Second-order Quasilinear Partial Differential Equations Research Articles

The [formula omitted] transformation for solving quasi-linear partial differential equations

The Z1-Z4 transformations proposed in our previous papers have acquired remarkable success in the field of linear partial differential equations, general solutions of a large number of typical equations were obtained for the first time, and exact solutions of many definite solution problems of general significance were also successfully obtained. In this paper we will present the Z5 transformation and use it to get the general solutions of a variety of first-order quasi-linear partial differential equations and, for the first time, to obtain analytical solutions of a class of second-order quasi-linear partial differential equations containing arbitrary functions. Using the general solutions, we obtain exact solutions of two definite solution problems. By comparison we find that the general solutions of some first-order quasi-linear partial differential equations that can be obtained using the characteristic equation method are incomplete.

Classical field theory of the Von Mises equation for irrotational polytropic inviscid fluids

The Von Mises second-order quasi-linear partial differential equation describes the dynamics of an irrotational, compressible and barotropic classical perfect fluid through a scalar function only, i.e. the velocity potential. It is shown here how to derive it in the case of a polytropic equation of state starting from a least action principle. The Lagrangian density is found to coincide with pressure. Once re-expressed in terms of the velocity potential, the action integral presents some similarities with other classical and quantum field theories. Aided by the Legendre transformation tool, we show that the nonlinear equation is completely integrable in the case of a non-steady parallel flow dynamics. A numerical solution of the equation in a critical shock wave forming scenario allows one to analyze then such a particular dynamics by using the so-called analogue gravity formalism which impressively links ordinary perfect fluids to curved spacetimes. Finally, implications for fluid dynamics and field theories are discussed.

Thoughts on the Busemann equation

A second-order quasi-linear partial differential equation of mixed elliptic-hyperbolic type in two independent variables, which mimics one introduced by A. Busemann in gas dynamics, and arises in the study of Minkowski spaces and in other theories, is considered.

Signature change, mixed problems and numerical relativity

Classical general relativity takes place on a manifold with a metric of fixed, Lorentzian, signature. However, attempts to amalgamate general relativity with quantum theory frequently involve manifolds with metrics whose signatures are Lorentzian in some regions and Euclidean in others. (Indeed even more exotic possibilities are discussed frequently.) Most theoretical calculations rely on analyticity arguments to continue variables from the Euclidean to the Lorentzian regime and vice versa. This paper examines models of signature change. It looks at a single second-order quasi-linear partial differential equation on a fixed background, whose principal part is elliptic in one regime and hyperbolic in another, i.e. a mixed problem. It introduces some examples, explains heuristically the concept of a well-posed problem and then discusses the issues involved in constructing a robust numerical algorithm to solve well-posed problems. The paper includes a worked example illustrating the proposed techniques, and a discussion of the role of the potential curvature singularity on the transition hypersurface.

Anti-plane shear deformations for non-Gaussian isotropic, incompressible hyperelastic materials

The purpose of this research is to investigate the mechanical response in anti-plane shear of a class of incompressible isotropic hyperelastic materials for which the strain-energy density depends only on the first invariant of the strain tensor. Our concern is with the subclass of these materials that exhibits hardening at large deformations. In the molecular theory of elasticity, these models are called non-Gaussian, since they introduce a distribution function which is not Gaussian for the end-to-end distance of the polymeric chain. Two classes of these materials are considered, namely those with limiting chain extensibility and power-law models. The governing partial differential equation of equilibrium in anti-plane shear (a single second-order quasilinear partial differential equation) is obtained for specific constitutive models of the above type. Some solutions are derived using group symmetry reduction methods. Applications to crack problems and spatial decay of end effects are described. The results are applicable to rubber-like and biological materials.

Spatial decay estimates for a coupled system of second-order quasilinear partial differential equations arising in thermoelastic finite anti-plane shear

The spatial decay behavior of solutions of a coupled system of second-order quasilinear partial differential equations, in divergence form, defined on a two-dimensional semi-infinite strip, is investigated. Such equations arise in the theory of anti-plane shear deformations for isotropic nonlinearly thermoelastic solids. Differential inequality techniques are employed to obtain exponential decay estimates. The results are illustrated by several examples. The results are relevant to Saint-Venant principles for nonlinear thermoelasticity as well as to theorems of Phragmen-Lindelof type.

Magnetohydrodynamic equilibria of a cylindrical plasma with poloidal mass flow and arbitrary cross sectional shape

The equilibrium of a cylindrical plasma with purely poloidal mass flow and cross section of arbitrary shape is investigated within the framework of the ideal MHD theory. For the system under consideration it is shown that only incompressible flows are possible and, consequently, the general two-dimensional flow equilibrium equations reduce to a single second-order quasilinear partial differential equation for the poloidal magnetic flux function , in which four profile functionals of appear. Apart from a singularity occurring when the modulus of Mach number associated with the Alfvén velocity for the poloidal magnetic field is unity, this equation is always elliptic and permits the construction of several classes of analytic solutions. Specific exact equilibria for a plasma confined within a perfectly conducting circular cylindrical boundary and having (i) a flat current density and (ii) a peaked current density are obtained and studied.

The effect of constitutive law perturbations on finite antiplane shear deformations of a semi-infinite strip

This paper is concerned with assessing the effects of small perturbations in the constitutive laws on antiplane shear deformation fields arising in the theory of nonlinear elasticity. The mathematical problem is governed by a second-order quasilinear partial differential equation in divergence form. Dirichlet (or Neumann) boundary-value problems on a semi-infinite strip, with nonzero data on one end only, are considered. Such problems arise in investigation of Saint-Venant end effects in elasticity theory. The main result provides a comparison between two solutions, one of which is a solution to a simpler equation, for example Laplace’s equation. Three examples involving perturbations of power-law material models are used to illustrate the results.

Decay Estimates for a Class of Nonlinear Boundary Value Problems in Two Dimensions

This paper is concerned with second-order quasilinear partial differential equations in two independent variables of the form ${\operatorname {div}}[\rho ({\bf x},u,{\operatorname {grad }}u){\operatorname {grad }}u] = 0$. Previous work of the authors, establishing exponential decay estimates for Dirichlet problems on a semi-infinite strip subject to nonzero data on the finite end, is extended to include regions of arbitrary shape, and, in the case of unbounded regions, the a priori assumption that solutions must decay to zero as $| {\bf x} | \to \infty $ is removed. The results have application to Saint-Venant principles for nonlinear elasticity as well as to theorems of Phragmén–Lindelöf type.

On the asymptotic behavior of solutions of inhomogeneous second-order quasilinear partial differential equations

This paper is concerned with the asymptotic behavior of solutions of a class of inhomogeneous second-order quasilinear partial differential equations in two independent variables defined over rectangular plane domains whose lengths greatly exceed their widths. Solutions to a Dirichlet problem for such equations are shown to be well approximated, away from the ends of the rectangle, by solutions to the corresponding one-dimensional problem for an ordinary differential equation on the cross-section of the rectangle. Applications to problems in geometry and nonlinear continuum mechanics are discussed.

General two-dimensional magnetohydrodynamic equilibria with mass flow

A set of reduced ideal MHD equations is derived to investigate equilibria of plasmas with mass flow in general two-dimensional geometry. These equations provide a means of investigating the effects of flow on self-consistent equilibria in a number of new two-dimensional configurations, such as helically symmetric configurations with helical axis, which are relevant to stellarators, as well as axisymmetric configurations. It is found that, as in the axisymmetric case, general two-dimensional flow equilibria are governed by a second-order quasilinear partial differential equation for a magnetic flux function, which is coupled to a Bernoulli-type equation for the density. The equation for the magnetic flux function becomes hyperbolic at certain critical flow speeds which follow from its characteristic equation. When the equation is hyperbolic, shock phenomena may exist. As a particular example, unidirectional flow along the lines of symmetry is considered. In this case, the equation mentioned above is always elliptic. An exact solution for the case of helically symmetric unidirectional flow is found and studied to determine flow effects on the magnetic topology.

Decay estimates for second-order quasilinear partial differential equations

This article is concerned with second-order quasilinear equations (not necessarily elliptic) in two independent variables of the form div[ ϱ(x, u, grad u)grad u] = 0. For Dirichlet (or Neumann) problems on a semi-infinite strip, with non-zero data on one end only, differential inequality techniques are employed to establish exponential decay estimates for quadratic (“energy”) integrals and to obtain pointwise estimates. The results have application to Saint-Venant principles for nonlinear elasticity and compressible fluid flows as well as to theorems of Phragmén-Lindelöf type.

Variational principles for second-order quasi-linear scalar equations

The problem of constructing variational principles for a given second-order quasi-linear partial differential equation is considered. In particular, we address the problem of finding a first-order function f whose product with the given differential operator is the Euler-Lagrange operator derived from some Lagrangian. Two sets of equations for such a function f are obtained. Necessary and sufficient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set. In certain special cases, such as the case when the operator is elliptic, the problem is completely solved. The utility of our results is illustrated by a variety of examples.

Steady Flow in the Axially Symmetric Torus Using the Guiding-Center Equations

General steady flow in an axially symmetric torus is investigated using the guiding-center equations. Several constants of the macroscopic and microscopic motion are obtained, and a second-order quasilinear partial differential equation is derived for the self-consistent magnetic flux function. The condition specifying that this equation be hyperbolic or elliptic is given. For the former case, jump conditions are given when weak (discontinuous) solutions are admitted.