MSE2094
Analytic Methods in Material Science

Assignment #3: Temperature gradients: Analytic-Numeric Comparison

DUE: Tuesday February 24, 1998

Objective: This assignment is designed introduce you to both analytical and numerical solution of a simple second order homogeneous differental equation..

Given:

• Problem definition:

Differential equation and schematic of Temperature Gradient Problem.

• Class notes and hand-outs on problem definition and Gauss elimination numerical method.
• Link to various Fortran77 and Fortan90 programs need for this assignment.

Part I (300 pts): Deliverables

• Cover page with your name, date and name of the assignment. (15pnts)

• Solve differential equation analytically for the unique and exact solution. Show complete derivation. (60pnts)

• Solve differential equation numerically by using Gauss elimination to solve simultaneous equations when h=1/4 and h=1/8 . When h=1/8 you will be required to recalculate the [A] amd {B} matrices. Can you generalize this calculation? Include a print out of your fortran program and results. (75pnts)

• Compare numerical results with exact analytic solution by using Microsoft Excel graphics similar to class handout. Print a copy or your excel table and graphical comparison and attach it to your fortran program and exact solution. Make sure all of the graphs axes are properly labeled with legend, schematic, governing differential equation and exact solution all on the same Excel print out. (90pnts)

• Does reducing the step-size, h, from h=1/4 to h=1/8 improve the results? How much? Why? If we continue to reduce the step-size will the accuracy continue to improve?. Attach your discusion together with the other documents. (60pnts)

Bonus Points: