Search results for "Independent equation"
showing 8 items of 8 documents
On new ways of group methods for reduction of evolution-type equations
2005
AbstractNew exact solutions of the evolution-type equations are constructed by means of a non-point (contact) symmetries. Also we analyzed the discrete symmetries of Maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently.
A fully adaptive wavelet algorithm for parabolic partial differential equations
2001
We present a fully adaptive numerical scheme for the resolution of parabolic equations. It is based on wavelet approximations of functions and operators. Following the numerical analysis in the case of linear equations, we derive a numerical algorithm essentially based on convolution operators that can be efficiently implemented as soon as a natural condition on the space of approximation is satisfied. The algorithm is extended to semi-linear equations with time dependent (adapted) spaces of approximation. Numerical experiments deal with the heat equation as well as the Burgers equation.
Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations
1998
This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted.
Infinite sets of conservation laws for linear and nonlinear field equations
1984
The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the ‘coupling constant’) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant u…
Removability of a Level Set for Solutions of Quasilinear Equations
2005
In this paper, we study the removability of a level set for the solutions of quasilinear elliptic and parabolic equations of the second order. We show, under rather general assumptions on the coeff...
The relaxation-time limit in the quantum hydrodynamic equations for semiconductors
2006
Abstract The relaxation-time limit from the quantum hydrodynamic model to the quantum drift–diffusion equations in R 3 is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are…
On the structure of the set of solutions of nonlinear equations
1971
Let T be a mapping from a subset of a Banach space X into a Banach space Y. The present paper investigates the nature of the set of solutions of the equation T(x) = y for a given y E Y, i.e. when T-l(y) # 0 ? What are the topological properties of T-l(y)? A prototype for an answer to these questions is given by Peano existence theorem on the connectedness of the set of solutions of an ordinary differential equation in the real case. In its general setting, this problem was first attacked by Aronszajn [l] and Stampacchia [l 11; recently, by Browder-Gupta [5], Vidossich [12] and, above all, Browder [3, Sec. 51 who gives several interesting results in an excellent treatment. Customary, the str…
Comparison results for Monge - Ampère type equations with lower order terms
2003
In this paper we deal with Monge-Ampère type equations in two dimensions and, using the symmetrization with respect to the perimeter, we prove some comparison results for solutions of such equations involving the solutions of conveniently symmetrized problems.