Partial Differential Equations Commons^™

Reconstruction Of Multiple Cracks From Experimental, Electrostatic Boundary Measurements, Kurt M. Bryan, Valdis Liepa, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We demonstrate the viability of using Electrical Impedance Tomography (EIT) for the reconstruction of multiple macroscopic cracks in a conductive medium.

Time-Discretization Of Hamiltonian Dynamical Systems, Yosi Shibberu

Mathematical Sciences Technical Reports (MSTR)

Difference equations for Hamiltonian systems are derived from a discrete variational principle. The difference equations completely determine piecewise-linear, continuous trajectories which exactly conserve the Hamiltonian function at the midpoints of each linear segment. A generating function exists for transformations between the vertices of the trajectories. Existence and uniqueness results are present as well as simulation results for a simple pendulum and an inverse square law system.

Electrostatic Positioning Of Droplets In Turbulent Flows (Lstm 375/Te/93), Nihad E. Daidzic, Adrian Melling

Aviation Department Publications

Report LSTM 375/TE/93, Lehrstuhl fuer Stroemungsmechanik Universitaet Erlangen-Nuernberg Cauerstr. 4, 8520 Erlangen Germany.

A Bifurcation Theorem And Applications, Alfonso Castro, Jorge Cossio

All HMC Faculty Publications and Research

In this paper we give a sufficient condition on the nonlinear operator N for a point (λ, u) to be a local bifurcation point of equations of the form u + λL^-1(N(u)) = 0, where L is a linear operator in a real Hilbert space, L has compact inverse, and λ ∈ R is a parameter. Our result does not depend on the variational structure of the equation or the multiplicity of the eigenvalue of the linear operator L. Applications are made to systems of differential equations and to the existence of periodic solutions of nonlinear second order …

Nonnegative Solutions To A Semilinear Dirichlet Problem In A Ball Are Positive And Radially Symmetric, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We prove that nonnegative solutions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular, this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the domain which ensures that nonnegative solutions are positive in the interior.

Multiple Solutions For A Dirichlet Problem With Jumping Nonlinearities Ii, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

No abstract provided for this article.

Nonnegative Solutions For A Class Of Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

In the recent past many results have been established on non-negative solutions to boundary value problems of the form

-u''(x) = λf(u(x)); 0 < x < 1,

u(0) = 0 = u(1)

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.

Uniqueness Of Positive Solutions For A Class Of Elliptic Boundary Value Problems, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

Uniqueness of non-negative solutions conjectured in an earlier paper by Shivaji is proved. Our methods are independent of those of that paper, where the problem was considered only in a ball. Further, our results apply to a wider class of nonlinearities.

Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro

All HMC Faculty Publications and Research

In this paper, we study the existence of weak solutions of the problem

□u + ∇G(u) = f(t,x) ; (t,x) є Ω ≡ (0,π)x(0,π)

u(t,x) = 0 ; (t,x) є ∂Ω

where □ is the wave operator ∂²/∂t² - ∂²/∂x², G: Rⁿ→R is a function of class C² such that ∇G(0) = 0 and f:Ώ→R^n is a continuous function having first derivative with respect to t in (L₂,(Ω))ⁿ and satisfying

f(0,x) = f(π,x) = 0

for all x є [0,π].

Critical Point Theory And The Number Of Solutions Of A Nonlinear Dirichlet Problem, Alfonso Castro, A. C. Lazer

All HMC Faculty Publications and Research

No abstract provided.

A Semilinear Dirichlet Problem, Alfonso Castro

All HMC Faculty Publications and Research

Let Ω be a bounded region in R^n. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem:

Δu(x) + g(x, u(x)) + f(x, u(x), ∇u(x)) = 0 ; x є Ω

u(x) = 0 ; x є ∂Ω

where Δ is the Laplacian operator, g : Ω x R → R and f : Ω x Rⁿ⁺¹ → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

Graphical Representations Of Singular Solutions Of Differential Equations, Kathryn Lois Pitman

Bachelors’ Theses

The first men to detect singular solutions of differential equations were Leibniz, Brook Taylor and Clairaut. The direct method of attack used by each man in attaining these solutions is not known, but a short history of each man and what he has contributed to mathematics can be given. Gottfried Wilhelm von Leibniz (1646-1716) was born in Leipsic, Germany, and was the son of a professor of law in a nearby university. He was well-educated in law himself, expecting to follow in the profession of his father. But not being under the necessity of earning his living, he enrolled at …