In the following examples, we will adopt the notation in the textbook. The unit grid size ("x" in our
lecture) is denoted as h. It is also understood that, for a uniform grid, f(x) f(xi), f(x+h) f(xi+
Introduction to Partial Differential Equation - II. Some simple analytic solutions
We will restrict the discussion here to the method of separation of variables. Let's start with some simple
examples
MAE 384 Homework 5 Discussion
Prob. 1 The result for (b) is much more accurate than those for (a) and (c), because the
scheme in (b) is "4th order" with a truncation error of O(h4), while the other tw
MAE 384 Homework 6
This problem set has 2 pages plus a list of the useful finite difference schemes in page 3.
1. Solve the initial value problem,
du
= u 2 sin x , u(0) = 0 ,
dx
to obtain u(0.4) by us
MAE 384 Homework 5
1. (a) Given the function
f(x) = ex sin(x) ,
evaluate the first derivative, f '(x), at x = 1 using the three-point backward scheme (2nd formula from
the top of Table 6-1 in p. 248 o
Examples of numerical solutions for boundary value problems
(See Sec. 9.3 of G&S textbook for more examples)
Example 1
d 2u/dx2 = 1 , u(0) = 0, u(1) = 0
Analytic solution is u(x) = x2/2 x/2. To obtain
MAE384 Homework 4
Discussion
Prob 1
Note that the Lagrange function, Li(x), should satisfy
Li(x) = 0, at x = xj , j i , and
Li(x) = 1, at x = xi
For example, L1, L3, and L4 should all be zero at x = x
MAE 384 Homework #1
1. (a) Find the largest positive number and the smallest positive number that are allowable on your
calculator. Describe the process through which you find these two limits. (b) Pr
MAE 384 HW2
Comments
Prob 1 is straightforward. Note that there are two solutions. See attached sample solution.
Prob 2
There are many ways to rewrite the original equation into the form, x = g(x), fo
MAE 384 Homework #2
1. Solve the equation,
0.5 x2 sin(x) + 0.1 = 0 ,
using Newton's method. Find all of the solutions. Please show your work by providing
the detail of the intermediate steps of the it
MAE384 Homework 4
1. The set of the following 4 data points is given:
x
0
1
2
3
y
1
6
25
76
(a) Perform interpolation by determining the 3rd order polynomial that passes through the points, using
Lagr
MAE384 Homework #3
1. Solve the equation,
3 8
6 2
x1
= 9
4
x2
,
using the LU decomposition method. No credit if you use any other methods, even with the correct
final solution. As always, please sho
MAE384 HW1
Comments:
Prob.1
We will discuss this problem in class.
Prob. 2
Many of you attempted to divide 256.1875 by 265 before proceeding to the standard binary
decomposition. In this case, the cor