Applications of variable-order fractional operators: a review

Patnaik, Sansit; Hollkamp, John P.; Semperlotti, Fabio

doi:10.1098/rspa.2019.0498

Cited by 164 publications

(117 citation statements)

References 187 publications

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“…A detailed review of the application of fractional calculus in solving applied problems is given in [9, Vols. 6-8], [19]. More detailed information as well as a bibliography related to the theory of fractional integro-differentiation, including the Hilfer fractional derivative, can be found in the recently published monograph [24].…”

Section: Problem Statementmentioning

confidence: 99%

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Yuldashev

Kadirkulovich

2020

UMJ

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In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

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Section: Problem Statementmentioning

confidence: 99%

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Yuldashev

Kadirkulovich

2020

UMJ

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“…we present the countable systems of ordinary integro-differential Equations (10) and (11) as follows…”

Section: Expansion Of the Solution Of The Direct Problem (1)-(4) Intomentioning

confidence: 99%

“…A detailed review of the applications of fractional calculus in solving practical problems is given in [6] (vol. [6][7][8], [10]. More detailed information, as well as a bibliography related to the theory of fractional integro-differentiation and fractional derivatives, can also be found in [11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

Yuldashev

Karimov

2020

Axioms

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The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.

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“…An introduction to variable-, fractional-order differences can be found in [ 11 , 12 , 13 ], and an example algorithm of finding parameters of FVOPID controllers is presented in [ 14 ]. One can find more about potential applications in modelling in [ 15 , 16 , 17 ], where authors provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods, and their applications of variable-order fractional differential equations. We emphasise that in our paper we use the digital version of fractional controllers, and we can compare obtained results to those used for example in [ 18 ].…”

Section: Introductionmentioning

confidence: 99%

Discrete-Time Fractional, Variable-Order PID Controller for a Plant with Delay

Oziablo

Mozyrska

Wyrwas

2020

Entropy

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In this paper, we discuss the implementation and tuning algorithms of a variable-, fractional-order Proportional–Integral–Derivative (PID) controller based on Grünwald–Letnikov difference definition. All simulations are executed for the third-order plant with a delay. The results of a unit step response for all described implementations are presented in a graphical and tabular form. As the qualitative criteria, we use three different error values, which are the following: a summation of squared error (SSE), a summation of squared time weighted error (SSTE) and a summation of squared time-squared weighted error (SST2E). Besides three types of error values, obtained results are additionally evaluated on the basis of an overshoot and a rise time of the output signals achieved by systems with the designed controllers.

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Applications of variable-order fractional operators: a review

Cited by 164 publications

References 187 publications

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

Discrete-Time Fractional, Variable-Order PID Controller for a Plant with Delay

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