Fractional Derivative Method Research Articles

A Tikhonov-type regularization method for Caputo fractional derivative

A Tikhonov-type regularization method for Caputo fractional derivative

Pointwise-in-time $ \alpha $-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients

<abstract><p>In this paper, a fully-discrete alternating direction implicit (ADI) difference method is proposed for solving three-dimensional (3D) fractional subdiffusion equations with variable coefficients, whose solution presents a weak singularity at $ t = 0 $. The proposed method is established via the L1 scheme on graded mesh for the Caputo fractional derivative and central difference method for spatial derivative, and an ADI method is structured to change the 3D problem into three 1D problems. Using the modified Grönwall inequality we prove the stability and $ \alpha $-robust convergence. The results presented in numerical experiments are in accordance with the theoretical analysis.</p></abstract>

Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard hydraulic conductivity

Aquitards significantly affect groundwater flow in multi-aquifer systems through adjacent aquifer leakage. Despite this, studies focusing on their heterogeneity and the non-conventional diffusion patterns of their flow are still limited. In this study, a factional derivative approach was first extended to explore the time-dependent behavior of flow transport in the aquitard. Two analytical solutions were derived for specific discharges in independent aquitards under different boundary conditions. The findings revealed that aquitard flow exhibits obvious anomalous diffusion behaviors, characterized by slower decay and heavy-tailed specific discharge data. The fractional derivative model provided a more accurate representation of this behavior than traditional models, as evidenced by its superior agreement with experimental data. Moreover, a transient model for pumping tests in a leaky aquifer system was developed, incorporating the memory effect of anomalous flow and vertical heterogeneity in aquitards. Relevant semi-analytical solutions were derived to explore the impacts of memory factor β and decay exponent of aquitard hydraulic conductivity (K) on the leakage aquifer system. Theoretical results demonstrated that stronger memory effect reduces drawdowns in the aquitard and confined aquifer during mid-to-late times. A larger dimensionless decay exponent (Ad) decreases aquitard drawdown and increases aquifer drawdown at late times. Sensitivity analysis showed aquitard drawdown experiences two peaks in sensitivity to β and Ad at early- or mid-times, affected by memory effect and decay exponent of aquitard K, signifying maximal impact at these specific intervals. This study provides a practical model to effectively manage groundwater resources by accurately reflecting aquitard memory and heterogeneity effects.

Lie symmetries and conserved quantities of fractional nonconservative singular systems

AbstractIn this paper, according to the fractional factor derivative method, we study the Lie symmetry theory of fractional nonconservative singular Lagrange systems in a configuration space. First, fractional calculus is calculated by using the fractional factor, and the fractional equations of motion are derived by using the differential variational principle. Second, the determining equations and the limiting equations of Lie symmetry under an infinitesimal group transformation are obtained. Furthermore, the fractional conserved quantity form of singular Lagrange systems caused by Lie symmetry is obtained by constructing a gauge‐generating function that fulfills the structural equation, which conforms to the Noether criterion equation. Finally, we present an example of a calculation. The results show that the Lie symmetry condition of nonconservative singular Lagrange systems is more strict than conservative singular systems, but because of increased invariance restriction, the nonconservative forces do not change the form of conserved quantity; meanwhile, the fractional factor method has high natural consistency with the integral calculus, so the theory of integer‐order singular systems can be easily extended to fractional singular Lagrange systems.

Explore deep network for a class of fractional partial differential equations

In this paper, we present a novel approach for solving a class of fractional partial differential equations (FPDEs) and their inverse problems using deep neural networks (DNNs). Our proposed framework utilizes the discrete Caputo fractional derivative method to approximate fractional partial derivatives, while leveraging automatic differentiation of neural networks to obtain integer derivatives. This approach offers several advantages, including avoiding the direct solution of the original FPDEs and overcoming the limitations faced by traditional numerical methods in handling FPDEs. To validate our approach, we provide numerical examples with known analytical solutions, accompanied by graphical and numerical results. Our findings demonstrate that the proposed method is easily implementable, exhibits fast convergence, robustness, and effectiveness in solving multidimensional FPDEs and their inverse problems.

Improved Grey Wolf Optimizer Tuned Active Disturbance Rejection Control for Ship Heading

The ship heading control is very important to the shipping industry. The Active Disturbance Rejection Control (ADRC) method is ideal for ship heading control, but it is highly correlated with parameters. In this brief, we employed the chaotic mapping and fractional derivative update method to enhance the Grey Wolf Optimizer (GWO), namely Improved GWO (IGWO). The IGWO is proposed to tune the parameters of ADRC adaptively, then the IGWO-tuned ADRC is utilized to control the ship heading. As the comparison, four other optimizers are also employed to tune the parameters of ADRC. Five different disturbance situations are considered to validate the stability of these tuning methods. The robustness of the proposed method for ship parameters perturbation is also analyzed by Monte Carlo experiments. The results indicate that the performance of proposed methods is superior to others.

Experimental and Theoretical Investigation of Viscoelastic Damper by Applying Fractional Derivative Method and Internal Variable Theory

Viscoelastic dampers are conventional passive vibration control devices with excellent energy dissipation performance. The fractional derivative has a simple form and high accuracy in the modelling of viscoelastic materials/dampers. The internal variables reflect the internal state evolution of materials, and are often used to analyze the deformation and thermal process of materials. In the present work, the mechanical properties of a plate-shear-type viscoelastic damper at room temperature are tested under sinusoidal displacement excitations. The impacts of frequency and displacement amplitude on the dynamic properties of the viscoelastic damper in a wide frequency domain (0.1–25 Hz) are investigated. The higher-order fractional derivative model and the temperature–frequency equivalent principle are employed to characterize the frequency and temperature influence, and the internal variable theory considering the internal/microscale structure evolutions is introduced to capture the displacement affection. The higher-order fractional derivative model modified with the internal variable theory and temperature–frequency equivalent principle (ITHF) is accurate enough in describing the dynamic behaviors of viscoelastic dampers with varying frequencies and displacement amplitudes.

He-Laplace method for time fractional burgers-type equations

The time fractional Burgers-type equations with He?s fractional derivative by He-Laplace method. It is a numerical approach coupled the Laplace transformation and HPM. The approximations to the initial value problem with different fractional orders are given without any discretization and complicated computation. Numerical results are provided to confirm its efficiency.

Nonrelativistic quark model for mass spectra and decay constants of heavy-light mesons using conformable fractional derivative and asymptotic iteration method

In this paper, we use the concept of conformable fractional derivative to study the nonrelativistic radial Schrödinger equation. We suggest an extended version of the Cornell potential as the quark–antiquark interaction of light and heavy mesons. We generalize the asymptotic iteration method to the fractional domain. The latter is used to calculate the energy eigenvalues, as well as the effect of the fractional order [Formula: see text] on energy spectra. To test the applicability of our model, we use the obtained results to reproduce the mass spectra of some light and heavy mesons such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] quarks. The mass spectra are obtained at different values of the fractional order parameter [Formula: see text] and were compared with experimental results and other relevant theoretical works. Using the wave function, we calculated the decay constants for heavy-light [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] mesons. Our results are found to be in good agreement with the experimental data, and improved in comparison with other theoretical previsions.

Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

<abstract> <p>In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.</p> </abstract>

An Efficient Design Procedure to Implement the Fractional-Order Chaotic Jerk Systems with the Programmable Analog Platform

An effective design procedure has been introduced for implementing the fractional order integrator structures with a modified low pass filters (LPFs) and its functionality is verified by realizing a fractional-order chaotic system. In these applications, the state variables of the fractional-order Sprott’s Jerk system are emulated by these first order LPFs. Since the discrete device based designs have the hard adjustment features and the circuit complexities; the realizations of these LPFs are carried out with the Field Programmable Analog Arrays (FPAAs), sensitively. Hence, the introduced LPF based method has been applied to the fractional order Sprott’s Jerk systems and these fractional-order systems, which are built by the several nonlinear functions, have been implemented with a programmable analog device. In this context, the minimum fractional-orders of the Sprott’s Jerk systems are calculated by considering the stability of the fractional-order nonlinear systems. After that, these systems are simulated by employing the Grünwald-Letnikov (G-L) fractional derivative method by using a common fractional-order. Thus, the stability analyses of the fractional-order Sprott’s Jerk system are supported by the numerical simulation results. After the numerical simulation stage, the design procedures of the FPAA based implementations of the Sprott’s Jerk systems have been dealt with in detail. Finally, thanks to the introduced first-order LPF method, the hardware realizations of the Sprott’s Jerk systems have been achieved successfully with a single FPAA device.

Application of a New Generalized Fractional Derivative and Rank of Control Measures on Cholera Transmission Dynamics

In this study, the mathematical model of the cholera epidemic is formulated and analyzed to show the impact of Vibrio cholerae in reserved freshwater. Moreover, the results obtained from applying the new fractional derivative method show that, as the order of the fractional derivative increases, cholera-preventing behaviors also increase. Also, the finding of our study shows that the dynamics of Vibrio cholerae can be controlled if continuous treatment is applied in reserved freshwater used for drinking purposes so that the intrinsic growth rate of Vibrio cholerae in water is less than the natural death of Vibrio cholerae. We have applied the stability theory of differential equations and proved that the disease-free equilibrium is asymptotically stable if R 0 < 1 , and the intrinsic growth rate of the Vibrio cholerae bacterium population is less than its natural death rate. The center manifold theory is applied to show the existence of forward bifurcation at the point R 0 = 1 and the local stability of endemic equilibrium if R 0 > 1 . Furthermore, the performed numerical simulation results show that, as the rank of control measures applied increases from no control, weak control, and strong control measures, the recovered individuals are 55.02, 67.47, and 674.7, respectively. Numerical simulations are plotted using MATLAB software package.

Fractional derivative method for describing solitons on the surface of deep water

Fractional derivative method for describing solitons on the surface of deep water

Fractional derivative method for describing solitons on the surface of deep water

Метод дробных производных используется в волновом уравнении для учета дисперсии волн при описании распространения гравитационных солитонных волн на поверхности глубокой воды. Этот подход аналогичен тому, который применяется при получении уравнения Кортевега-де Фриза для солитонов на поверхности мелкой воды, где дисперсионное слагаемое волнового уравнения представляет собой третью производную скорости. Он является альтернативой известному подходу Захарова и др., основанному на модели нелинейного уравнения Шредингера. Получаемое нелинейное интегральное уравнение решается численно.

LTI Kapasitor ve Konformal Kesirli Mertebeden Türev Kullanılarak Modellenmiş Kapasitör ile İki Kapasitör Problemi

Fractional order circuit elements have been started to model different types of circuit elements, circuits and systems in the last decades. There are different types of fractional derivatives. Recently, a new simple fractional derivative method called“conformable fractional derivative” has been brought out. It is simpler than other fractional derivatives and has already been used tomodel supercapacitors. It is important to model the new circuit elements and analyze the circuits containing them so that they can be exploited at their full potential. Two capacitor problem is a famous problem in physics and circuit theory. In this study, a new two capacitor problem a circuit which consists of an LTI capacitor and a supercapacitor which has been modelled with conformable fractional derivative have been examined. The differential equations which describe the circuit have been derived. The circuit current is found explicitly however the voltages of the capacitors do not have analytical solutions. That’s why they are solved numerically.

A high-order numerical scheme for solving nonlinear time fractional reaction-diffusion equations with initial singularity

We propose a high-order numerical scheme for nonlinear time fractional reaction-diffusion equations with initial singularity, where L2-1σ scheme on graded mesh is used to approximate Caputo fractional derivative and Legendre spectral method is applied to discrete spatial variable. We give the priori estimate, existence and uniqueness of numerical solution. Then the unconditional stability and convergence are proved. The rate of convergence is O(M−min⁡{rα,2}+N−m), which is obtained without extra regularity assumption on the exact solution. Numerical results are given to confirm the sharpness of error analysis.

Effectiveness of damaged viscoelastic dampers in seismic protection of structures under main shocks and aftershocks

It has been demonstrated through uniaxial reversal loading tests that damaged viscoelastic dampers are to some extent still able to possess stiffness and energy dissipation capabilities. Based on past performance test results, some analytical models originating from the fractional derivative method have been provided to account for the design, beyond-design, and residual performances of viscoelastic dampers. In this study, these models are employed for modelling the actual dynamic behavior of supplemental viscoelastic dampers installed at different stages in the numerical model of a practical high-rise steel building. Scenarios of main shocks and subsequent aftershocks are assumed as seismic inputs. The numerical results indicate that, even though viscoelastic dampers suffer damage during main shocks, the damaged dampers are still effective in reducing the structural acceleration and displacement responses compared with the undamped structural responses. It is assumed in the analytical model of viscoelastic dampers that the shear force will be immediately reduced to the beyond-design performance without hardening behavior when suffering damage at the beginning half-cycle. Thus, the damped structural acceleration control performance at that instant, particularly of the floors installed with the damaged dampers, might not be satisfactory but still superior to the undamped structural performance. Although some doubts are clarified in this study, it is emphasized that damaged viscoelastic dampers must still be appropriately repaired or even replaced before the next major earthquake occurs.

New analytical solutions for the inextensible Heisenberg ferromagnetic flow and solitonic magnetic flux surfaces in the binormal direction

Maxwellian electromagnetism describes the wave features of the light and related subjects. Its original formulation was established 150 years ago. One of the four Maxwell’s equations is Gauss’s law, which states significant facts regarding magnetic flux through surfaces. It was also observed that optical media provides surface electromagnetism around 60 years ago. This observation leads to improve new techniques on nano-photonics, metamaterials, and plasmonics. The goal of this manuscript is to suggest novel accurate and local conditions for defining magnetic flux surfaces for the inextensible Heisenberg ferromagnetic flow in the binormal direction. The theoretical accuracy of the methodology is verified through the evolution of magnetic vector fields and the anti-symmetric Lorentz force field operator. On the other hand, the numerical accuracy and efficiency is developed in detail by considering the conformable fractional derivative method when these fields are transformed under the traveling wave hypothesis.

New fractional Heisenberg antiferromagnetic model and solitonic magnetic flux surfaces with normal direction

In this paper, we study applications of fractional Heisenberg antiferromagnetic model associated with the magnetic [Formula: see text]-lines in the normal direction. Evolution equations of magnetic [Formula: see text]-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are examined in terms of magnetic and geometric quantities via the conformable fractional derivative method. Finally, we obtain new numerical fractional solutions for nonlinear fractional Schrödinger system with the inextensible Heisenberg antiferromagnetic flow model.

Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.