The impulse response of a fractional-order system with the transfer function $$s^{\delta }/{[(s^{... more The impulse response of a fractional-order system with the transfer function $$s^{\delta }/{[(s^{\alpha }-a)^2+b^2]^n}$$ s δ / [ ( s α - a ) 2 + b 2 ] n , where $$n \in \mathbb {N}$$ n ∈ N , $$a \in {\mathbb {R}}$$ a ∈ R , $$b \in {\mathbb {R}}^+$$ b ∈ R + , $$\alpha \in {\mathbb {R}}^+$$ α ∈ R + , $$\delta \in {\mathbb {R}}$$ δ ∈ R , is derived via real and imaginary parts of two-parameter Mittag-Leffler functions and their derivatives. With the aid of a robust algorithm for evaluating these derivatives, the analytic formulas can be used for an effective transient analysis of fractional-order systems with multiple complex poles. By some numerical experiments it is shown that this approach works well also when the popular SPICE-family simulating programs fail to converge to a correct solution.
Communications in Nonlinear Science and Numerical Simulation, 2021
Abstract This paper discusses some properties of solutions to fractional neutral delay differenti... more Abstract This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag-Leffler functions; their stability properties are hence studied by using use Rouche’s theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified.
The theory of functional connections, an analytical framework generalizing interpolation, was ext... more The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying some constraints expressed in terms of integrals and derivatives of non-integer order. The objective of these expressions was to solve fractional differential equations or other problems subject to fractional constraints. Although this work focused on the Riemann–Liouville definitions, the method is, however, more general, and it can be applied with different definitions of fractional operators just by changing the way they are computed. Three examples are provided showing, step by step, how to apply this extension for: (1) one constraint in terms of a fractional derivative, (2) three constraints (a function, a fractional derivative, and an integral), and (3) two co...
Several approaches to the formulation of a fractional theory of calculus of "variable order&... more Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.
In the paper titled "New numerical approach for fractional differential equations" by A... more In the paper titled "New numerical approach for fractional differential equations" by A. Atangana and K.M. Owolabi [Math. Model. Nat. Phenom., 13(1), 2018], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations.
This paper focuses on the numerical solution of initial value problems for fractional differentia... more This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized Mittag-Leffler function. Then suitable quadrature rules are devised and order conditions of algebraic type are derived. Theoretical findings are validated by means of numerical experiments and the effectiveness of the proposed approach is illustrated by means of comparisons with other standard methods.
This article investigates the neuroadaptive optimal fixed-time synchronization and its circuit re... more This article investigates the neuroadaptive optimal fixed-time synchronization and its circuit realization along with dynamical analysis for unidirectionally coupled fractional-order (FO) self-sustained electromechanical seismograph systems under subharmonic and superharmonic oscillations. The synchronization model of the coupled FO seismograph system is established based on drive and response seismic detectors. The dynamical analysis reveals this coupled system generating transient chaos and homoclinic/heteroclinic oscillations. The test results of the constructed equivalent analog circuit further testify its complex nonlinear dynamics. Then, a neuroadaptive optimal fixed-time synchronization controller integrated with the FO hyperbolic tangent tracking differentiator (HTTD), interval type-2 fuzzy neural network (IT2FNN) with transformation, and prescribed performance function (PPF) together with the constraint condition is developed in the backstepping recursive design. Furthermore, it is proved that all signals of this closed-loop system are bounded, and the tracking errors fall into a trap of the prescribed constraint along with the minimized cost function. Extensive studies confirm the effectiveness of the proposed scheme.
Abstract This paper presents an extension of the trapezoidal integration rule, that in the presen... more Abstract This paper presents an extension of the trapezoidal integration rule, that in the present work is applied to devise a pseudo-recursive numerical algorithm for the numerical evaluation of fractional-order integrals. The main benefit of pseudo recursive implementation arises in terms of higher accuracy when the algorithm is run in the “short memory” version. The rule is suitably generalized in order to build a numerical solver for a class of fractional differential equations. The algorithm is also specialized to derive an efficient numerical algorithm for the on-line implementation of linear fractional order controllers. The accuracy of the method is theoretically analyzed and its effectiveness is illustrated by simulation examples.
Several fractional-order operators are available and an in-depth knowledge of the selected operat... more Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.
The paper describes different approaches to generalize the trapezoidal method to fractional diffe... more The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.
Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable a... more Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines
Fractional integrals and derivatives based on the Prabhakar function are useful to describe anoma... more Fractional integrals and derivatives based on the Prabhakar function are useful to describe anomalous dielectric properties of materials whose behaviour obeys to the Havriliak–Negami model. In this work some formulas for defining these operators are described and investigated. A numerical method of product-integration type for solving differential equations with the Prabhakar derivative is derived and its convergence properties are studied. Some numerical experiments are presented to validate the theoretical results.
Exponential integrators are powerful and well{established methods particularly suited for the tim... more Exponential integrators are powerful and well{established methods particularly suited for the time{integration of semilinear systems of ordinary dierential equations (ODEs) with linear stiness. By solving exactly the sti term, exponential integrators allow to integrate the remaining non{sti part of the system by means of explicit schemes without calling for severe restrictions on the step-size. In this talk we discuss the generalization of exponential integrators to problems of non integer orders, namely fractional dierential equations (FDEs), which are nowadays used in several areas, including biology, nance, physics
The impulse response of a fractional-order system with the transfer function $$s^{\delta }/{[(s^{... more The impulse response of a fractional-order system with the transfer function $$s^{\delta }/{[(s^{\alpha }-a)^2+b^2]^n}$$ s δ / [ ( s α - a ) 2 + b 2 ] n , where $$n \in \mathbb {N}$$ n ∈ N , $$a \in {\mathbb {R}}$$ a ∈ R , $$b \in {\mathbb {R}}^+$$ b ∈ R + , $$\alpha \in {\mathbb {R}}^+$$ α ∈ R + , $$\delta \in {\mathbb {R}}$$ δ ∈ R , is derived via real and imaginary parts of two-parameter Mittag-Leffler functions and their derivatives. With the aid of a robust algorithm for evaluating these derivatives, the analytic formulas can be used for an effective transient analysis of fractional-order systems with multiple complex poles. By some numerical experiments it is shown that this approach works well also when the popular SPICE-family simulating programs fail to converge to a correct solution.
Communications in Nonlinear Science and Numerical Simulation, 2021
Abstract This paper discusses some properties of solutions to fractional neutral delay differenti... more Abstract This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag-Leffler functions; their stability properties are hence studied by using use Rouche’s theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified.
The theory of functional connections, an analytical framework generalizing interpolation, was ext... more The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying some constraints expressed in terms of integrals and derivatives of non-integer order. The objective of these expressions was to solve fractional differential equations or other problems subject to fractional constraints. Although this work focused on the Riemann–Liouville definitions, the method is, however, more general, and it can be applied with different definitions of fractional operators just by changing the way they are computed. Three examples are provided showing, step by step, how to apply this extension for: (1) one constraint in terms of a fractional derivative, (2) three constraints (a function, a fractional derivative, and an integral), and (3) two co...
Several approaches to the formulation of a fractional theory of calculus of "variable order&... more Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative.
In the paper titled "New numerical approach for fractional differential equations" by A... more In the paper titled "New numerical approach for fractional differential equations" by A. Atangana and K.M. Owolabi [Math. Model. Nat. Phenom., 13(1), 2018], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations.
This paper focuses on the numerical solution of initial value problems for fractional differentia... more This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized Mittag-Leffler function. Then suitable quadrature rules are devised and order conditions of algebraic type are derived. Theoretical findings are validated by means of numerical experiments and the effectiveness of the proposed approach is illustrated by means of comparisons with other standard methods.
This article investigates the neuroadaptive optimal fixed-time synchronization and its circuit re... more This article investigates the neuroadaptive optimal fixed-time synchronization and its circuit realization along with dynamical analysis for unidirectionally coupled fractional-order (FO) self-sustained electromechanical seismograph systems under subharmonic and superharmonic oscillations. The synchronization model of the coupled FO seismograph system is established based on drive and response seismic detectors. The dynamical analysis reveals this coupled system generating transient chaos and homoclinic/heteroclinic oscillations. The test results of the constructed equivalent analog circuit further testify its complex nonlinear dynamics. Then, a neuroadaptive optimal fixed-time synchronization controller integrated with the FO hyperbolic tangent tracking differentiator (HTTD), interval type-2 fuzzy neural network (IT2FNN) with transformation, and prescribed performance function (PPF) together with the constraint condition is developed in the backstepping recursive design. Furthermore, it is proved that all signals of this closed-loop system are bounded, and the tracking errors fall into a trap of the prescribed constraint along with the minimized cost function. Extensive studies confirm the effectiveness of the proposed scheme.
Abstract This paper presents an extension of the trapezoidal integration rule, that in the presen... more Abstract This paper presents an extension of the trapezoidal integration rule, that in the present work is applied to devise a pseudo-recursive numerical algorithm for the numerical evaluation of fractional-order integrals. The main benefit of pseudo recursive implementation arises in terms of higher accuracy when the algorithm is run in the “short memory” version. The rule is suitably generalized in order to build a numerical solver for a class of fractional differential equations. The algorithm is also specialized to derive an efficient numerical algorithm for the on-line implementation of linear fractional order controllers. The accuracy of the method is theoretically analyzed and its effectiveness is illustrated by simulation examples.
Several fractional-order operators are available and an in-depth knowledge of the selected operat... more Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.
The paper describes different approaches to generalize the trapezoidal method to fractional diffe... more The paper describes different approaches to generalize the trapezoidal method to fractional differential equations. We analyze the main theoretical properties and we discuss computational aspects to implement efficient algorithms. Numerical experiments are provided to illustrate potential and limitations of the different methods under investigation.
Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable a... more Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines
Fractional integrals and derivatives based on the Prabhakar function are useful to describe anoma... more Fractional integrals and derivatives based on the Prabhakar function are useful to describe anomalous dielectric properties of materials whose behaviour obeys to the Havriliak–Negami model. In this work some formulas for defining these operators are described and investigated. A numerical method of product-integration type for solving differential equations with the Prabhakar derivative is derived and its convergence properties are studied. Some numerical experiments are presented to validate the theoretical results.
Exponential integrators are powerful and well{established methods particularly suited for the tim... more Exponential integrators are powerful and well{established methods particularly suited for the time{integration of semilinear systems of ordinary dierential equations (ODEs) with linear stiness. By solving exactly the sti term, exponential integrators allow to integrate the remaining non{sti part of the system by means of explicit schemes without calling for severe restrictions on the step-size. In this talk we discuss the generalization of exponential integrators to problems of non integer orders, namely fractional dierential equations (FDEs), which are nowadays used in several areas, including biology, nance, physics
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