System Of First-order Partial Differential Equations Research Articles

Accessibility Conditions for PDE Systems and the Adjoint System

A group-theoretical approach is used to tackle the problem of (local) accessibility along a trajectory of systems described by partial differential equations (PDEs). In particular, a class of first-order PDE systems with boundary conditions and boundary control is applied to illustrate the key ideas of our approach. The methods mainly rely on a coordinate-free formulation of PDE systems, which also incorporates boundary conditions. The accessibility is discussed in general and an approach based on (pointwise, continuous) transformation groups and their invariants is motivated. Using an infinitesimal criterion for invariance we study special group invariants to derive (local) conditions on accessibility. It is highlighted that the basic questions lead to the investigation of a particular adjoint system. A nonlinear example demonstrates the methods and results.

Reciprocity for electromagnetic–acoustic fields in compressible magnetoplasmas with anisotropic pressure

Maxwell's curl equations and the linearized balance equations for momentum and pressure tensor for a plasma in an external magnetic field are combined in a system of first-order partial differential equations for the field quantities. With the corresponding adjoint system, the Lagrange identity is established, leading to a symmetry relation between the Green functions for the original and the adjoint fields. With temporal mapping the adjoint field can be transformed into a ‘Lorentz-adjoint’ (physical) field, leading to a reciprocity relation between the Green functions for the original and for the Lorentz-adjoint fields.

A method for finding exact solutions to hyperbolic systems of first-order PDEs

Differential constraints are used as a means of developing a systematic method for finding exact solutions to quasilinear nonautonomous hyperbolic systems of first-order partial differential equations (PDEs) involving two independent variables. The leading assumption of the hyperbolicity of the basic system together with a strict compatibility argument permits characterization of the most general class of quasilinear first-order constraint equations, which can be appended to the governing mathematical model under consideration. Furthermore, the exact solutions can be found by integrating the resulting overdetermined and consistent system of PDEs. When the number of auxiliary constraint equations is equal to that of the field equations, the solutions provided by the ‘nonclassical’ group method for PDEs are recovered; for a very special form of the appended constraint equations simple wave solutions to hyperbolic systems of PDEs are also obtained. Within the present theoretical framework a model describing rate-type media is considered and several classes of initial- and/or boundary-value problems are solved.

A Coupled Local-Mode Analysis of Surface-Wave Propagation In A Laterally Heterogeneous Waveguide

SUMMARY We consider surface-wave propagation in a laterally heterogeneous waveguide based upon coupled local modes, which are the eigenfunctions of a laterally homogeneous waveguide with a vertical structure identical to that of the laterally heterogeneous waveguide at the horizontal location of interest. Energy transfer from one overtone to another is expressed in terms of local-mode amplitude coefficients which determine the surface-wave amplitude. The amplitude coefficients can be determined from a system of first-order partial-differential equations. In particular, we investigate the nature of forward-coupled Love and Rayleigh waves along a given surface-wave trajectory. We demonstrate that overtone coupling is determined by lateral gradients in density, elastic parameters, and topography on discontinuities, and that the coupling strength is on the order of the ratio of the lateral change in the model parameters to the squared wavenumber spacing and fluctuates with a wavelength on the order of 2n over this spacing. The theory allows for topography on solid-solid as well as fluid-solid boundaries. When overtone coupling is ignored the results are reduced to traditional surface-wave JWKB theory.

Implicit Finite-Difference Procedures for the Computation of Vortex Wakes

Implicit finite-difference procedures for the primitive form of the incompressible Navier-Stokes and the compressible Euler equations are used to compute vortex wake flows. The partial differential equations in strong conservation-law form are transformed to cluster grid points in regions with large changes in vorticity. In addition to clustering, fourth-order accurate, spatial difference operators are used to help resolve the flowfield gradients. The use of implicit time-differencing permits large time steps to be taken since temporal variations are typically small. Computational efficiency is achieved by approximate factorization. Both two-dimensional and preliminary three-dimensional calculations are described. I. Introduction T HE concentrated vorticity in the near wake of a large aircraft can pose a destructive threat to smaller aircraft within the same airspace. Consequently, experimental and theoretical efforts have been under way to understand, predict (for use in avoidance systems), and possibly reduce the vortex wake hazard. Most theoretical models developed to treat this problem rely on tracing discrete vortices,13 but an alternate and potentially more powerful approach is to use finitedifference procedures.4'5 Computer programs based on such methods can ultimately account for flowfield nonlinear effects with few ad hoc assumptions. Implicit finite-difference procedures are developed here to solve the incompressible Navier-Stokes equations and compressible Euler equations for simplified two- and threedimensional, unsteady vortex wake flows. This paper is divided into five interdependent sections. The flowfield and its numerical implications are discussed in Sec. II. The incompressible equations are developed for simulation as a system of first-order partial-differential equations in Sec. Ill, and the finite-difference algorithms are described in Sec. IV. Simulation based on the compressible flow equations is discussed in Sec. V. and, finally, simple wake-vortex flow calculations using both kinds of modeling are presented in Sec. VI.