Now the 1-D
initial-value problem (2.1) is given in the following form
Instead of calculating the transverse dynamic spin susceptibility from a Dyson-type integral equation with a dynamic exchange-correlation kernel, as is usually done, the key idea advanced here is to reformulate it as the solution of an
initial-value problem in the time domain.
In this study we briefly review the nonlinear nonlocal model and its approximate models for unidirectional wave propagation and then compare numerically the solutions of the exact model with the solutions of the approximate models for an
initial-value problem. In particular, we consider an
initial-value problem for the nonlocal model with initial data strictly compatible with the solitary wave solution of the KdV equation or the BBM equation and then use a finite-difference scheme to solve the
initial-value problem numerically.
In our ONR-funded study, we use a semianalytic Fourier-Laplace method to solve the complete
initial-value problem for linear waves forced by an idealized tsunami at the lower boundary.
There exists some [t.sub.1] such that the
initial-value problem,
to approximate the solution y(x) of the
initial-value problem at [x.sub.0] + h, the end of one step of length h.
This kind of
initial-value problem of ODEs can be solved using numerical computation.
Denote q(t) = q(t; [r.sub.m], [[epsilon].sub.m]) the solutions of the
initial-value problemEquation (2) was first introduced in [12] and both global existence and blow-up results for solutions of the
initial-value problem with initial data in appropriate function spaces were established.
Taylor's expansion of a second-order
initial-value problemWhen an
initial-value problem is stiff, one will typically observe that a code based on an explicit scheme will need to use extremely small stepsizes in order to compute a stable solution as opposed to one based on an implicit scheme.
Now the 1D
initial-value problem (2.2) is in the following form