false)} \int_{0}^{x} \frac{f^{false(n false)} false(s false)}{{false(x - s false)}^{α + 1 - n}} normald s, n - 1 < α \leq n, n \in double-struckN, f \in C_{- 1}^{m} .

…”

Section: Some Preliminaries In Fractional Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

Self Cite

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“…Therefore, we must use of the approximate and numerical methods. There are several numerical methods for solving such equations, see [8][9][10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

A numerical scheme to solve variable order diffusion-wave equations

2019

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Original scientific paper https://doi.org/10.2298/TSCI190729371MIn this work, we consider variable order diffusion-wave equations. We choose variable order derivative in the Caputo sense. First, we approximate the unknown functions and its derivatives using Bernstein basis. Then, we obtain operational matrices based on Bernstein polynomials. Finally, with the help of these operational matrices and collocation method, we can convert variable order diffusion-wave equations to an algebraic system. Few examples are given to demonstrate the accuracy and the competence of the presented technique.

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“…Integro-differential equations play an important role in the modeling of numerous of physical phenomena from science and engineering. Hence, searching the exact and approximate solutions of integro-differential equations have attracted appreciable attention for scientists and applied mathematicians (Dzhumabaev 2018;Fairbairn & Kelmanson 2018;Hendi & Al-Qarni 2017;Kürkçü et al 2017;Rahimkhani et al 2017;Rohaninasab et al 2018;Yüzbaşı & Karaçayır 2017). The fractional calculus represents a powerful tool in applied mathematics to study a myriad of problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering (Abbas et al 2015).…”

Section: Introductionmentioning

confidence: 99%

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

Daşçıoğlu¹,

Bayram²

2019

JSM

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The main purpose of this study was to present an approximation method based on the Laguerre polynomials to obtain the solutions of the fractional linear Fredholm integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivative of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed. ABSTRAKTujuan utama kajian ini adalah untuk mengemukakan kaedah penghampiran berdasarkan polinomial Laguerre untuk mendapatkan penyelesaian pecahan linear persamaan pembezaan-kamiran Fredholm. Kaedah ini menjelmakan persamaan pembezaan-kamiran ke sistem persamaan aljabar linear dengan menggunakan titik-titik kolokasi. Di samping itu, hubungan matriks untuk terbitan pecahan Caputo polinomial Laguerre juga diperoleh. Selain itu, beberapa contoh dibentangkan untuk menggambarkan ketepatan kaedah dan hasilnya dibincangkan.Kata kunci: Persamaan pembezaan-kamiran Fredholm; persamaan pembezaan-kamiran pecahan; polinomial Laguerre

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Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations

Cited by 57 publications

References 35 publications

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

A numerical scheme to solve variable order diffusion-wave equations

Solving Fractional Fredholm Integro-Differential Equations by Laguerre Polynomials

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