First-order Partial Differential Equations Research Articles

Rigorous solution for drained analysis of spherical cavity expansion in soils of finite radial extent

This paper revisits the expansion problem of a spherical cavity in an elastic–plastic soil mass with a finite radial extent under drained conditions, which proves to be non-self-similar. Rigorous semi-analytical solutions for the analysis of this problem are developed adopting several well-known soil models. With the small strain assumption, analytical expressions for both elastic stresses and displacements are obtained considering the stress dependency of soil moduli and influences of the outer boundary. In the plastic zone, the non-self-similar expansion process is formulated into a set of first-order partial differential equations (PDEs) based on the innovative combination use of the Lagrangian and Eulerian descriptions, and an efficient algorithm for solving these equations is developed. The solution method is quite general to be extended to many other constitutive models and is validated by comparing with existing solutions in some special cases. Then the cavity expansion behaviours in both clay and sand are discussed with a particular focus on the boundary effect. It is shown that, when the ratio of the outer to inner radius is small, the boundary effect may significantly influence the cavity expansion response. The present solution may provide a useful theoretical method for the validation of relevant finite element calculations and interpretation of some typical geotechnical problems such as cone penetration tests in small-size chambers, considering the boundary effect.

Solving a System of First-Order Partial Differential Equations Using Decomposition Methods

Solving a System of First-Order Partial Differential Equations Using Decomposition Methods

A Mingled Tau-Finite Difference Method for Stochastic First-Order Partial Differential Equations

A Mingled Tau-Finite Difference Method for Stochastic First-Order Partial Differential Equations

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

In this paper, a harmonic balance method is introduced to solve the intrinsic 1D nonlinear equations presented by Hodges for analyzing initially curved and twisted anisotropic rotating beams. First, the nonlinear first-order partial differential equations are discretized in space domain using the Galerkin approach. Then, the time response of the system of equations is approximated by using first and the second harmonic terms under the harmonic excitations applied to the structure. The harmonic balance algorithm has been developed for the linearized system of equations about the steady-state solution and for fully nonlinear equations. The differences between these two systems have been addressed under consideration of different levels of external loads. Fully analytical formulae have been presented for the HB solution of the linearized equations in this work. In the proposed approach for the nonlinear system, the Jacobian matrix utilized in the numerical solution part is obtained analytically to increase the time efficiency of the solution method. The results of the proposed approach for an initially twisted and curved blade are compared with a different algorithm based on a finite difference approach to discretize the equations and a time marching method to solve the system of equations in time domain. The comparison between the outputs of the harmonic balance method and the time marching method demonstrate high correlations between these distinct methods. It should be noted that the accuracy of the time marching method solution has been evaluated in another publication by the authors using the output of FLIGHTLAB for a blade with different types of boundary conditions. The proposed harmonic balance-based method delivers a very time-efficient approach for time analysis of twisted/curved beams and blades under harmonic loads in comparison with other employed methods. The developed approach can be a powerful computational tool for design optimization of composite blades.

Chaos in a class of first-order partial difference equations with delay controllers

Chaos in a class of first-order partial difference equations with delay controllers

Similarity transformations for modified shallow water equations with density dependence on the average temperature

Abstract The Lie symmetry analysis is applied for the study of a modified one-dimensional Saint–Venant system in which the density depends on the average temperature of the fluid. The geometry of the bottom we assume that is a plane, while the viscosity term is considered to be nonzero, as the gravitational force is included. The modified shallow water system is consisted by three hyperbolic first-order partial differential equations. The admitted Lie symmetries form a four-dimensional Lie algebra, the A 3,3 ⊕ A 1. However, for the viscosity free model, the admitted Lie symmetries are six and form the A 5,19 ⊕ A 1 Lie algebra. For each Lie algebra we determine the one-dimensional optimal system and we present all the possible independent reductions provided by the similarity transformations. New exact and analytic solutions are calculated for the modified Saint–Venant system.

Existence of chaos in controlled first-order partial difference equations with general delay controllers

This paper is concerned with chaos in a class of controlled first-order partial difference equations with general delay controllers. Three criteria of chaos are established by applying snap-back repeller theory. All the equations are proved to be chaotic in the sense of both Devaney and Li-Yorke when they satisfy some weak conditions. Three examples are provided with computer simulations.

Space-Time Finite Element Method for Fully Intrinsic Equations of Geometrically Exact Beam

In this paper, a space-time finite element method based on a Galerkin-weighted residual method is proposed to solve the nonlinear fully intrinsic equations of geometrically exact beam which are first-order partial differential equations about time and space. Therefore, it is natural to discretize it in time and space simultaneously. Considering the continuity and intrinsic boundary conditions in the spatial direction and the continuity and periodic boundary conditions in the time direction, the boundary value scheme of space-time finite element for solving the full intrinsic equations is derived. This method has been successfully applied to the static analysis and dynamic response solution of the fully intrinsic equations of nonlinear geometrically exact beam. The numerical results of several examples are compared with the analytical solution, existing algorithms, and literature to illustrate the applicability, accuracy and efficiency of this method.

THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collaborators recently. We show that this potential function agrees with the <italic>generalized Lyapunov function</italic> of the deterministic part of the stochastic dynamical system. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also be interpreted as the proof of existence of Lyapunov function for a general dynamical system.

АППРОКСИМАЦИЯ ТЕЧЕНИЯ СЛОЯ ОГРАНИЧЕННОЙ ТОЛЩИНЫ МНОГОСВЯЗНОГО КОНТУРА СОВОКУПНОСТЬЮ ТЕЧЕНИЙ ОДНОСВЯЗНЫХ ОБЛАСТЕЙ

The calculations are based on the one proposed by A.A. Ilyushin. theory of flow in a thin plastic layer, enclosed between two surfaces of tool bodies approaching according to a given law, under the assumption that: the material of the spreading layer is ideally plastic and volumetrically incompressible; elastic deformations of the tool are small and can be neglected; on the contact surfaces of the metal layer with the tool, the classical L. Prandtl flow law is accepted. To study the characteristics of the flow, we use the well-known boundary value problem in the “ideal fluid” formulation, which is described by nonlinear first-order partial differential equations with respect to the contact pressure and components of the flow velocity vector. In the general case, the quantities under study are complex functions that depend on the shape of the deformation zone, the magnitude of the contact pressure at the considered point of the contact surface, the presence, composition and method of applying a lubricant, the roughness of the contacting surfaces and other factors. The paper proposes a method for solving problems of flow of plastic layers in a new formulation. Using the proposed mathematical apparatus, a solution algorithm and a computer model of the flow were constructed.

Lie Symmetry Analysis of the Aw–Rascle–Zhang Model for Traffic State Estimation

We extend our analysis on the Lie symmetries in fluid dynamics to the case of macroscopic traffic estimation models. In particular we study the Aw–Rascle–Zhang model for traffic estimation, which consists of two hyperbolic first-order partial differential equations. The Lie symmetries, the one-dimensional optimal system and the corresponding Lie invariants are determined. Specifically, we find that the admitted Lie symmetries form the four-dimensional Lie algebra A4,12. The resulting one-dimensional optimal system is consisted by seven one-dimensional Lie algebras. Finally, we apply the Lie symmetries in order to define similarity transformations and derive new analytic solutions for the traffic model. The qualitative behaviour of the solutions is discussed.

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.

The NNLO QCD analysis of gluon density at small-x

In this paper, a next-to-next-to-leading order (NNLO) quantum chromodynamics (QCD) calculation of gluon distribution function at small-[Formula: see text] is presented. The gluon distribution function is explored analytically in the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi approach by a Taylor expansion at small-[Formula: see text] as two first-order partial differential equations in two variables: Bjorken [Formula: see text] and [Formula: see text][Formula: see text]. We have solved the system of equations at LO, NLO and NNLO, respectively, by Lagrange’s method. The resulting analytical expressions are compared with the available global parton distribution function fits as well as with the results of the Block–Durand–McKay model. We have further performed an [Formula: see text] test to check the compatibility of our predictions and observed that our results can be consistently described in the context of perturbative QCD. A comparative analysis of the obtained results at LO, NLO and NNLO reveals that the NNLO approximation has a significant contribution to the gluon distribution function particularly in the small-[Formula: see text] region.

A ray perturbation method for the acoustic attenuating transversely isotropic eikonal equation

Modeling complex-valued traveltimes is helpful for developing attenuation-associated techniques for seismic data processing. Transverse isotropy can explain the directional variation of velocity and attenuation anisotropy of long-wavelength seismic waves in many sedimentary rocks. The acoustic attenuating transversely isotropic eikonal equation can be used to accurately calculate P-wave complex-valued traveltimes under such a geologic condition. However, no ray-tracing system for this eikonal equation could be found in the literature until now. We have developed a ray perturbation method to solve this eikonal equation. Unlike all existing ray perturbation methods, our newly proposed method does not perturb the exact ray-tracing system but splits the acoustic attenuating transversely isotropic eikonal equation into a nonlinear partial differential equation (PDE) and a first-order PDE. These two PDEs can be solved by the method of characteristics. This gives rise to two sets of ray-tracing equations for the real and imaginary parts of the complex-valued traveltimes, respectively. Both sets of ray-tracing equations share the same raypath, which allows us to merge them into a complete ray-tracing system for complex-valued traveltimes. Numerical examples are used to demonstrate the high accuracy of the newly proposed ray-tracing system, analyze the complex-valued traveltimes of diving P waves, and compare the modeled complex-valued traveltimes with those extracted from constant-[Formula: see text] viscoelastic waveforms.

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.

Time-dependent finite-difference model for transient and steady-state analysis of thermoelectric bulk materials

A mathematical model of coupled thermoelectricity is presented to investigate the transient and steady-state behaviour of thermoelectric bulk material. Governing partial differential equations (PDEs) for the coupled thermal and electrical behaviour of the thermoelectric model are discretised using the explicit finite-difference method. Differencing schemes like Upwind and Lax–Wendroff methods are employed to obtain solutions for the first-order hyperbolic PDEs, whereas FTCS (Forward Time, Centred Space) scheme is employed to solve second-order parabolic PDEs. Courant-Friedrichs-Lewy and Von Neumann stability analyses are done to ensure the stability and convergence of the model. The model considers the temperature dependency of thermal conductivity, electrical conductivity, and Seebeck coefficient of the P/N materials separately. and accounts for the Seebeck, Peltier, and Joule-Thomson effects in thermoelectric materials. The new model is practically useful to predict the transient and steady-state behaviours of a thermoelectric device with multiple P-N elements. The results of the presented finite-difference model are proven to agree well with experimental values as well as 3D simulations with ANSYS®.

Solution of a Semi-Boundary-Value Problem for a First-Order Degenerate Partial Differential Equation

A first-order partial differential equation with constant irreversible coefficients in a Banach space is considered. In the particular case of a finite-dimensional space, the initial-boundary-value problem with irreversible matrix coefficients has no solution; hence, we pose Showalter-type conditions. Due to the regularity of the operator pencil, the equation splits into differential equations in subspaces and given conditions lead to initial conditions in subspaces. A solution to the problem is constructed and an example is provided.

Singular Perturbations in First-Order Partial Differential Equations with Matrix Differential Operators

In this paper, we consider differential equations whose degeneracy is due to the presence of a matrix differential operator coefficient of the derivative perturbed by a small parameter. Properties of this operator are applied to the analysis of the presence of a boundary layer in solutions of initial-boundary-value problems for equations considered. Regularity conditions of degeneracy are determined.

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.

On constraint preservation and strong hyperbolicity

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the free-evolution approach, which consists in obtaining the solutions of the entire system by solving only the evolution equations. Certainly, this is valid only when the chosen initial data satisfies the constraints and the constraints are preserved in the evolution. In this paper, we establish the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation. This is achieved by considering quasi-linear first-order PDEs, assuming the sufficient condition and deriving strongly hyperbolic first-order partial differential evolution equations for the constraints. We show that, in general, these constraint evolution equations correspond to a family of equations parametrized by a set of free parameters. We also explain how these parameters fix the propagation velocities of the constraints. As application examples of this framework, we study the constraint conservation of the Maxwell electrodynamics and the wave equations in arbitrary space–times. We conclude that the constraint evolution equations are unique in the Maxwell case and a family in the wave equation case.