1,352 research outputs found
Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities
We study the second order nonlinear differential equation \begin{equation*}
u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j}
b_{j}(x)k_{j}(u) = 0, \end{equation*} where ,
are non-negative Lebesgue integrable functions defined in
, and the nonlinearities are
continuous, positive and satisfy suitable growth conditions, as to cover the
classical superlinear equation , with . When the positive
parameters are sufficiently large, we prove the existence of at
least positive solutions for the Sturm-Liouville boundary value
problems associated with the equation. The proof is based on the Leray-Schauder
topological degree for locally compact operators on open and possibly unbounded
sets. Finally, we deal with radially symmetric positive solutions for the
Dirichlet problems associated with elliptic PDEs.Comment: 23 pages, 6 PNG figure
Existence of positive solutions of a superlinear boundary value problem with indefinite weight
We deal with the existence of positive solutions for a two-point boundary
value problem associated with the nonlinear second order equation
. The weight is allowed to change its sign. We assume
that the function is
continuous, and satisfies suitable growth conditions, so as the case
, with , is covered. In particular we suppose that is
large near infinity, but we do not require that is non-negative in a
neighborhood of zero. Using a topological approach based on the Leray-Schauder
degree we obtain a result of existence of at least a positive solution that
improves previous existence theorems.Comment: 12 pages, 4 PNG figure
A Time-Dependent Dirichlet-Neumann Method for the Heat Equation
We present a waveform relaxation version of the Dirichlet-Neumann method for
parabolic problem. Like the Dirichlet-Neumann method for steady problems, the
method is based on a non-overlapping spatial domain decomposition, and the
iteration involves subdomain solves with Dirichlet boundary conditions followed
by subdomain solves with Neumann boundary conditions. However, each subdomain
problem is now in space and time, and the interface conditions are also
time-dependent. Using a Laplace transform argument, we show for the heat
equation that when we consider finite time intervals, the Dirichlet-Neumann
method converges, similar to the case of Schwarz waveform relaxation
algorithms. The convergence rate depends on the length of the subdomains as
well as the size of the time window. In this discussion, we only stick to the
linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and
Engineering, Vol. 98, Springer-Verlag 201
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