Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 7 docx
Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 7 docx
... a
2
φ
xx
, a
2
=
k
cρ
, (39.2)
so that it is valid for diffusion in a non-homogeneous medium for which c and k are functions of x and φ and so that
it is valid for a geometry in which A is a function of ... dξ =
x
a
h
x
(x, ξ) dξ + h(x, x).
Hint 39.6
1834
Solution 37. 37
We will assume that both α and β are nonzero. The cases of real and pure imaginary have already been cove...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 7 docx
... d
2
n
=
∞
n=1
n
2
a
2
n
+ b
2
n
+ c
2
n
+ d
2
n
14 15
-1
-0 .5 0 .5
1
-0.2
-0.1
0.1
0.2
-1
-0 .5 0 .5
1
-0.2
-0.1
0.1
0.2
Figure 28.13: The odd and even periodic extension of x(1 − x), 0 ≤ x ≤ 1.
The ... the form
p
2
y
+ p
1
y
+ p
0
y = µy, for a ≤ x ≤ b,
α
1
y(a) + α
2
y
(a) = 0, β
1
y(b) + β
2
y
(b) = 0,
where the p
j
are real and continuous and p
2
> 0 on [...
Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 2 docx
... error
5 24 23.6038 0.01 65
15 8 .71 783 · 10
10
8.66 954 · 10
10
0.0 055
25 6.20448 · 10
23
6.18384 · 10
23
0.0033
35 2. 952 33 · 10
38
2.9 453 1 · 10
38
0.0024
45 2. 658 27 · 10
54
2. 653 35 · 10
54
0.0019
In ... J
−n
(z) and J
n
(z) are not linearly independent for integer n.
16 25
2
4
6 8 10
12
14
-0.4
-0.2
0.2
0.4
0.6
0.8
1
5
10
15
20
-0.2
-0.1
0.1
0.2
0.3
Figure...
Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt
... This is zero if and only if u
0
= u
1
= u
2
= u
3
= 0. Thus there
2 17
-10
-5 5
10
- 15
-10
-5
5
Figure 6.23: (z) − (z) = 5
Solution 6.16
|
e
ıθ
−1| = 2
e
ıθ
−1
e
−ıθ
−1
= 4
1 −
e
ıθ
−
e
−ıθ
+1 ... Figure 7. 10. The modulus and principal argument of f(z) = z are
plotted in Figure 7. 11.
Example 7 .5. 2 Consider the function f(z) = z
2
. In Cartesian co ordinates and...
Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 7 pdf
... +
z
4
90
+ ···
=
1
z
5
1 −
7
45
z
4
+ ···
=
1
z
5
−
7
45
1
z
+ ···
Thus we see that the residue is −
7
45
. N ow we can evaluate the integral.
C
cot z coth z
z
3
dz = −ı
14
45
π
13.2 Cauchy ... union of
C
p
and C
. (C
p
stands for Principal value Contour; C
i
stands for Indented Contour.) C
i
is an indented contour that
avoids the first order pole at z = 1. Figur...
Advanced Mathematical Methods for Scientists and Engineers Episode 3 Part 7 pps
... Equations
10 17
19.6 Hints
The Constant Coefficient Equation
Normal Form
Hint 19.1
Transform the equation to normal form.
Transformations of the Independent Variable
Integral Equations
Hint 19.2
Transform ... that satisfy the left and right boundary conditions are
c
1
(x − a) and c
2
(x − b).
Thus the Green’s function has the form
G(x|ξ) =
c
1
(x − a), for x ≤ ξ
c
2
(x − b), for x ≥...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 1 docx
... difference equation for b
n
that is of order one less than the
equation for a
n
.
1 178
1
2
3
4
5
6
-1
-0 .5
0 .5
1
1 .5
1
2
3
4
5
6
-1
-0 .5
0 .5
1
1 .5
Figure 23.3: The graph of approximations and numerical ... −
1
√
5
r
1
=
2
1 −
√
5
1 −
1
√
5
1 +
√
5
2
= −
2
1 −
√
5
1 −
√
5
2
√
5
= −
1
√
5
Thus the n
th
term in the Fibonacci sequence has the formu...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 4 docx
... eigenfunction
φ. Green’s formula states
φ|L[φ] − L[φ]|φ = 0
φ|λφ − λφ|φ = 0
(λ − λ)φ|φ = 0
Since φ ≡ 0, φ|φ > 0. Thus λ = λ and λ is real.
1319
27. 7 Hints
Hint 27. 1
13 27
2. For k = n, φ
k
, ... the
formula to obtain information about the eigenvalues before we solve a problem.
Example 27. 4.2 Consider the self-adjoint eigenvalue problem
−y
= λy, y(0) = y(π) = 0.
1...
Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 9 docx
... ıa
∞
0
, for (s) > 0
=
1
ı2
1
s − ıa
−
1
s + ıa
L[sin(at)] =
a
s
2
+ a
2
for (s) > 0
150 7
Since
ˆ
f
(s) is defined for (s) > 0,
ˆ
f(s) is analytic for (s) > 0.
Let σ be real and ... have
1
ı2π
B
+
L
+
+
C
+
L
−
e
st
π
s
1/2
e
−2(as)
1/2
ds = 0.
152 7
By inspection we see that this is satisfied for f(t) = 1 for 0 < t < T . We c...
Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 1 pdf
... b.
159 7
Hint 32.10
Hint 32.11
The left side is the convolution of u(x) and
e
−ax
2
.
Hint 32.12
Hint 32.13
Hint 32.14
Hint 32. 15
Hint 32.16
Hint 32. 17
Hint 32.18
Hint 32.19
Hint 32.20
1 57 9
Solution ... f(−x)])
where F, F
c
and F
s
are respectively the Fourier transform, Fourier cosine transform and Fourier sine transform.
Hint, Solution
1 57 6
... Since
159 5
Exercise 32.2...
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