17.3 Use leastsquares regression to t a straight line to
3p024691112151710
@5070987101212
lling with the slope and intercept, compute the standard elror of
the estimate and the correlation coefcient. Plot the data and the
aggression line. Then repeat the
Chapter 7
Differential Equations
4y
dx
3
dx
= x(y3)
= (y3)dy
=
4
=
1
y
x
y dy = 4 x
R
R
1 y3 dy = 4 dx
x = y 3 ln |y| = 4 ln |x| + C
3 +4y
(y2 +2y+4)dy
dy
1
2
dx
(b) dx
= x(yy2 +2y+4)
=
+
=
=
dy =
3
2
x
y
y +4y
y +4
R 1
R
2
dx
1 y = ln |x| + C
y + y 2 +4
L5 MATLAB and Error
Analysis
MATLAB
Is a flagship software which was originally
developed as a matrix laboratory. A variety of
numerical functions, symbolic computations,
and visualization tools have been added to the
matrix manipulations.
MATLAB is clo
25.1 Solve the following initial value problem over the interval from
t = 0 to 2 where y(0) = 1. Display all your results on the same graph.
dy 2
= t - 1.1
dt y y
(a) Analytically.
(b) Eulers method with h = 0.5 and 0.25.
(d) Fourth-order RK method with
L9 Trapezoidal Rule
Newton-Cotes Integration Formulas
The Newton-Cotes formulas are the most common
numerical integration schemes.
They are based on the strategy of replacing a
complicated function or tabulated data with an
approximating function that i
L7 LU Decomposition
LU Decomposition and Matrix Inversion
Provides an efficient way to compute matrix
inverse by separating the time consuming
elimination of the Matrix [A] from
manipulations of the right-hand side cfw_B.
Gauss elimination, in which the
CHAPTER 9.
PARTIAL DIFFERENTIAL EQUATIONS
(8) Write u (x, t) = X (x) T (t) and substitute into the partial differential equation:
X (x) T (t) = c2 X (x) T (t)
Hence, the two differential equations are:
T (t) + c2 T (t) = 0
X (x) + X (x) = 0, X (0) = 0, X
348 CHAPTER 9.
PARTIAL DIFFERENTIAL EQUATIONS
1. Verify that the functions u1(m, t) = cos x-
cos ct and u2(z,t) 2 sins: - sin ct are so-
lutions of the wave equation utt = c2u.
Deduce that cos(a: + ct) and cos(m ct)
are solutions of the same PDE.
2. S
1
CHAPTER 10
10.2 (a) The coefficient a21 is eliminated by multiplying row 1 by f21 = 3/10 = 0.3 and subtracting the
result from row 2. a31 is eliminated by multiplying row 1 by f31 = 1/10 = 0.1 and subtracting the result from
row 3. The factors f21 and f
Chapter 9
Partial Differential Equations
(2) Write u (x, t) = X (x) T (t) and substitute into the partial differential equation:
X (x) T (t) = c2 X (x) T (t)
Hence, the two differential equations are:
X (x) + X (x) =0, X (0) = 0, X (L) = 0
(SLP)
T (t) = c
L10 Simpsons Rules
Simpsons Rules
More accurate estimate of an integral is
obtained if a high-order polynomial is used to
connect the points. The formulas that result
from taking the integrals under such
polynomials are called Simpsons rules.
2
Simpsons
L8 Least Squares Methods
CURVE FITTING
Describes techniques to fit curves (curve fitting) to discrete
data to obtain intermediate estimates.
There are two general approaches to curve fitting:
Data exhibit a significant degree of scatter. The strategy i
CHAPTER 9.
PARTIAL DIFFERENTIAL EQUATIONS
(3) Write u (x, t) = v (x, t) + l (x) and substitute into the partial differential equation:
vt (x, t) = c2 vxx (x, t) + l (x)
The boundary conditions imply that
u (0, t) = v (0, t) + l (0) = A, u (L, t) = v (L, t
1
CHAPTER 25
25.1 (a) The analytical solution can be derived by separation of variables
dy
t 2 1.1 dt
y
ln y
t3
1.1t C
3
Substituting the initial conditions yields C = 0. Taking the exponential gives the final result
y
t3
1.1t
e3
The result can be plot
L9 Trapezoidal Rule
Newton-Cotes Integration Formulas
The Newton-Cotes formulas are the most common
numerical integration schemes.
They are based on the strategy of replacing a
complicated function or tabulated data with an
approximating function that i
L12 Runga-Kutta Methods
Improvements of Eulers method
A fundamental source of error in Eulers
method is that the derivative at the beginning
of the interval is assumed to apply across the
entire interval.
Two simple modifications are available to
circum
17.3 The results can be summarized as
y versus x
y = 4.851535 + 0.35247x
1.06501
0.914767
Best fit equation
Standard error
Correlation coefficient
x versus y
x = 9.96763 + 2.374101y
2.764026
0.914767
We can also plot both lines on the same graph
y
12
8
y
L6 Iterative Methods
1 Solutions of Non-linear Equations
Why?
2
4ac
b
b
2
ax + bx + c = 0 x =
2a
But
ax 5 + bx 4 + cx 3 + dx 2 + ex + f = 0 x = ?
sin x + x = 0 x = ?
2
Open Methods
Open methods
are based on
formulas that
require only a
single starting
v
r 10x1+2X2*13:27
3.X] _ 6X2 + 2153 : 761.5
x1+ I2 + 5123 : 21.5
Use LU decomposition to solve the system.
all the steps in the computation. Also solve the system for
s alternative right-hand-side vector: cfw_Bf : [12 18 *6].
0.3
a) Solve the follo
L9 Ordinary Differential
Equations
Ordinary Differential Equations
Equations which are composed of an unknown
function and its derivatives are called differential
equations.
Differential equations play a fundamental role in
engineering because many phys
L13 Finite Difference Method
Partial Differential Equations
2
Finite Difference: Elliptic Equations
Solution Technique
Elliptic equations in engineering are typically used to
characterize steady-state, boundary value problems.
For numerical solution of
L11 Eulers Method
Ordinary Differential Equations
Equations which are composed of an unknown
function and its derivatives are called differential
equations.
Differential equations play a fundamental role in
engineering because many physical phenomena ar