arXiv subjects
Publications and source records attributed to Dieter S. Schmidt.
Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.
For a class of oscillatory resonant problems, involving Dirichlet problems for semilinear PDE's on balls and rectangles in $R^n$, we show the existence of infinitely many solutions, and study the global solution set. The first harmonic of the right hand side is not required to be zero, or small. We also derive asymptotic formulas in terms of the first harmonic of solutions, and illustrate their accuracy by numerical computations. The numerical method is explained in detail.
We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ Δu +λ\left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \] with a positive parameter $λ$, and $1 \frac{n}{n+2}$. This was already observed by I. Flores [6], who proved the existence of infinitely many ground state solutions. We study properties of infinitely many solution curves of this problem that are separated by these ground state solutions. We also study singular solutions (where $u(0)=\infty$), and again the Lin-Ni equation plays a special role. \medskip Super-critical equations are very challenging computationally: solutions exist only for very large $λ$, and curves of positive solutions make turns at very large values of $u(0)=||u||_{L^{\infty}}$. We overcome these difficulties by developing new results on singular solutions, and by using some delicate capabilities of {\em Mathematica} software.
This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as its different branches, appear automatically on the computer screen. For radial $p$-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman [15], and use this transformation for more accurate numerical computations. While for $p>2$ the solutions are not of class $C^2$, we show that they are of the form $w(r^{\frac{p}{2(p-1)}})$, where $w(z)$ is of class $C^2$. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.