Nonlinear First-order Partial Differential Equations Research Articles

Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffusion or spurious oscillations, and predicts correct reflection characteristics at the boundaries. For the resonant vibrations problem, the CESE method captures the linear-to-nonlinear evolution of the resonant waves and the distribution of wave energy among multiple modes in the nonlinear regime.

Evolution of a Mixing Zone in Granular Avalanches

A nonlinear first-order partial differential equation in two space variables and time describes the process of kinetic sieving in an avalanche, in which larger particles tend to rise to the surface while smaller particles descend, quickly leading to completely segregated layers. The interface between layers is a shock wave satisfying its own nonlinear equation. When the interface becomes vertical, it loses stability, and develops a mixing zone. The mixing zone is described explicitly under idealized initial conditions, and verified with numerical simulation. The problem and its solution are similar to twodimensional Riemann problems for scalar first-order conservation laws; the difference here is that the equation is not scale-invariant, due to shear in the avalanche, an essential ingredient of kinetic sieving.

Towards recovery of complex shapes in meshes using digital images for reverse engineering applications

When an object owns complex shapes, or when its outer surfaces are simply inaccessible, some of its parts may not be captured during its reverse engineering. These deficiencies in the point cloud result in a set of holes in the reconstructed mesh. This paper deals with the use of information extracted from digital images to recover missing areas of a physical object. The proposed algorithm fills in these holes by solving an optimization problem that combines two kinds of information: (1) the geometric information available on the surrounding of the holes, (2) the information contained in an image of the real object. The constraints come from the image irradiance equation, a first-order non-linear partial differential equation that links the position of the mesh vertices to the light intensity of the image pixels. The blending conditions are satisfied by using an objective function based on a mechanical model of bar network that simulates the curvature evolution over the mesh. The inherent shortcomings both to the current hole-filling algorithms and the resolution of the image irradiance equations are overcome.

Output regulation of nonaffine nonlinear systems using singular perturbation theory

A control synthesis method for output regulation based on singular perturbation theory combined with inverting design is considered for a class of nonaffine nonlinear systems. The resulting control signal is defined as a solution to “fast” dynamics which inverts a series error model, whose state is exponentially stable. It is shown that, under sufficient conditions being consistent with the assumptions of singular perturbation theory, this problem is solvable with (ɛ) tracking error if and only if a set of first-order nonlinear partial differential equations are solvable. The control law can be easily constructed and the simulations show the feasibility and effectiveness of the controller.

Estimation of the time a perturbed Lagrangian system stays in an assigned region

The problem of estimating the mean time a weakly perturbed dynamical system stays in a fixed region of the phase variables is investigated. The motion is described by Lagrange's equations with an attractive force potential and in the presence of additive dissipative forces. The corresponding Cauchy problem is obtained in Hamiltonian variables for a non-linear first-order partial differential equation. Its classical positive solution specifies the action functional and the estimate sought for the time interval. The structure of the equations that allows of an explicit solution in terms of expressions for the kinetic and potential energy, as well as dissipative and dispersion matrices for a random Wiener-type perturbation, is established. The phenomenon of the escape of a phase point from different parts of the boundary of the region is investigated. Interesting problems of estimating the time for the inversion of the inner gimbal of a gyroscope, the time taken to reach an assigned level or a potential barrier of a multidimensional oscillatory system that has central symmetry, and the time a non-linear system with two degrees of freedom takes to escape over a potential barrier for a Henon–Heiles potential are investigated as examples.

On entire solutions of a family of nonlinear partial differential equations of prime degree

We characterize entire solutions of a new family of nonlinear first-order partial differential equations of prime degree p (equation 1). In particular, we show that in the Borel family (defined in the article) of entire functions in the entire solutions of (1) are of exponential form. Then, we address the general case when p = 3, and show that under some natural conditions the entire solutions are also of exponential form.

Numerical methods for nonlinear first-order partial differential equations with deviated variables

In the article classical solutions of initial problems for nonlinear differential equations with deviated variables are approximated by solutions of quasilinear systems of difference equations. Interpolating operators on the Haar pyramid are used. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. This new approach to solving nonlinear equations with deviated variables numerically is based on a method of linearization for initial problems. Numerical examples are given. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Dimension of measures invariant with respect to the Ważewska partial differential equation

We study asymptotic properties of a nonlinear first-order partial differential equation which describes the reproduction of blood cells. This equation under conditions proposed by Ważewska generates a semigroup of transformations with highly chaotic behaviour of trajectories. We show that this semigroup has invariant measures with arbitrary large dimension.

Generalized Euler Method for Nonlinear First-Order Partial Differential Equations

Classical solutions of nonlinear first-order partial differential equations are approximated by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on the comparison method. Nonlinear estimates of the Perron type are assumed for increment functions.

Un modèle non-linéaire de prolifération cellulaire : extinction des cellules et invariance

This Note analyses a nonlinear age-maturity structured system which arises as a model of the blood cellular production in the bone marrow. The resulting model is a nonlinear first-order partial differential equation in which there is a distributed temporal delay and a retardation in the maturation variable. We prove that uniqueness of solutions depends only on small maturity cells (stem cells) and we give a result of invariance. To cite this article: M. Adimy, F. Crauste, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

WELL-POSEDNESS OF AN INITIAL BOUNDARY VALUE PROBLEM FROM LASER DYNAMICS

In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

Paraxial Eikonal Solvers for Anisotropic Quasi-P Travel Times

The first-arrival quasi-P wave travel-time field in an anisotropic elastic solid solves a first-order nonlinear partial differential equation, the qP eikonal equation, which is a stationary Hamilton–Jacobi equation. The solution of the paraxial quasi-P eikonal equation, an evolution Hamilton–Jacobi equation in depth, gives the first-arrival travel time along downward propagating rays. We devise nonlinear numerical algorithms to compute the paraxial Hamiltonian for quasi-P wave propagation in general anisotropic media. A second-order essentially nonoscillatory (ENO) Runge–Kutta scheme solves this paraxial eikonal equation with a point source as an initial condition in O(N) floating point operations, where N is the number of grid points. Numerical experiments using 2-D transversely isotropic models with inclined symmetry axes demonstrate the accuracy of the algorithms.

New symmetry solution techniques for first-order non-linear PDEs

Using recent extensions of work of S. Lie (Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, Teubner, 1891) and É. Cartan (Leçons sur les invariants intégreaux, Hermann, Paris, 1922; Les systèms differentials extérieures et leurs applications géométrique, Hermann, Paris, 1945) for integrating Frobenius integrable vector field distributions via symmetry, we examine some symmetry techniques for finding local solutions of first-order non-linear partial differential equations (PDEs). In the language of exterior differential systems, we develop a technique for solving first-order quasilinear PDEs that is then applied to general, first-order non-linear partial differential equations. Our results are significant inasmuch as we give algorithms for solving first-order PDEs in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system.

Solutions near singular points to the eikonal and related first-order nonlinear partial differential equations in two independent variables

A detailed study of solutions to the first-order partial differential equation $H(x,y,z_x,z_y)=0$, with special emphasis on the eikonal equation $z_x^2+z_y^2=h(x,y)$, is made near points where the equation becomes singular in the sense that $dH=0$, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function $H$ and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.

An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations

Solutions of the optimal control and $H_\infty$-control problems for nonlinear affine systems can be found by solving Hamilton--Jacobi equations. However, these first-order nonlinear partial differential equations can, in general, not be solved analytically. This paper introduces an iterative algorithm which solves these equations numerically for points near the origin. The procedure converges to the stabilizing solution exponentially with respect to the iteration variable. The algorithm is implemented on both illustrative and comparative examples.

Mathematical analysis of long-wave breaking on open channels with bottom friction

In this paper we study the breaking of long waves propagating along an open channel with linear friction on the bottom. The equations governing the wave propagation consist of a pair of first-order nonlinear hyperbolic partial differential equations (PDEs). We first transformed the PDEs into a pair of ordinary differential equations (ODEs) along the characteristic directions by means of a pair of Riemann invariants. By analyzing the ODEs, we found that the breaking of waves can be identified by the singularity of the derivative of the Riemann invariants. Thus, we derived an analytical solution for the derivative of the Riemann invariants. Then, a breaking criterion and an analytical formula for the estimation of breaking time were developed and validated through numerical experiments. It is also shown in the paper that the present model includes the previous model neglecting bottom friction as a special case.

Doped-fiber amplifier dynamics: a system perspective

Sun et al. [see Electron. Lett., vol. 32, p. 1490, Aug. 1996] succeeded in reducing the set of coupled first-order nonlinear partial differential equations determining the wavelength-dependent, time-varying amplifier gain into a single ordinary differential equation (ODE). In this paper, we further simplify the ODE bringing into greater evidence the physical meaning of the amplification process, and greatly enhancing the utility of the ODE as an analysis and design tool. We find that the gain dynamics of a doped-fiber amplifier are completely specified by its total number of excited ions r, whose time behavior is described by a simple first-order differential equation. We exploit this new understanding of amplifier gain dynamics: 1) to develop an equivalent circuit model for amplifier gain dynamics, 2) to identify that channel addition causes much faster transients than channel dropping in wavelength division multiplexing networks, and 3) to demonstrate that gain excursions can be significant in multichannel packet switching applications, which unlike time-multiplexed signals are characterized by bursts and lulls in communications. We are also able to revisit the most significant previously published results on both steady-state and dynamic analysis of doped-fiber amplifiers with a much more concise and more intuitive derivation.

A New Model for Interest Rates

There are many theories and models underlying the valuation of fixed income security portfolios. This work addresses the problem from a new perspective: the objective is to find a lower bound for the value of a portfolio of cash flows. We set up conditions for the evolution of a short-term interest rate and value a liability using its present value. We formulate a first-order nonlinear hyperbolic partial differential equation for the value, V, of the portfolio. We explore the solution of this equation and then hedge our portfolio with market-traded zero-coupon bonds of known value. We include some salient examples — generating the Yield Envelope and valuing caps, floors and bond options.

Entire solutions of first-order nonlinear partial differential equations

Entire solutions of first-order nonlinear partial differential equations

COLLATION AND UPWINDING FOR THERMAL FLOW IN PIPELINES: THE LINEARIZED CASE

Simulating thermal effects in pipeline flow involves solving a coupled non-linear system of first-order hyperbolic equations. The advection term has two large eigenvalues of opposite signs, corresponding to the propagation of high-speed sound waves, and one eigenvalue close to or even equal to zero, representing the much slower fluid flow velocity, which transports temperature. Standard collocation methods work well for isothermal flow in pipelines, but the stagnating eigenvalue causes difficulties when thermal effects are included. In a companion paper we formulate and analyse a new numerical method for the non-linear system which arises in thermal modelling. The new method applies to general coupled systems of non-linear first-order hyperbolic partial differential equations with one degenerate eigenvalue. In the present paper we focus on a linearized constant coefficient form of the thermal flow equations. This substantially simplifies presentation of the error analysis for the numerical scheme. We also include numerical results for the method applied to the fully non-linear system. Both the error analysis and the numerical experiments show that the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution.