Project 13 introduces a two point boundary value problem, that is, a physical situation that can be modeled as a function f(x) defined over a finite interval, for which the values of f(x) are known at the two endpoints, and a differential equation determines the shape of f(x) in between.
A common physical example is a long thin insulated metal rod, which is heated at one end, with the other end placed in ice water. The temperature at each point of the rod is described by something called "the steady state heat equation", which is a second order differential equation.
This situation is different from the typical differential equation we have solved before. Instead of having a starting time and starting values, we have two locations (the ends of the rod) and one specified value at each place. This is the characteristic of problems called two point boundary value problems.
In order to describe the function f(x), we will consider two related techniques, the finite difference method, which replaces derivatives by difference quotients, and the finite element method, which concentrates on describing the solution as a sum of simple basis functions.
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