First-order Linear Partial Differential Equations Research Articles

МОДЕЛЬ АДСОРБЦИОННОЙ ОЧИСТКИ СТОКОВ С УЧЕТОМ ХИМИЧЕСКОЙ РЕАКЦИИ

The article poses and solves the general problem of the dynamics of adsorption from a liquid in a fixed layer of adsorbent, taking into account a first-order chemical reaction and a linear adsorption isotherm. The adsorption model is based on the mass balance equations of the target component for an infinitesimal layer element and adsorption kinetics, written as a system of linear first-order partial differential equations with corresponding boundary conditions. The problem is solved by the Laplace transform method. This made it possible to reduce the joint solution of a system of equations in the originals to an independent solution in images of the Cauchy problem for an ordinary differential equation and an algebraic equation. As a result, expressions were obtained for the concentrations of the target component in the adsorbent layer and in the flow at any time and in any section of the layer. This made it possible to simulate a practical problem of the adsorption of chloroform by zeolites for the treatment of wastewater from a real swimming pool and obtain interesting graphic materials based on the solution and the developed algorithm. The results of the solution can be translated into numbers by programming in any available high-level algorithmic language. It is shown that the shape of the curves of changes in the concentrations of the target component in the adsorbent and adsorbent over the thickness of the adsorbent layer and over time does not change, although the numerical values of these parameters change in a fairly wide range. It is also noted that in the range of concentration changes considered in the article, the concentration of the adsorbent initially decreases sharply, and then smoothly along the entire length of the fixed adsorbent layer decreases slightly. It has been revealed that the use of methods, techniques and theorems of the Laplace transform method in relation to the problem of adsorption dynamics makes it possible to deeply penetrate the mechanism of the problem being solved, which is not possible with other solution methods. This allows, with a known set of initial data, to find the change in the concentrations of the target component in both phases and select the required thickness of the adsorbent layer.

First-order linear Marcus SPDEs

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)].

Some remarks on the solution of the cell growth equation

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al . 2015 Solutions to an advanced functional partial differential equation of the pantograph type ( Proc. R. Soc. A 471 , 20140947 ( doi:10.1098/rspa.2014.0947 )). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.

APPLICATION OF THE FOURIER TRANSFORMATION FOR SOLVING A FIRST ORDER PARTIAL DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

This work is devoted to the solution of a linear first-order hyperbolic partial differential equation with constant coefficients. The specific objective is to prove the existence and uniqueness of the solution of the proposed PDE. The existence and uniqueness of the solution have been proved. To demonstrate the existence of the solution, the Fourier transformation was used. The variational formulation was used to prove the uniqueness of the solution. The combination of the Fourier transformation and the variational formulation yielded the expected results: the existence and uniqueness of the solution.

Robust stabilization of 2 × 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of 2 × 2 first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.

On a system of differential equations with random parameters

An explicit formula for the mathematical expectation and second moment functions of a solution to a linear system of ordinary differential equations with a random parameter and a vector random righthand side is obtained. The problem is reduced to the deterministic Cauchy problem for systems of first-order linear partial differential equations. We obtain an explicit formula for a solution of linear systems of partial differential equations of the first order with constant coeffcients. An example is given showing that random factors can have a stabilizing effect on a linear system of differential equations.

Robustification of stabilizing controllers for ODE-PDE-ODE systems: A filtering approach

In this paper, we present a filtering technique that robustifies stabilizing controllers for systems composed of heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) interconnected with Ordinary Differential Equations (ODEs) through their boundaries. The actuation is either available through one of the ODE or at the boundary of the PDE. The proposed framework covers a broad general class of interconnected systems. Assuming that a stabilizing controller is available, we derive simple sufficient conditions under which appropriate low-pass filters can be combined with the control law to robustify the closed-loop system. Our approach is based on a rewriting of the distributed dynamics as a delay-differential algebraic equation and an analysis in the Laplace domain. The proposed technique will simplify the design of stabilizing controllers for the class of systems under consideration (which can now be done only on a case-by-case basis due to the complexity and generality of the underlying interconnections) since it dissociates the stabilization problem from the robustness aspects. Indeed, it becomes possible to use convenient (but non-robust) techniques for the stabilization of such systems (as the cancellation of the boundary coupling terms or the inversion of the ODE dynamics), knowing that the resulting control law can be made robust (to delays and uncertainties) using the proposed filtering methodology.

A Population Pyramid Dynamics Model and Its Analytical Solution. Application Case for Spain

This paper presents the population pyramid dynamics model (PPDM) to study the evolution of the population pyramid of a determined country or society, deducing as a crucial objective its exact analytical solution. The PPDM is a first-order linear partial differential equation whose unknown variable is the population density (population per age unit) depending on time and age, jointly an initial condition in the initial time and a boundary condition given by the births in the zero age. In addition, the dynamical patterns of the crude birth, death, immigration and emigration rates depending on time, jointly with the mathematical pattern of the initial population pyramid depending on ages, take part of the PPDM. These patterns can be obtained from the historical data. An application case of the PPDM analytical solution is presented: Spain, in the 2007–2021 period for its validation, and in the 2021–2026 period for its future forecasting. This application case also permits to obtain the forecasting limits of the PPDM by comparing the historical data with those provided by the PPDM. Other variables that can be obtained from the historical population pyramids data, such as the dependency ratio and the life expectancy at birth, are considered.

Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations

Abstract The research of differential equation models in music creation and audio is a current research hotspot. This paper uses homogeneous first-order linear partial differential equations to analyze the relevant music signals in music creation. This paper focuses on analyzing the expression of music signals based on time series. This paper discusses its characteristic function and conducts partial differential equation modeling of music characteristic signals. The method successfully established the string vibration model and the time-series autoregressive model of the music signal. This paper becomes a new differential equation to solve the music signal processing method in music creation.

General solutions’ laws of linear partial differential equations I

Z2 and Z3 transformations are proposed for the first time in this paper, combined with the previously proposed Z1 transformation, the general solutions of many kinds of n-ary m-order linear partial differential equations (PDEs), including the famous transport equation are obtained for the first time. Some first-order linear PDEs that cannot be solved by the characteristic equation method can be solved by Z transformations. Using the general solutions, we also obtain the exact solutions of some definite solution problems.

General solutions of the Laplace equation

With the purpose of concisely and effectively obtaining the general or exact solutions of partial differential equations (PDEs), we put forward some universal Z 1 transformations in present paper. Not only many linear equations can be solved, but also analytical solutions of some nonlinear equations can be obtained by utilizing this method, and many solutions contain arbitrary functions. Taking as the typical case, we gain the general solution of Laplace equation for the first time. During the solving process, we find that the form of the general solution of some PDEs is not unique. On the basis of the practical cases, we also find that general solutions of some first-order linear PDEs obtained by the characteristic equation method are incomplete.

Robust state-feedback stabilization of an underactuated network of interconnected [formula omitted] hyperbolic PDE systems

We detail in this article the development of a delay-robust stabilizing state feedback control law for an underactuated network of two subsystems of heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the two subsystems is actuated. The proposed approach is based on the backstepping methodology. A backstepping transform allows us to construct a first feedback to tackle in-domain couplings present in the actuated PDE subsystem. Then, we introduce a predictive tracking controller to stabilize the second PDE subsystem. The stabilization of this subsystem implies the stabilization of the whole network. Finally, the proposed control law is combined with a low-pass filter to become robust with respect to small delays in the control signal and uncertainties on the system parameters.

Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

First-order linear partial differential equations using the GeoGebra and GeoGebra 3D graphing calculator

In this paper we will show, through various examples of how are the animated solutions of certain linear partial differential equations of the first order with constant homogeneous coefficients through GeoGebra applets designed for a first course of Partial Differential Equations for the engineering careers of the University of Antofagasta-Chile

ReLOPE: Resistive RAM-Based Linear First-Order Partial Differential Equation Solver

Data movement between memory and processing units poses an energy barrier to Von-Neumann-based architectures. In-memory computing (IMC) eliminates this barrier. RRAM-based IMC has been explored for data-intensive applications, such as artificial neural networks and matrix-vector multiplications that are considered as “soft” tasks where performance is a more important factor than accuracy. In “hard” tasks such as partial differential equations (PDEs), accuracy is a determining factor. In this brief, we propose ReLOPE, a fully RRAM crossbar-based IMC to solve PDEs using the Runge–Kutta numerical method with 97% accuracy. ReLOPE expands the operating range of solution by exploiting shifters to shift input data and output data. ReLOPE range of operation and accuracy can be expanded by using fine-grained step sizes by programming other RRAMs on the BL. Compared to software-based PDE solvers, ReLOPE gains $31.4\times $ energy reduction at only 3% accuracy loss.

On a matrix-valued PDE characterizing a contraction metric for a periodic orbit

<p style='text-indent:20px;'>The stability and the basin of attraction of a periodic orbit can be determined using a contraction metric, i.e., a Riemannian metric with respect to which adjacent solutions contract. A contraction metric does not require knowledge of the position of the periodic orbit and is robust to perturbations. <p style='text-indent:20px;'>In this paper we characterize such a Riemannian contraction metric as matrix-valued solution of a linear first-order Partial Differential Equation. This enables the explicit construction of a contraction metric by numerically solving this equation in [<xref ref-type="bibr" rid="b7">7</xref>]. In this paper we prove existence and uniqueness of the solution of the PDE and show that it defines a contraction metric.

Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

We detail in this article the development of a robust stabilizing output feedback control law for an underactuated cascade network of n systems of two heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the subsystems is actuated. The proposed approach combines successive backstepping transformations that present a specific cascade structure. With these transformations it is possible to rewrite the original network system as a simple system for which all the in-domain coupling terms have been removed. One can then design a strictly proper stabilizing control law. The proposed control law is proved to be robust to small delays in the actuation. Finally, a boundary observer is designed, enabling stabilization by output feedback.

Absolute stimulated Raman side scatter in direct-drive laser-produced plasmas

Production of suprathermal electrons by stimulated Raman scattering (SRS) is a principal concern for contemporary direct-drive inertial confinement fusion experiments at the National Ignition Facility and similar systems since such electrons penetrate and preheat the target core, preventing efficient implosion. The higher temperatures and longer scale lengths in these experiments favor SRS over two-plasmon decay, which predominated in earlier experiments. In particular, current experiments are expected to exceed the threshold for absolute Raman side scatter, which would then dominate the interaction since it grows temporally until saturated by nonlinear mechanisms such as hot-electron production. Until recently, analyses of SRS side scatter have treated the case of a single laser beam incident on a plasma, but the direct-drive approach to laser fusion employs a multitude of beams to drive the implosion. Here, we present an analysis that can be applied to an arbitrary number of beams with varied angles of incidence and polarizations. In the case of a single beam, it allows a physically motivated derivation and verification of an analytic threshold formula. In the general case of multiple beams and arbitrary orientation and polarizations, the threshold is found by numerical integration of a set of first-order linear partial differential equations.

Stability Analysis for a Class of Linear $2\times 2$ Hyperbolic PDEs Using a Backstepping Transform

In this paper, we develop a sufficient stability condition for a class of coupled first-order linear hyperbolic partial differential equations (PDEs) with constant coefficients that appear when considering target systems for backstepping boundary control.Using a backstepping transform, the problem is reformulated as a stability problem for a difference equation with distributed delay. Finding the explicit solution to the backstepping kernels, we derive an explicit sufficient condition depending on the plant coefficients. This stability condition is compared to an existing stability result based on a Lyapunov analysis. Both the proposed and existing sufficient conditions are then contrasted in some examples to a (computationally expensive) numerical approximation of a necessary and sufficient condition for exponential stability to illustrate their conservatism.

Steady states of quantum Brownian motion and decomposition of quantum states into ensembles of Gaussian packets having a uniform position variance

After replacing the reduced density matrix of a quantum harmonic oscillator linearly coupled to a thermal bath by its ‘characteristic function’ (a non-negative definite function defined on the Heisenberg-Weyl group), the master equation of the reduced density matrix can be transformed into a first-order linear partial differential equation of the characteristic function. We derive stationary solutions for the Caldeira–Leggett (CL) master equation as well as for the late-time Hu–Paz–Zhang master equation, and examine the positivity problem for these solutions. In order to illustrate to what extent classical properties of quantum states of the damping oscillator emerge dynamically due to coupling to the environment, we discuss the problem of how and under what conditions a steady state predicted by the CL master equation could be expressed as a statistical mixture of Gaussian packets having a uniform position variance. Similar studies are carried out for the solutions of a Lindblad-form generalization of the CL master equation, where the positivity problem is certainly absent. Our results show that in the low-temperature regime the steady states predicted by the Lindblad-type master equation may depend on the damping strength of the bath and there are cases which may not show Gaussian-packet decomposition.