Derivative Terms Research Articles

Testing and simulation of the hysteresis characteristics of magnetostrictive materials under harmonic excitation

As various power electronic devices are widely used in the power grid, the current flowing through the drive winding of the magnetostrictive actuator inevitably contains harmonic components. The flux density within the hysteresis material is non-sinusoidal with harmonic components, which distorts the hysteresis characteristics of the material and affects the accurate calculation of losses. This paper builds a harmonic magnetic performance testing platform for magnetostrictive materials to test and analyze the magnetic properties under different excitation conditions (magnetic density amplitude, harmonic order, and harmonic content). To complete the mathematical description of the static hysteresis model containing closed minor hysteresis loops, two pinning coefficients k1(Bn,kTHD) and k2(Bn,kTHD), which are related to the degree of harmonic distortion, are used to express the rate of change of the irreversible magnetization components. Fractional order derivative and harmonic frequency-dependent terms are used to characterize the eddy losses and excess losses mathematically. Finally, the dynamic hysteresis model under harmonic excitation is developed. Comparative analysis of model calculation results and experimental test data shows that the model can not only accurately simulate the distorted hysteresis characteristics but also predict the loss of magnetostrictive materials. This research is of significance to the optimal design and temperature rise analysis of magnetostrictive devices.

A Fractional Reduced Differential Transform Method for Solving Multi-Fractional Telegraph Equations

This paper presents a novel modification of the Fractional Reduced Differential Transform Method (FRDTM) to solve space-time multi-fractional telegraph equations. The telegraph equation is crucial in modeling voltage and current distribution in electrical transmission lines, and its solutions have applications in physics, economics, and applied mathematics. The proposed method effectively simplifies the fractional differential equations by omitting one fractional derivative term, allowing for the transformation of the remaining terms using the FRDTM. The solutions demonstrate the method’s accuracy and efficiency in fractional partial differential equations. This study advances the analytical solutions of fractional telegraph equations by providing a straightforward yet powerful approach to fractional differential problems.

Incremental Nonlinear Dynamics Inversion Control with Nonlinear Disturbance Observer Augmentation for Flight Dynamics

A flight controller formulation based on incremental nonlinear dynamics inversion (INDI) control with nonlinear disturbance observer (NDO) is proposed. INDI control is a nonlinear controller based on incremental dynamics. Aimed to attain robustness for nonlinear dynamics inversion (NDI)-based controller, incremental dynamics are derived using the first-order Talyor series expansion to nonlinear systems. The incremental dynamics-based controller requires information on state derivative terms to strengthen the robustness property of the nonlinear controller. The proposed controller utilizes the first-order low-pass filter to obtain the state derivative estimate to implement incremental dynamics into the system. Because the incremental form creates uncertainty term which is an aftermath of the Taylor series expansion, the proposed controller adopts the NDO to eliminate this effect. The controller is applied to the generic transport model which was developed by NASA for simulation purposes. The proposed NDO-based INDI control underwent simulations, together with an INDI controller without disturbance observer, and showed that the developed method results in better performances, providing important advantages where it compensates the uncertainties, removes the steady-state error, and shows less oscillating longitudinal body rate response than the baseline controller, desirable for aerodynamics applications with faster system response.

Analysis of Bioheat Transfer with Thermoelastic Effect

The development of thermal therapy always requires more reasonable temperature distribution predictions. Controlling the amount of heating is a common practice within general thermal therapy operations. This paper used a modified three-phase lag (TPL) bioheat transfer equation to describe the behavior of heat conduction in tissue with thermoelastic effect. To explore the effect of thermal load, the tissue was subjected to a constant surface temperature and a pulsed surface heat flux, respectively. The modified TPL bioheat transfer equation involves mixed derivative terms and higher-order time derivatives of temperature. In analyzing such problems, there are mathematical difficulties. Therefore, the hybrid numerical scheme based on the Laplace transform and an improved discrete method was proposed to solve the present problem. The influence of thermoelastic parameters on the behavior of heat transfer in tissue has been investigated. The results depict the effect of thermal load on thermal response within heat transfer medium is obvious. The thermoelastic effect excites the thermal response oscillation, which is intensified for the reduction of material constant characteristic [Formula: see text] and phase lag [Formula: see text], in the heat conduction medium.

Global Kato smoothing and Strichartz estimates for Schrödinger type equations with rough decay potentials

Let [Formula: see text] be a higher-order elliptic operator on [Formula: see text], where [Formula: see text] is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with [Formula: see text]. In particular, we first establish sharp global Kato smoothing estimates for [Formula: see text], based on uniform resolvent estimates of Kato–Yajima type for the absolutely continuous part of [Formula: see text]. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig–Ruiz–Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

Numerical solutions of PDEs for corporate bond pricing: A computational finance approach

Abstract. In this paper, we will see how one can use numerical algorithms to solve PDEs, containing partial derivative terms, for corporate bond pricing. We utilise the Black-Scholes framework which involves adding stochastic interest rates and credit risk into a PDE model. Then, we will take a quick look at some different numerical methods to solve the PDE model, like finite difference methods (FDM), finite element methods (FEM) as well as standard Monte Carlo (MC) simulations. Above all, we provide some simulations to show how well they work for PDEs, and how much time it takes. We would like to highlight that numerical methods lead to more accurate and efficient bond pricing with these methods, and also that we can analyse together how well each method works under different market scenarios. The paper will end with future directions for numerical methods, which should allow including more sophisticated algorithms and machine-learning techniques that improve scalability and can be used in a real-time manner in financial markets.

A solving new method for the urea-selective catalytic reduction (SCR) system in a diesel engine using coupled hyperbolic-parabolic partial differential equations (PDEs)

To control the diesel engine urea SCR system with high accuracy, firstly, the partial differential equations of the SCR system are simplified through variable substitution and the method of characteristic lines to eliminate the partial derivative terms of the hyperbolic partial differential equations in the flow direction. The backward difference method is used to solve the problem, and the adaptive time step is adjusted to improve computational efficiency. Secondly, the Levenberg-Marquardt algorithm is applied to identify the model parameters per second based on the 1800-s test bench data. By combining the experimental data with the parameter identification results, this paper calculated the downstream NOx concentration with 99.5 % accuracy. Finally, the 1800s transient test data was applied to a commonly used single-state SCR control model, and cell numbers 1–4 of the cases were numerically simulated. It was found that the reduced-order model had a computation time of 1 s but was less accurate. When the test data was applied to the model presented in this study, the calculation time was 27s, and the model's calculation results show that the average error of the downstream NOx concentration is 16.95 ppm, which is 14.3 ppm lower than that of the two-cell one-state model.

D-Brane effective Lagrangian in spacetimes with boundaries

In this study, we explore the transformation of Dp-branes to Dp-1-branes under T-duality when the D-brane is embedded in a spacetime with a boundary. Our goal is to derive the higher-derivative corrections to the Dirac–Born–Infeld (DBI) Lagrangian for both the bulk and boundary terms. For the bulk terms, we calculate the α′ corrections for the massless open string fields, up to the 8th order in the dimensionless Maxwell field strength. We demonstrate that the bulk Lagrangian can satisfy the T-duality constraint without residual total derivative terms in the base space. This determines the most general independent couplings of the massless open string fields in the bulk Lagrangian, encompassing 145 coupling constants, up to three parameters. Two of these parameters are physical and are determined by disk-level S-matrix elements, while the third is unphysical and can be eliminated by field redefinitions and integration by parts. The final result for the bulk Lagrangian consists of 49 couplings. For the boundary terms, the application of T-duality symmetry to massless open and closed string fields enables us to extend the DBI Lagrangian to include the boundary unit normal vector field nμ and its first derivative.

Interfacial dynamic impermeable crack analysis in dissimilar piezoelectric materials by a new interaction integral

Dynamic intensity factors (IFs) provide an important parameter for assessing the failure risk of an interfacial crack for piezoelectric composites exposed to electromechanical impact loadings. In the present research, a new dynamic interaction integral (I-integral) is built for computing the dynamic stress IFs (SIFs) and electric displacement IF (EDIF) of an interfacial crack located in dissimilar inhomogeneous piezoelectric media. Through proper selection of auxiliary variables, the domain expression of I-integral for the inhomogeneous piezoelectric bi-materials does not need to take into account any derivative term for any piezoelectric material property. Further, after rigorous theoretical derivation, the resulting I-integral is still valid for complex models where the integration domain contains other multiple interfaces, regardless of whether the extra materials are straight or curved. Incorporating the modified extended finite element method, the accuracy is confirmed by checking the dynamic IFs extracted from the I-integral with referenced results. The domain-independence is tested by varying ranges of integration domains for inhomogeneous piezoelectric bi-materials and multi-interface piezoelectric composites. Finally, typical examples are employed to discuss the influences of the direction and magnitude of electric impact loading, combination of polarizations, inhomogeneous degree of piezoelectric bi-materials and complex distribution of diverse material properties on the dynamic IFs.

Lattice Boltzmann method for tempered time-fractional diffusion equation

Tempered fractional calculus, as an extension of fractional calculus, has been successfully applied in numerous scientific and engineering fields. Although several traditional numerical methods have been improved for solving a variety of tempered fractional partial differential equations, solving these equations by the lattice Boltzmann (LB) method is an unresolved issue. This paper is dedicated to presenting a novel LB method for the tempered time-fractional diffusion equation. The tempered time-fractional diffusion equation is first transformed into an integer-order partial differential equation by approximating the tempered fractional derivative term. Then the LB method is proposed to solve the transformed objective equation. The Chapman-Enskog procedure is conducted to confirm that the present LB method can accurately recover the objective equation. Some numerical examples with an analytical solution are employed to validate the present LB method, and a strong consistency is observed between the numerical and analytical solutions. The numerical simulations indicate that the LB method is a second-order accurate scheme. The proposed LB method presents a new approach to solving the tempered time-fractional diffusion equation, which is beneficial for the widespread application of the tempered time-fractional diffusion equation in addressing complex transport problems.

Fermionic Casimir energy in Horava–Lifshitz scenario

In this work, we investigate the violation of Lorentz symmetry through the Casimir effect, one of the most intriguing phenomena in modern physics. The Casimir effect, which represents a macroscopic quantum force between two neutral conducting surfaces, is widely regarded as a triumph of Quantum Field Theory. In this study, we present new results for the Casimir effect, focusing on the contribution of mass associated with fermionic quantum fields confined between two large parallel plates, in the context of Lorentz symmetry violation within the Horava–Lifshitz formalism. To calculate the Casimir energy and pressure, we impose a MIT bag boundary condition on the two plates, compatible with the higher-order derivative term in the modified Dirac equation. Our results reveal a strong influence of Lorentz violation on the Casimir effect. We observe that the Casimir energy is significantly affected, both in intensity and sign, potentially resulting in a repulsive or attractive force between the plates, depending on the critical exponent associated with the Horava–Lifshitz formalism.

Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations*

Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations.

Development of a smoothed particle hydrodynamics model for porous media flows with enhanced volume conservation and the revisit of the mass conservation equation

Porous media exist extensively in hydraulic and coastal engineering structures, while the modeling of wave/flow interaction with porous media remains challenging. This work develops a smoothed particle hydrodynamics (SPH) model for accurately simulating wave/flow interaction with porous media. The mass and momentum conservation equations incorporating the mixture theory are adopted. The resistant forces of the solid skeleton of porous media on fluid flows are described by the nonlinear empirical formula. The research contributions of the work lie in two aspects. First, two categories of mass conservation equations for porous media flow are revisited and analyzed to examine the influences of the local time derivative term of fluid volume fraction on simulation results. Second, the Volume Conservation Shifting scheme is, for the first time, introduced into SPH to enhance volume conservation for simulating porous media flows. The developed SPH model is validated by an analytical case of seepage flows in a U-tube with porous media and then applied to study four benchmark examples involving both saturated and unsaturated porous media, i.e., dam-break flow through a crushed stone dam, rapid seepage flow through a rockfill dam, solitary wave propagation over a porous seabed, and solitary wave propagation over a submerged porous breakwater. The morphological features and dynamic pressure heads of the porous media flows have been satisfactorily predicted, demonstrating the good accuracy and enhanced volume conservation of the developed SPH model.

O(9, 25) symmetry of heterotic string theory at orders α′, α′2

In a recent study, we have observed that by imposing a truncated T-duality transformation on the circular reduction of the bosonic couplings in the heterotic theory at four- and six-derivative orders, we can calculate these couplings in a particular YM gauge where the YM potential vanishes but its field strength remains non-zero. Importantly, the coupling constants are independent of the gauge choice, so these results are valid across different YM gauge choices.In this work, we explore the cosmological reduction of these couplings when the YM gauge fields belong to the Cartan subalgebra of SO(32) or E8 × E8. We demonstrate that after applying appropriate one-dimensional field redefinitions and total derivative terms, the couplings can be expressed in a proposed O(9, 25)-invariant canonical form, which is the extension of the canonical O(9, 9)-invariant form for just the NS-NS fields proposed by Hohm and Zwiebach. This O(9, 25)-invariant expression is in terms of the trace of the first time derivative of the generalized metric, which encompasses both the YM field and the NS-NS fields.

An improved affine mixed‐grid method for frequency‐domain finite‐difference elastic modelling

AbstractIn seismic frequency‐domain finite‐difference modelling, the affine mixed‐grid method effectively eliminates the spatial sampling restriction associated with square meshes of the rotated mixed‐grid method. Nevertheless, the affine mixed‐grid method makes a weighted average of the entire elastic wave equations, resulting in reduced accuracy compared to the average‐derivative method in the case of rectangular meshes. It is worth noting, however, that the average‐derivative method is presently inapplicable to free‐surface scenarios, whereas the affine mixed‐grid method is applicable. By performing weighted averages of the derivative terms instead of the entire elastic wave equations in Cartesian and affine rotated coordinate systems, we have developed an improved affine mixed‐grid method for elastic‐wave frequency‐domain finite‐difference modelling. The proposed improved affine mixed‐grid method 9‐point scheme overcomes the drawback that the accuracy of affine mixed‐grid method is lower than that of average‐derivative method for unequal directional grid intervals. Moreover, the improved affine mixed‐grid method 6‐point scheme provides much higher numerical accuracy than the affine mixed‐grid method 6‐point scheme at either equal or unequal directional grid intervals. On the other hand, the proposed improved affine mixed‐grid method simplifies the coding complexity for implementing free‐surface condition in elastic‐wave frequency‐domain finite‐difference modelling by modifying the elastic parameters of the free‐surface layer and thus constructing the impedance matrix containing the free‐surface condition directly.

A robust proportional filtered integral controller based on backpropagation neural network

This paper introduces a novel design of a proportional filtered-integral (P-FI) controller whose parameters are auto-tuned using the backpropagation neural networks (BPNN) algorithm. The proposed controller is designed to address the main challenges posed by a class of complex systems characterized by uncertain high-gain and pure integrator dynamics, including high-frequency noise amplification and poor robustness against parameter variations. Designing such a controller involves three main steps: First, a proportional–integral–derivative (PID) controller is designed, with its parameters auto-tuned online using the BPNN algorithm, resulting in the primary (BPNN-PID) controller. Second, the obtained parameters of the previous controller are utilized to compute those of a low-pass filter offline. This filter is then cascaded with the integral parts of a PID controller, forming a P-FI controller structure. This configuration introduces a phase lead within a specific frequency range without amplifying high-frequency noise, overcoming the primary disadvantage of the derivative term. Finally, the parameters of the resulting P-FI controller are again auto-tuned online using the BPNN algorithm, resulting in the final robust (BPNN-P-FI) controller. This novel controller structure and its parameters tuning procedure, based on a two-stage BPNN learning approach, constitute the main contributions of this paper. Simulation results demonstrate the superiority of the proposed controller in terms of time-domain performance, sensor noise attenuation, and closed-loop robustness compared to those obtained with the BPNN-PID and optimally tuned PID controllers.

A high‐order pure streamfunction method in general curvilinear coordinates for unsteady incompressible viscous flow with complex geometry

AbstractIn this paper, a high‐order compact finite difference method in general curvilinear coordinates is proposed for solving unsteady incompressible Navier‐Stokes equations. By constructing the fourth‐order spatial discretization schemes for all partial derivative terms of the pure streamfunction formulation in general curvilinear coordinates, especially for the fourth‐order mixed derivative terms, and applying a Crank‐Nicolson scheme for the second‐order temporal discretization, we extend the unsteady high‐order pure streamfunction algorithm to flow problems with more general non‐conformal grids. Furthermore, the stability of the newly proposed method for the linear model is validated by von‐Neumann linear stability analysis. Five numerical experiments are conducted to verify the accuracy and robustness of the proposed method. The results show that our method not only effectively solves problems with non‐conformal grids, but also allows grid generation and local refinement using commercial software. The solutions are in good agreement with the established numerical and experimental results.

Damping efficiency of the fractional Duffing system and an assessment of its solution accuracy

Dynamical systems with high damping efficiency in a wide frequency band can be useful on a small scale – for harvesting energy from ambient vibrations, and on a large scale – for damping harmful vibrations of mechanical structures. In this paper we present an assessment of the quality of solutions and damping efficiency of systems with fractional order derivatives. To simulate the fractional system the fourth-order Runge–Kutta method and Grünwald–Letnikov methods are used. We propose a coefficient for assessing the quality of solutions to fractional systems by reference to the quality of the calculated energy balance. As an exemplary system we study the Duffing model with embedded additional fractional-order derivative terms. Based on this coefficient, intervals of key numerical simulation parameters are determined to ensure the appropriate quality for the calculations of energy flows and energy balance. The determined values of these parameters are then used in tests of the damping efficiency of the studied system. Our results show that by modifying the fractional terms it is possible to configure a system that exhibits a “broadband effect”, i.e. a system that is characterized by high-amplitude vibrations and, consequently, high energy efficiency in a wide range of excitation frequencies.

Modulation instability, state transitions and dynamics of multi-peak rogue wave in a higher-order coupled nonlinear Schrödinger equation

The coupled nonlinear Schrödinger equation provides an effective description of the propagation of four optical fields in the fibers. The present work explores the dynamics of rogue waves that arise in a coupled nonlinear Schrödinger equation of higher-order that is acquired using the Ablowitz-Kaup-Newell-Segur method. The generalized (m,N−m)-fold Darboux transformation is used to determine the exact rogue wave solutions, which show multiple peaks and depressions. In addition, a detailed analysis is conducted of the modulation instability of three different kinds of plane-wave solutions. The results indicate that the coefficients of the second and third derivative terms, rather than just the coefficients of the third derivative term, have an impact on modulation instability. Lastly, the transition between rogue waves and solitons under the influence of higher-order dispersion effects is elucidated, with explicit soliton solutions provided within the modulation stability region.

Vibration suppression of a platform by a fractional type electromagnetic damper and inerter-based nonlinear energy sink

The theory of linear and nonlinear dynamic vibration absorbers is a well-established topic for many years. However, many recent contributions paid attention to the nonlinear vibration absorbers and different practical realizations of corresponding devices. Here, we propose a mechanical system constituted of the inerter-based nonlinear energy sink attached to the main body that is resting on an elastic foundation and is grounded through the fractional type electromagnetic damper. The two-degree-of-freedom system is described via two coupled differential equations with one of them having a fractional-order derivative term and the other one containing cubic stiffness nonlinearity. The incremental harmonic balance (IHB) method is employed to solve the equations and studies the strongly nonlinear periodic responses of the system. Applied approximated solution methodology is validated by the numerical Newmark method adapted to deal with the system of nonlinear fractional-order differential equations. The appropriate and necessary number of harmonics used in the IHB solution is commented and validated. This study can be a first step in understanding the dynamics and giving directions for the future design of vibration-isolating platforms based on inerter-based nonlinear vibration absorbers and electromagnetic dampers.