Numerical solutions for solving a class of fractional optimal control problems via fixed-point approach

Zeid, Samaneh Soradi; Kamyad, Ali Vahidian; Effati, Sohrab; Hosseinpour, Soleiman

doi:10.1007/s40324-016-0102-0

Cited by 15 publications

(6 citation statements)

References 36 publications

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“…(29) have been proven in Zeid et al (2017). According to the above discussions and Zeid et al (2017), Eq. (27) is equivalent to the following Volterra integral equations:…”

Section: Indirect Schemementioning

confidence: 78%

“…All methods for solving optimal control problems are divided mainly into two categories as direct and indirect in which the former methods describe the continuous FOCPs to a finite-dimensional nonlinear programming problem and others are based on the necessary optimality conditions of a FOCP. An up-to-date bibliography on numerical methods for solving FOCPs was recently reported by Zeid et al (2017).…”

Section: Introductionmentioning

confidence: 99%

“…This should also be evident in some form when using them for collocation. Also, using the methods of RBF as an extension of shape function-based method, an even more efficient method can be developed which is increasingly utilized in applied science (Bohannan 2008;Zamani et al 2017;Zeid et al 2017), generally give an accurate solution in the lower iteration.…”

Section: Introductionmentioning

confidence: 99%

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Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Soradi-Zeid

2019

Comp. Appl. Math.

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The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods. Keywords Optimal control problems • Radial basis functions • Direct and indirect methods • Fractional two-point boundary value problem Mathematics Subject Classification 49M37 • 49M05 • 49L99 • 65K05

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“…(29) have been proven in Zeid et al (2017). According to the above discussions and Zeid et al (2017), Eq. (27) is equivalent to the following Volterra integral equations:…”

Section: Indirect Schemementioning

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Soradi-Zeid

2019

Comp. Appl. Math.

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“…Furthermore, a free final time problem can be converted into a fixed final time problem using suitable transformation. In Zeid et al (2017) and Zhou and Xu (2016) has been studied the well-posedness of a kind of nonlinear coupled system of FDEs, stemming from a class of FOCPs.…”

Section: Fractional Optimal Control Problemsmentioning

confidence: 99%

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

2020

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The main purpose of this work is to develop effective spectral collocation methods for solving numerically general form of non-linear fractional two-point boundary value problems with two end-point singularities. Specifically, we define two classes of Müntz-Legendre polynomials to deal with the singular behaviors of the solutions at both end-points, which enhance greatly the accuracy of the numerical solutions. Important and practical formulas for ordinary derivatives of Müntz-Legendre polynomials are derived. Then using a three-term recurrence formula and Jacobi-Gauss quadrature rules, applicable and stable numerical approaches for computing left and right Caputo fractional derivatives of these basis functions are provided. One of the important features of the present method is to show how to recover spectral accuracy from the end-point singularities for the coupled systems of fractional differential equations with left and right fractional derivatives using the mentioned approximation basis. Moreover, we present a detailed analysis with accurate convergence rates of the singular behavior of the solution to a rather general system of nonlinear fractional differential equations including left and right fractional derivatives. Numerical results show the expected convergence rates. Finally, some numerical examples are given to demonstrate the performance of the proposed schemes.

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“…Because the next state of many systems depends on its current and historical states, there is a great demand to improve topical methods for the real life problems 4,5 . These problems happen in bioengineering, 6 solid mechanics, 7 anomalous transport, 8 continuum and statistical mechanics, 9 economics, 10 relaxation electrochemistry, 11 diffusion procedures, 12 complex networks, 13,14 and optimal control problems 15,16 . Fractional diffusion equations are largely used in describing abnormal convection phenomenon of liquid in medium.…”

Section: Introductionmentioning

confidence: 99%

A new approach to solving multiorder time‐fractional advection–diffusion–reaction equations using BEM and Chebyshev matrix

Khalighi

Matlob

Malek

2020

Math Methods in App Sciences

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In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two-dimensional multi-order time-fractional partial differential equations; nonlinear and linear in respect to spatial and temporal variables, respectively. Fractional derivatives are estimated by Caputo sense. Boundary element method is used to convert the main problem into a system of a multi-order fractional ordinary differential equation. Then, the produced system is approximated by Chebyshev operational matrix technique, ans its condition number is analyzed. Accuracy and efficiency of the proposed hybrid scheme are demonstrated by solving three different types of two-dimensional time fractional convection-diffusion equations numerically. The convergent rates are calculated for different meshing within the boundary element technique. Numerical results are given by graphs and tables for solutions and different type of error norms.

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Numerical solutions for solving a class of fractional optimal control problems via fixed-point approach

Cited by 15 publications

References 36 publications

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

A new approach to solving multiorder time‐fractional advection–diffusion–reaction equations using BEM and Chebyshev matrix

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