Time-dependent Term Research Articles

Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations

In this article, a new numerical technique is presented to obtain numerical solution of a system of fractional delay differential equations (FDDE’s) involving proportional and time dependent delay terms. The fractional derivative is used in Caputo sense. The proposed technique is the combination of fractional differential transform and Bell polynomial. The existence and uniqueness results are discussed for FDDE’s. Three numerical problems are discussed to show reliability and efficiency of the method. Numerical results are compared with exact and Matlab DDENSD solution. The main advantage of the present method is handing effectively the nonlinear terms present in the FDDEs by using Bell polynomial. The present method can deal with both linear and nonlinear FDDEs. The convergence result is discussed, and error analysis is presented in detail.

Distributed Disturbance Observer-Based Containment Control of Multi-Agent Systems with Event-Triggered Communications

This article investigates a class of multi-agent systems (MASs) with known dynamics external disturbances, where the communication graph is directed, and the followers have undirected connections. To eliminate the impacts of external disturbance, the technologies of disturbance observer-based control are introduced into the containment control problems. Additionally, to save communication costs and energy consumption, a distributed disturbance observer-based event-triggered controller is employed to achieve containment control and reject disturbance. Furthermore, designing the event-triggered function using an exponential function is beneficial for a time-dependent term while ensuring the exclusion of Zeno behavior. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis.

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

Origins of Electromagnetic Radiation from Spintronic Terahertz Emitters: A Time-Dependent Density Functional Theory plus Jefimenko Equations Approach.

Microscopic origins of charge currents and electromagnetic (EM) radiation generated by them in spintronic THz emitters-such as, femtosecond laser pulse-driven single magnetic layer or its heterostructures with a nonmagnetic layer hosting strong spin-orbit coupling (SOC)-remain poorly understood despite nearly three decades since the discovery of ultrafast demagnetization. We introduce a first-principles method to compute these quantities, where the dynamics of charge and current densities is obtained from real-time time-dependent density functional theory, which are then fed into the Jefimenko equations for properly retarded electric and magnetic field solutions of the Maxwell equations. By Fourier transforming different time-dependent terms in the Jefimenko equations, we unravel that in the 0.1-30THz range the electric field of far-field EM radiation by the Ni layer, chosen as an example, is dominated by charge current pumped by demagnetization, while often invoked magnetic dipole radiation from the time-dependent magnetization of a single magnetic layer is a negligible effect. Such an effect of charge current pumping by a time-dependent quantum system, whose magnetization is shrinking while its vector does not rotate, does not require any spin-to-charge conversion via SOC effects. In the Ni/Pt bilayer, EM radiation remains dominated by the charge current within the Ni layer, whose magnitude is larger than in the case of a single Ni layer due to faster demagnetization, while often invoked spin-to-charge conversion within the Pt layer provides an additional but smaller contribution. By using the Poynting vector and its flux, we also quantify the efficiency of conversion of light into emitted EM radiation, and its angular distribution.

Model order reduction for the input–output behavior of a geothermal energy storage

In this article, we consider a geothermal energy storage system in which the spatio-temporal temperature distribution is modeled by a heat equation with a time-dependent convection term. Such storage systems are often embedded in residential heating systems. The control and management of such systems requires knowledge of aggregated characteristics of the temperature distribution in the storage. These describe the input–output behavior of the storage, the associated energy flows, and their response to charging and discharging processes. Our aim is to derive an efficient, approximate description of these characteristics by using low-dimensional systems of ordinary differential equations (ODEs). This leads to a model order reduction problem for a large-scale linear system of ODEs resulting from the semidiscretization of the heat equation combined with a linear algebraic output equation. In a first step, we approximated the nonautonomous system of ODEs by a linear time-invariant system. Then, we applied Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system that has only a few state equations but almost the same input–output behavior. The results of our extensive numerical experiments show the efficiency of the applied model order reduction method. We found that only a few suitably chosen ODEs are sufficient to achieve good approximations of the input–output behavior of the storage.

Restoration of the merely time-dependent lowest term in a linear Bi-flux diffusion equation

Restoration of the merely time-dependent lowest term in a linear Bi-flux diffusion equation

Global existence and Blow-up for the 1D damped compressible Euler equations with time and space dependent perturbation

In this paper, we consider the 1D Euler equation with time and space dependent damping term −a(t,x)v. It has long been known that when a(t,x) is a positive constant or 0, the solution exists globally in time or blows up in finite time, respectively. In this paper, we prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient a satisfies the following condition |a(t,x)−μ0|≤a1(t)+a2(x),where μ0≥0 and a1 and a2 are integrable functions with t and x. Under this condition, we show the global existence and the blow-up with small initial data, when μ0>0 and μ0=0 respectively. The key of the proof is to divide space into time-dependent regions, using characteristic curves.

A fourth Order Pseudoparabolic Inverse Problem to Identify the Time Dependent Potential Term from Extra Condition

In this work, the pseudoparabolic problem of the fourth order is investigated to identify the time -dependent potential term under periodic conditions, namely, the integral condition and overdetermination condition. The existence and uniqueness of the solution to the inverse problem are provided. The proposed method involves discretizing the pseudoparabolic equation by using a finite difference scheme, and an iterative optimization algorithm to resolve the inverse problem which views as a nonlinear least-square minimization. The optimization algorithm aims to minimize the difference between the numerical computing solution and the measured data. Tikhonov’s regularization method is also applied to gain stable results. Two examples are introduced to explain the reliability of the proposed scheme. Finally, the results showed that the time dependent potential terms are successfully reconstructed, stable and accurate, even in inclusion of noise.

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Boundedness of solutions to a second-order periodic system with p-Laplacian and unbounded perturbation terms

The second-order periodic system with p-Laplacian and unbounded time-dependent perturbation terms is investigated. Using the principle integral method, it is shown that under certain assumptions on the unbounded and periodic terms, all solutions to the equation possess boundedness.

Asymptotically autonomous robustness of random attractors for the stochastic wave lattice equations with random viscosity

In this paper, we focus on the asymptotic behavior of solutions to the non-autonomous stochastic wave lattice systems with random viscosity and multiplicative white noise. Under proper conditions, we first prove the existence and uniqueness of the pullback random attractors and then show their backward compactness in a weighted space. Finally, we establish the asymptotically autonomous robustness of pullback random attractors when the nonlinearity is bounded and the time-dependent forcing term converges to the time-independent counterpart.

Effect of installing branch-shaped fin on cold energy saving during freezing considering nanomaterial

This research examines the solidification through a finned tank containing alumina nano-powders and water as PCM (phase change material). Mathematical models were developed, assuming a uniform concentration of nano-powders and neglecting convective term effects. The model includes three time-dependent terms, discretized using an implicit approach, with solutions obtained via the Galerkin method and convergence achieved using an adaptive mesh. The thermal conductivity of NEPCM improves with increased concentration (ϕ), resulting in a faster transition from liquid NEPCM to ice. For pure water, the complete freezing time is 6795.38 s; however, with additives, this time is reduced by 26.78 %. Additionally, using powders with a higher "m" value further accelerates the freezing process, decreasing the completion time by about 6.98 %. The effect of powder configuration becomes more pronounced with increasing concentration. These findings are crucial for enhancing the sustainability of natural resources by improving cold storage and solidification processes.

An inverse problem of reconstructing the unknown coefficient in a third order time fractional pseudoparabolic equation

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

The dimension coupling method for 3D transient heat conduction problem with variable coefficients

In this study, a dimension coupling method (DCM) is presented to study the variable coefficients transient heat conduction problems. The dimension splitting method (DSM) is introduced to split the 3D domain into some 2D domains, which are discretized by selecting the improved element-free Galerkin (IEFG) method; thus the corresponding discretized equation in each 2D domain is derived, and the finite element method (FEM) is employed to deal with these discretized equations in a third direction, and the time-dependent term is discretized by selecting the finite difference method (FDM), thus the final solved equation of the transient heat conduction problems is obtained. In the first example, the convergence is proved by increasing the nodes and meshes on the influence of relative errors. Some numerical results illustrate that the proposed method can substantially enhance the computational efficiency of the IEFG method. When an exact solution of the numerical example is a nonlinear function about splitting direction with natural boundary condition is given, the proposed method has shorter computing time than the hybrid element-free Galerkin (HEFG) method when dealing with this situation.

Properties of molecular clumps and cores in colliding magnetized flows

ABSTRACT We simulate the formation of molecular clouds in colliding flows of warm neutral medium with the adaptive mesh refinement code flash in eight simulations with varying initial magnetic field strength, between 0.01–5 μG. We include a chemical network to treat heating and cooling and to follow the formation of molecular gas. The initial magnetic field strength influences the fragmentation of the forming cloud because it prohibits motions perpendicular to the field direction and hence impacts the formation of large-scale filamentary structures. Molecular clump and core formation occurs anyhow. We identify 3D clumps and 3D cores, which are defined as connected, CO-rich regions. Additionally, 3D cores are heavily shielded. While we do not claim those 3D objects to be directly comparable to observations, this enables us to analyse their full virial state. With increasing field strength, we find more fragments with a smaller average mass; yet the dynamics of the forming clumps and cores only weakly depends on the initial magnetic field strength. The molecular clumps are mostly unbound, probably transient objects, which are weakly confined by ram pressure or thermal pressure, indicating that they are swept up by the turbulent flow. They experience significant fluctuations in the mass flux through their surface, such that the Eulerian reference frame shows a dominant time-dependent term due to their indistinct nature. We define the cores to encompass highly shielded molecular gas. Most cores are in gravitational-kinetic equipartition and are well described by the common virial parameter $\alpha _\mathrm{vir}$, while some undergo minor dispersion by kinetic surface effects.

Quasi-equilibrium chemical evolution in starless cores

ABSTRACT The chemistry of H2O, CO, and other small molecular species in an isolated pre-stellar core, L1544, has been assessed in the context of a comprehensive gas-grain chemical model, coupled to an empirically constrained physical/dynamical model. Our main findings are (i) that the chemical network remains in near equilibrium as the core evolves towards star formation and the molecular abundances change in response to the evolving physical conditions. The gas-phase abundances at any time can be calculated accurately with equilibrium chemistry, and the concept of chemical clocks is meaningless in molecular clouds with similar conditions and dynamical time-scales, and (ii) A comparison of the results of complex and simple chemical networks indicates that the abundances of the dominant oxygen and carbon species, H2O, CO, C, and C+ are reasonably approximated by simple networks. In chemical equilibrium, the time-dependent differential terms vanish, and a simple network reduces to a few algebraic equations. This allows rapid calculation of the abundances most responsible for spectral line radiative cooling in molecular clouds with long dynamical time-scales. The dust ice mantles are highly structured and the ice layers retain a memory of the gas-phase abundances at the time of their deposition. A complex (gas-phase and gas-grain) chemical structure therefore exists, with cosmic-ray induced processes dominating in the inner regions. The inferred H2O abundance profiles for L1544 require that the outer parts of the core and also any medium exterior to the core are essentially transparent to the interstellar radiation field.

IDENTIFICATION OF A TIME-DEPENDENT SOURCE TERM IN A NONLOCAL PROBLEM FOR TIME FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data.

The prediction of deformation in the process of erecting ship blocks based on inherent strain approach

Ship structures are referred to as plated and/or welded structures. Plates are sequentially welded using a high-temperature heat source to assemble the overall ship. The heat source used in the sequential welding inevitably generates imperfections in the ship structure, e.g., welding-induced deformation. Predicting welding-induced deformation is a critical task in design and control management at a shipyard, and much research has focused on the inherent strain method among possible efficient numerical methods. However, determining the inherent strain requires the synchronisation of all strain terms as a function of time and welding uncertainty because the welding position depends on the human worker. The present study thus derives time-dependent terms for inherent strain of the overall welding process and includes the welding position and cooling method in the function of the heat transfer coefficient. The inherent strain is derived through the detailed analysis of the overall welding process in terms of the application and utilisation, and it is simulated and reviewed for the erection of the hull block of a container ship and an LNG carrier. The inherent strain is expected to be used as a simple form of strain in the further study of various large welded structures and materials and in research on welding parameters.

SIMULTANEOUS IDENTIFICATION OF THE RIGHT-HAND SIDE AND TIME-DEPENDENT COEFFICIENTS IN A TWO-DIMENSIONAL PARABOLIC EQUATION

This paper investigates the simultaneous identification of time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from the additional measurements. To investigate the solvability of the inverse problem, we first examine an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Then, applying the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Furthermore, using the equivalency, the existence and uniqueness theorem for the classical solution of the original problem is obtained and some discussions on the numerical solutions for this inverse problem are presented including numerical examples.

Mixed-Delay-Based Augmented Functional for Sampled-Data Synchronization of Delayed Neural Networks With Communication Delay.

The synchronization control for delayed neural networks (DNNs) via a sampled-data controller considering communication delay is studied by input delay approach. Although few scholars have put forward the coexistence of transmission delay and communication delay in this problem, no report has clarified the interaction between transmission delay and communication delay. Also, the time-squared terms are underutilized. Thus, a novel augmented Lyapunov functional, which consists of a mixed-delay-based augmented part and a time-squared two-sided looped part, is proposed to fill this gap. In the mixed-delay-based augmented part, not only the information of transmission delay and communication delay themselves, but also the interaction between those two delays is considered. Time-dependent quadratic terms as well as the sampling integral states are introduced in the two-sided looped part, so that more characteristic information of the sampling pattern is encompassed and the relationship of the states at the sampling instant is enhanced. Then, this novel augmented functional is applied to the synchronization control of DNNs. A less conservative synchronization criterion is obtained in the form of linear matrix inequalities. A numerical example illustrates the validity and superiority of the presented synchronization criterion.