An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

Nemati, Somayeh; Lima, Pedro M.; Sedaghat, S.

doi:10.1016/j.apnum.2018.05.005

Cited by 43 publications

(31 citation statements)

References 34 publications

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“…, n − 1, with h = t f n . Then the generalized modified hat functions consist of a set of n + 1 linearly independent functions in L 2 [0, t f ] that are defined as follows [15,16]:…”

Section: Properties Of the Modified Hat Functionsmentioning

confidence: 99%

“…Now, we consider x n (t) = X T Ψ(t) as an approximation of the state function x(·), where X is given by (17). Then, by neglecting the error of the operational matrix and utilizing (16), (24), and (25), we obtain that…”

Section: Error Estimatementioning

confidence: 99%

“…The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems.…”

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confidence: 99%

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A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

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We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann-Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann-Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.Recently, numerical methods based on modified hat functions have shown to provide an effective way for solving fractional differential equations [15,16]. Here, for the first time in the literature, we introduce such a method for solving a general class of fractional optimal control problems. More precisely, in this work we begin by considering the following fractional optimal control problem (FOCP):

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“…, n − 1, with h = t f n . Then the generalized modified hat functions consist of a set of n + 1 linearly independent functions in L 2 [0, t f ] that are defined as follows [15,16]:…”

Section: Properties Of the Modified Hat Functionsmentioning

confidence: 99%

Section: Error Estimatementioning

confidence: 99%

mentioning

confidence: 99%

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A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati

Lima

Torres

2019

Communications in Nonlinear Science and Numerical Simulation

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“…This system is not linear, it is solved by numerical methods [16,17]. The solution of this system allows us to determine the flight distance for particles and points on the distribution zone surface they fall into.…”

Section: Advances In Engineering Research Volume 151mentioning

confidence: 99%

Mathematical Simulation of Defecate Distribution by a Centrifugal Working Tool

Kolesnikov¹,

Dyachkov²,

Shatskiy³

et al. 2018

International Scientific and Practical Conference "AgroSMART - Smart Solutions for Agriculture" (AgroSMART 2018)

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The article is devoted to theoretical studies of defecate distribution over the soil surface by a centrifugal working tool. Defecate has physical and mechanical properties which are similar to those of mineral fertilizers. Therefore, for its distribution, fertilizer spreaders with centrifugal working elements are used. However, without structural changes, their use is limited due to increasing defecate weight fed to the working tool (12-60 kg/s). It is necessary to determine parameters of the working tool for defecate distribution. To solve this problem, a method for calculating defecate distribution over the soil surface has been developed. The method is implemented in the form of a computer program used to study the nature of defecate distribution depending on the following variables: defecate particle size; a blade installation angle on the disk surface; blade lengths; disk conicity; disc installation height above the soil surface; disc rotation speed; distance between disks (for a two-disc spreader) and feed zone parameters (location, shape, material entry unevenness).

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“…Because of the computational complexity of fractional derivatives in the delay case, the exact analytical solution of the fractional delay differential equations is hardly available. Therefore, in the last decades many researchers have been attracted to deal with the numerical solution of this class of problems (see for example [23,24,25,26,27,28]).…”

Section: Introductionmentioning

confidence: 99%

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Nemati

Lima

Sedaghat

2020

Applied Numerical Mathematics

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In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss-Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss-Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.

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An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

Cited by 43 publications

References 34 publications

A numerical approach for solving fractional optimal control problems using modified hat functions

A numerical approach for solving fractional optimal control problems using modified hat functions

Mathematical Simulation of Defecate Distribution by a Centrifugal Working Tool

Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

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