# Numerical Methods for Partial Differential Equations

## Program for Day 1

#### Tuesday, May 08, 2001, 07:00-08:50 p.m., MP 401

1. 07:05-07:30
Effective Insulation of a Planting Box
Alexandra L. Ward
We consider the problem of planting a tree in a planting box located near a pipe line carrying hot steam. Heat is transfered from the steam line to the surrounding ground, including the planting box. We consider the effect that lining two sides of the box with insulation has on the temperature distribution over the area.

2. 07:30-07:55
Calcium Ion Diffusion in Cardiac Cells
Alexander L. Hanhart
The propagation of calcium ions in cardiac cells is modeled by a system of three reaction diffusion equations. Calcium ions are released at certain points in the cell according to a probability function that relies on the concentration of the ions. Unfortunately, this creates a discontinuous forcing term in our system that relies on the delta function. However, using finite elements, we can work our way around this problem. This talk serves as an introduction to the problem, as well as outlining a finite element solution method, and presenting results based on this method.

3. 07:55-08:20
Finite Element Method in Truss Analysis
Jiaqiao Hu
From the physical point of view, we will formulate the differential equation of a linear elastic bar in one dimension. Further, we will develop its variational formulation. Then by introducing the vector transformation and the finite element method, we will see how these can be appied in a 2-dimensional bridge truss analysis problem.

4. 08:20-08:45
Temperature Response at the Midplane of a Cylindrical Shell
Samuel G. Webster
The spread of heat energy throughout an isotropic solid is a common physical occurrence. It is known that heat diffuses across such a solid until it reaches a state of equilibrium. Mathematically, this evolutionary process can be represented by the diffusion equation. Using the finite element method, we seek to measure the temperature response at the midplane of a cylindrical shell after a sudden change in internal surface temperature. The accuracy of these parabolic solutions is then verified by a comparison to a theoretical prediction, as well as by the existence of second order convergence.

## Program for Day 2

#### Thursday, May 10, 2001, 07:00-08:50 p.m., MP 401

1. 07:05-07:30
Bifurcation from the Set of Equilibria of the Reaction-Diffusion Equation
Valeriy R. Korostyshevskiy
We study the behavior of steady-state solutions of the nonlinear reaction-diffusion equation. The main point of interest is how the system performs depending on the values of scalar parameter(s). To find that we use specific software packages that allow us to make significant conclusions.

2. 07:30-07:55
Putting a Freeze on the Concrete Island Drilling System
Tracy E. Thoma
This paper explores the application of the finite element method to the PDE known as the heat equation. The physical problem at hand involves a concrete island drilling system with embedded water baths, which supports a barge-mounted drilling rig. The entire platform is placed in the ocean, which can drop to temperatures of -60 degrees F in the Arctic. If the water baths in the concrete freeze, the walls will fracture. This situation can be avoided with the use of insulation, as will be shown by determining the temperature throughout the system via the finite element method.

3. 07:55-08:20
Use of Galerkin Method and AUTO to find Solution Branches of the Cahn-Hilliard Equation
Jennifer E. Deering
The Cahn-Hilliard Equation is introduced. The Galerkin method is applied to the equation to produce a system of ordinary differential equations. AUTO is then used to find solution branches. This work is intended to reproduce some results found by S. Maier-Paape and U. Miller.

4. 08:20-08:45
A Feasibility Study in Estimating Magnetic Field and Fluid Velocity Patterns Just Beneath the Core-Mantle Boundary from Surface Magnetic Field Measurements via the Radial Motional-Induction Equation
Terence J. Sabaka
We explore the feasibility of inferring fluid velocity and initial magnetic fields just beneath the core-mantle boundary (CMB) from magnetic measurements at Earth's surface via the radial motional-induction equation (RMIE), a hyperbolic transport equation describing magnetic field sustainment through field advection by highly conductive molten metal in the core. Given spectral field models are recovered exactly from a perturbed initial guess by nonlinear least-squares analysis of radial surface data synthesized from these models, where partial derivatives are obtained by solving systems of coupled ODEs in time resulting from spectral decompositions of the RMIE, and related equations, in space.