José Canosa

A new method for the solution of the Schrödinger equation

We approximate the potential in the one-dimensional Schrodinger equation by a step function with a finite number of steps. In each step, the resulting differential equation has constant coefficients and is integrated exactly in terms of circular or hyperbolic functions. The solutions are then matched at the interface of each layer to construct the eigenfunctions in the whole domain. Unique features of the numerical method are: (a) All the eigenfunctions and eigenvalues are obtained with the same absolute accuracy for the same number of steps in the potential;(b) any desired number of eigenvalues and eigenfunctions are obtained in one single pass without any need to supply initial guesses for the eigenvalues; (c) for any fixed number of steps in the potential, we obtain in principle the whole infinite spectrum of eigenvalues and eigenfunctions.

NUMERICAL SOLUTION OF FISHER'S EQUATION

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fishers equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration. NONLINEAR DIFFUSION; EPIDEMIC WAVES; NUMERICAL SIMULATION

Diffusion in Nonlinear Multiplicative Media

The time‐dependent behavior of the nonlinear distributions defined by the diffusion equation with several nonlinear source terms is studied. The nonlinear diffusion equation is solved by an eigenfunction‐expansion method, which is in principle independent of geometry or number of dimensions. The qualitative time behavior of the distributions and their steady states can be ascertained from a simple analysis of the fundamental mode approximation only. Explicit solutions are presented in one‐ and two‐dimensional geometries.

A direct solution of the radiative transfer equation: Application to Rayleigh and Mie atmospheres

Abstract A direct method is given for the solution of the spherical harmonics approximation to the equation of radiative transfer in plane-parallel atmospheres. Although the method is formulated theoretically for non-homogeneous atmospheres with an arbitrary phase function, at present it has only been implemented for homogeneous atmospheres. Test computations performed for Rayleigh and Mie scattering phase functions show that the direct method is unconditionally stable and solves efficiently problems both for optically thin and very thick atmospheres. Timing comparisons with the method of Chandrasekhar for Rayleigh atmospheres and with an integral-equation iterative method for Mie atmospheres are quite favorable to the proposed method.

The recurrence of the initial state in the numerical solution of the Vlasov equation

Abstract The approximate recurrence of the initial state, observed recently in the numerical solution of Vlasovs equation by a finite-difference Eulerian model, is shown to be a property of three independent numerical methods. Some of the methods have exponentially growing modes (Dawsons beaming instabilities), and some others do not. The recurrence is in fact a manifestation of the finite velocity resolution of the numerical methods—a property which is independent of the approximation of a plasma by a finite number of electron beams. The recurrence is shown explicitly in the numerical simulation of Landau damping by three different methods: Fourier-Hermite, the recent variational method of Lewis, and the Eulerian finite-difference method.

Numerical solution of Mathieu's equation

Abstract A method is presented for the numerical solution of Mathieus equation. The power of the method lies in the fact that it can be used equally for ordinary and extremely asymptotic problems, making possible the computation of Mathieu functions for large values of the parameter with an accuracy heretofore unattainable.

Replacement Free Energy in the Thermodynamics of Small Systems

The object of this study is to present an analysis of the origin and calculation of the replacement free energy in the thermodynamics of small systems. An exact expression is obtained for the replacement free energy of a one‐dimensional “crystal” with free ends.

Threshold conditions for electron trapping by nonlinear plasma waves

Numerical solutions of the nonlinear Vlasov equation show that the long time behavior of an electrostatic plasma wave depends critically on the value of the parameter γτ (γ is Landaus damping coefficient and τ is the electron bounce time), both for large and small initial wave amplitudes. When the initial wave amplitude is large, the initial nonlinear damping is stronger than Landau damping; for smaller initial amplitudes, the behavior is initially described by Landaus theory. For both types of problems if γτ > 0.5, the wave damps monotonically; if γτ < 0.5, the numerical results indicate that the plasma performs amplitude oscillations. When γτ ≈ 0.5, the plasma displays its critical behavior; here, after some initial damping the wave amplitude levels off at a constant value. The critical behavior observed for mild nonlinear problems is in good agreement with the experimental results of Franklin, Hamberger, and Smith. For a moderately strong nonlinear wave displaying the critical behavior, the wave perf...

Electrostatic Oscillations in Plasmas with Cutoff Distributions

When numerically solving the linear Vlasov equation with cutoff electron distributions, the well‐known Landau‐damped oscillation of the electric field was observed first but, after a short transition time, a constant‐amplitude harmonic oscillation with a frequency distinctly higher than Landaus frequency was found. This coexistence of two distinct modes of oscillation—one damped and one undamped—has not been observed before. The behavior is explained quantitatively by comparison of the dispersion equation solution with the results obtained from the numerical solution of Vlasovs equation.

ASYMPTOTIC BEHAVIOR OF CERTAIN NONLINEAR BOUNDARY-VALUE PROBLEMS.

The asymptotic properties of a class of nonlinear boundary‐value problems are studied. For large values of a parameter, the differential equation is of the singular‐perturbation type, and its solution is constructed by means of matched asymptotic expansions. In two special cases, very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.