User:Eml5526.s11.team2.stewart/HW2

=Problem 2.1: "Derive the First Order Differential Heat Equation for a 1-D Finite Element"=

Problem Statement:
To derive the first order differential equation for heat was introduced in Meeting 6 Page 2 Using a free-body diagram apply the appropriate heat balance and verify that the following differential equation holds for the case of a 1-D finite element with heat applied:

The following figure is the free body diagram of the finite element heat problem:

Solution:
The first order differential equation can derived by summing the heat flux, heat flow, and heat outflow:

where

Summing all heat leaving and entering the finite element:

Expanding and dividing through by dx

From initial and boundary conditions, the final solution is equal to equation 1.1:

CHANGE EQUATION NUMBERS, NEATEN UP

=Problem 2.5: "Proof of Equivalent Equations"=

Problem Statement:
Complete part B of proof of equation

Such that:

Proof:
Equation 2 from Lecture 7-2 states that:

Choice 1: $$\displaystyle \alpha_1=1$$
Therefore:

Choice 2: $$\displaystyle \alpha_2=1$$
Therefore:

Choice n: $$\displaystyle \alpha_n=1$$
Therefore:

Identity
Multiplying equation 5.9 through by $$\alpha_n$$ :

Repeating for $$\alpha_1$$ and $$\alpha_2$$, etc and adding their respective equations:

Since equation 5.2 states that $$\underline w$$ is the sum of the product of $$\alpha _i$$ and $$\underline b _i$$, the following is proven: