Order Time Derivatives Research Articles

Flat Filtering Cascade Control of Fourth Order Systems

Abstract In this article, the analysis and implementation of an alternative Flat Filtering Control for a class of partially known fourth order flat systems is given. The Flat filtering control uses the cascade property of the system, which leads to a simplified control design in which high order time derivatives are algebraically simplified in terms of low order measurable states and the subsequent integral compensation. The control proposal is implemented and validated experimentally in a fourth order mechanical system (rotatory flexible joint) with accurate tracking results.

On key properties of the Lion’s and Kreisselmeier’s adaptation algorithms

The paper revises properties of two identification/adaptation algorithms proposed by Lion (1967) and Kreisselmeier (1977) more than 40 years ago to accelerate parametric convergence under regressor persistency of excitation (PE) condition. First, being motivated by paper Aranovskiy et al. (2017) it is demonstrated that these algorithms can provide asymptotic (not exponential) parametric convergence under simple condition which is weaker than requirement of PE. Second, it is shown that via some condition these schemes can be used for generating the high order time derivatives (HOTD) of the adjustable parameters that are necessary for solution of a wide range of problems of identification and adaptive control including backstepping design procedure.

Approximation of the disturbance dynamics by Extended State Observer Using an Artificial Delay

A disturbance estimator based on the design of an extended state observer (ESO) often considers that the time-derivative of the perturbation (or higher order time-derivatives) is very small and can be taken equal to zero in the observer design (standard extended system). In this paper, a better approximation of the disturbance dynamics is proposed using a backward difference method. A new extended system is designed based on this approximation. Any observer that makes the error dynamics exponentially stable for the standard extended system can then be used to estimate the state of the new extended system. The efficiency of the method is illustrated through an example.

A novel generalized thermoelasticity with higher-order time-derivatives and three-phase lags

PurposeIn this work, a modified thermoelastic model of heat conduction, including higher order of time derivative, is constructed by extending the Roychoudhuri model (TPL) (Choudhuri, 2007). In this new model, Fourier’s law of heat conduction is replaced by using Taylor series expansions, including three different phase lags for the heat flux, the thermal displacement and the temperature gradient. The generalized thermoelasticity models of Lord–Shulman (Lord and Shulman, 1967), Green and Naghdi (1991), dual-phase lag (Tzou, 1996) and three-phase lag (TPL) (Choudhuri, 2007) are obtained as special cases. The paper aims to discuss these issues.Design/methodology/approachThe aim of this work is to establish a new generalized mathematical model of thermoelasticity that includes TPL in the vector of heat flux, and in the thermal displacement and temperature gradients extending TPL model (Li et al., 2019e). In this model, Fourier law of heat conduction is replaced by using Taylor series expansions to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient, and the thermal displacement gradient and keeping terms up with suitable higher orders.FindingsThe established high-order three-phase-lag heat conduction model reduces to the previous models of thermoelasticity as special cases.Originality/valueIn this paper, a TPL thermoelastic model is developed by extending the Roychoudhuri (Sherief and Raslan, 2017) model (TPL) considering the Taylor series approximation of the equation of heat conduction. This model is an alternative construction to the TPL model. The new model includes high order of TPL in the vector of heat flux, and in the thermal displacement and temperature gradients.

Inverse Lyapunov Theorem for Linear Time Invariant Fractional Order Systems

This paper investigates the inverse Lyapunov theorem for linear time invariant fractional order systems. It is proved that given any stable linear time invariant fractional order system, there exists a positive definite functional with respect to the system state, and the first order time derivative of that functional is negative definite. A systematic procedure to construct such Lyapunov candidates is provided in terms of some Lyapunov functional equations.

Solutions of fractional gas dynamics equation by a new technique

In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section.

Uniqueness in the inversion of distributed orders in ultraslow diffusion equations

This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Neumann) boundary condition. We first prove the wellposedness of the forward problem by means of eigenfunction expansion, which ensures that the weak solution has the classical derivatives. We next give a Harnack type inequality of the solution in the frequency domain under the Laplace transform. Finally we show a uniqueness result for the inverse problem in determining the weight function in the distributed order time derivative from one point observation.

A new analytical technique of the [formula omitted]-type difference schemes for time fractional mixed sub-diffusion and diffusion-wave equations

A time fractional mixed sub-diffusion and diffusion-wave equation involving at least two Caputo time derivatives of order γ∈(1,2) and α∈(0,1) is considered. A new analytical technique is introduced to analyze the standard finite difference method based on L1 scheme. Both stability and convergence of the scheme are proved rigorously. The final convergence result shows clearly that the temporal accuracy arrives at the order of O(τmin{2−α,3−γ}) in a discrete H1-norm. This novel analytical technique can provide new insights in analyzing other multi-term mixed time fractional differential equations.

Two-temperature thermoelastic model without energy dissipation including higher order time-derivatives and two phase-lags

In the current analysis, we have constructed a new general thermoelastic heat conduction model with two-temperature including higher order time-derivatives and two phase-lags. The thermoelastic models established by Chen and Gurtin Chen, Gurtin 1968. Z. Angew. Math. Phys. 19:614–627, Green and Naghdi of type III Green and Naghdi 1993. J. Elast. 31:189–208 and dual-phase-lag model proposed by Tzou 1995 . ASME J. Heat. Transf. 117:8–16 are extended. The proposed model has been restructured by the theory of fading memory, where the heat flow vector depends on the history of the gradient of thermal displacement. The problem of the infinite medium with a cylindrical cavity exposed to a decaying exponentially laser pulse is treated based on the resulting formulation. To explain the validity and accuracy of the derived model, the impacts of various physical parameters on the field variables are discussed in detailed and presented in tabular and graphical forms.

Modeling nonlinear fractional diffusion-wave equations with Burgers-type evolution equations

Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions under which the FPDE can be modeled accurately with a Burgers-type evolution equation for progressive wave motion. The method of lines in combination with a general Runge-Kutta method for forward integration is used for numerical analysis. [B.E.S. is supported by the ARL:UT McKinney Fellowship in Acoustics.]Fractional partial differential equations (FPDEs) with a time derivative of fractional order are used to describe wave motion in complex viscoelastic media with non-traditional equations of motion. Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] derived a fractional diffusion-wave equation for a nonlinear Lucassen wave propagating along an elastic layer coupled to a viscous substrate. The fractional time derivative of order 3/2 in the linear form of this equation lies midway between order 1 for a diffusion process described by a parabolic equation, and order 2 for the traditional hyperbolic wave equation. The inclusion of nonlinear elasticity tends to inhibit purely progressive wave motion that is associated with classical nonlinear plane waves in fluids and solids, and which is described accurately by parabolic-like, Burgers-type evolution equations. In this work, a general FPDE is analyzed in a parameter space consisting of varying nonlinearity and time fractional orders. The focus is on conditions ...

Signal propagation in electromagnetic media described by fractional-order models

In this paper, signal propagation is analysed in electromagnetic media described by fractional-order (FO) models (FOMs). Maxwell’s equations with FO constitutive relations are introduced in the time domain. Then, their phasor representation is derived for one-dimensional case of the plane wave propagation. With the use of the Fourier transformation, the algorithm for simulation of the non-monochromatic wave propagation is introduced. Its implementation in Matlab allows for generation of time-domain waveforms of signals propagating in the media described by FOMs. It is demonstrated that despite high attenuation, a small perturbation of the time-derivative orders in Maxwell’s equations allows for tuning of the time of signal arrival to the observation point. In all the cases studied, the rate of pulse advancement increases, with simultaneous decrease of the value of the time-derivative orders in FO Maxwell’s equations.

Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution

In this work, the time fractional KdV equation with Caputo time derivative of order \begin{document}$ \alpha \in (0,1) $\end{document} is considered. The solution of this problem has a weak singularity near the initial time \begin{document}$ t = 0 $\end{document} . A fully discrete discontinuous Galerkin (DG) method combining the well-known L1 discretisation in time and DG method in space is proposed to approximate the time fractional KdV equation. The unconditional stability result and O \begin{document}$ (N^{-\min \{r\alpha,2-\alpha\}}+h^{k+1}) $\end{document} convergence result for \begin{document}$ P^k \; (k\geq 2) $\end{document} polynomials are obtained. Finally, numerical experiments are presented to illustrate the efficiency and the high order accuracy of the proposed scheme.

A Collocation Method for the Numerical Solution of Nonlinear Fractional Dynamical Systems

Fractional differential problems are widely used in applied sciences. For this reason, there is a great interest in the construction of efficient numerical methods to approximate their solution. The aim of this paper is to describe in detail a collocation method suitable to approximate the solution of dynamical systems with time derivative of fractional order. We will highlight all the steps necessary to implement the corresponding algorithm and we will use it to solve some test problems. Two Mathematica Notebooks that can be used to solve these test problems are provided.

On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation

This paper presents the development of a new iterative method for solving the two-dimensional hyperbolic telegraph fractional differential equation (2D-HTFDE) which is central to the mathematical modeling of transmission line satisfying certain relationship between voltage and current waves in specific distance and time. This equation can be obtained from the classical two-dimensional hyperbolic telegraph partial differential equation by replacing the first and second order time derivatives by the Caputo time fractional derivatives of order 2α and α respectively, with 1/2< alpha < 1 . The iterative scheme, called the fractional skewed grid Crank–Nicolson FSkG(C-N), is derived from finite difference approximations discretized on a skewed grid rotated clockwise 450 from the standard grid. The skewed finite difference scheme combined with Crank–Nicolson discretization formula will be shown to be unconditionally stable and convergent by the Fourier analysis. The developed FSkG(C-N) scheme will be compared with the fractional Crank–Nicolson scheme on the standard grid to confirm the effectiveness of the proposed scheme in terms of computational complexities and computing efforts. It will be shown that the new proposed scheme demonstrates more superior capabilities in terms of the number of iterations and CPU timings compared to its counterpart on the standard grid but with the same order of accuracy.

A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equations

Approximate analytical solution of a fractional Black–Scholes pricing model is much relevant due to its practical importance to financial markets. In this article, a powerful approximate iterative mathematical scheme based on residual power series (RPS) algorithm is presented to achieve the numerical results of the nonlinear time fractional Black–Scholes (BS) equations based on European options. The Caputo-type fractional derivatives are considered in the present article. The residual power series method (RPSM) developed by Arqub (2013) is the novel technique for finding the analytical Taylor series solutions of systems of linear and nonlinear ODEs and PDEs. The residual power series technique supplies the approximate analytical solutions of the problem in truncated series form using residual error concept along with given initial conditions. The numerical procedures reveal that only a few iterations are sufficient for better approximations of the solutions, which clearly exhibits the reliability and effectiveness of this iterative scheme. This article shows that the adopted scheme is quite systematic as well as computationally attractive regarding solution procedure. In addition, effects of fractional order of time derivatives on the solutions for various particular cases are also depicted through graphs.

Modeling of evolution nonlinear ecosystem

The term «synergy», introduced in scientific literature by G. Haken, means «joint action, selforganization, the special effects of the joint action in complex systems». Synergetics as a methodology considers irreversibility as a condition for the development of systems, and the "imbalance" of systems as an imperative of development, and this is its fundamental difference from cybernetics, from the theory of operations research and from the general theory of systems. Synergy is usually called mathematical dialectics. The theory of synergetics and the theory of self-organization, in contrast to dialectics, have a powerful formalization apparatus, a set of basic and derivative models, including in the form of differential equations in time derivatives of the second order, in the form of fractals with fractional dimension, hamiltonians, taking into account the prehistory and long memory, mathematical models taken from the theory of bifurcations. The purpose of synergetics – the identification of general ideas, general methods and general laws in various fields of natural science.At the same time new concepts have been developing about dissipative structures generating under no equilibrium conditions as a result of the exchange of energy (and substance) with the environment when approaching the external energy to the material. This scientific direction was headed by I.R. Prigogine. Nonlinear Science opens up new possibilities in the study of the conduct of real complex dynamic ecosystems. Nonlinear dynamics is a new science studying the evolution of the real nonlinear ecosystems, where along with determinism appears dynamic chaos. Shows the role of mathematical modeling in the study of nonlinear ecologic systems. Evolutionary change in complex ecosystems due to the phenomenon of self-organization and are associated with a decrease in entropy, which is achieved by the increasing complexity of the structural organization of the ecosystem.

Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative

We obtain some Holder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈ (0, 1) cast by a Caputo derivative. The Holder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $$s \in (0,1)$$ , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.

A high accuracy method for long-time evolution of acoustic wave equation

This paper makes a first attempt to investigate the long-time behaviour of solutions of 2D acoustic wave equation by integrating strengths of the Krylov deferred correction (KDC) method in temporal direction and the meshless generalized finite difference method (GFDM) in space domain. The KDC approach introduces the second order time derivative of acoustic pressure as a new unknown variable, and then converts the acoustic wave equation into the Helmholtz-type equation with the new variable through the temporal discretization. We finally solve the transformed equation in space domain by employing the meshless GFDM. Numerical experiments are provided to verify the accuracy of the proposed method, and numerical results illustrate the good performance of the method for long-time numerical simulations of acoustic wave problems.

HAUSDORFF DERIVATIVE MODEL FOR CHARACTERIZATION OF NON-FICKIAN MIXING IN FRACTAL POROUS MEDIA

This study investigates the scalar dissipation rate (SDR) and dilution index of non-Fickian mixing by using the Hausdorff derivative model for conservative and first-order decaying tracers under different boundary conditions. The expressions of SDR and dilution index are derived based on the analytical solution of the Hausdorff derivative model, in which the time and space Hausdorff derivative orders, respectively capture the complexity in transport trajectory and transport scale. The properties of SDR and dilution index are discussed for different cases of Peclet number, decaying rate of the radioactive tracer, and Hausdorff derivative order, respectively. We find that the SDR of non-Fickian mixing decays more slowly than that of the Fickian diffusion, and the time scale deviates from [Formula: see text]. The evolution of the SDR has a sharp peak and decays very fast to zero when Peclet number is large. For the radioactive tracer, the larger values of decay rate, the smaller values of SDR, which decays faster to zero. The Hausdorff derivative model with larger Peclet number leads to larger dilution index. The dilution index is larger for smaller decay rate before reaching the equilibrium state. Consequently, the two metrics can be satisfactorily used to describe non-Fickian mixing based on the Hausdorff derivative model. Future studies should be designed to examine the evolution of SDR and dilution index in real geological and hydrological systems undergoing structure changes and chemical reactions.