1a. Consider the problem 2-15 from Ozisik (1993). Starting with the heat diffusion equation, Eq. 1-11a in Ozisik (1993), show how it can be reduced to: a²T ²T + ax² ay² in [0 < x
1a. Consider the problem 2-15 from Ozisik (1993). Starting with the heat diffusion equation, Eq. 1-11a in Ozisik (1993), show how it can be reduced to: a²T ²T + ax² ay² in [0 < x
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question
This is a multiple part question and i need help with only part D, part D might need some help from above parts that is provided in the table and the equations avaliable.
![5
6
TABLE 2-2 The Solution X(), the Norm NB) and the Eigenvalues of the Differential Equation
Subject to the Boundary Conditions Shown in the Table Below
Boundary
Condition
at x=0
No.
1
2
7
3
4
dX
-=+H₁X=0
dx
9
-
dX
dx
dx
dX
dx
dX
ö
dX
-=0
X=0
X=0
+ H₂X=0
+H₁X=0
dX
dx
dx
dX
i
X=0.
dX
dx
A
dX
dx
dX
Boundary
Condition
at x = L
dx
+H₂X=0
= 0
X=0
d²X(x)
dx²
+ H₂X=0
0
+ H₂X=0
=0
+8²x(x)=0 in
Bcos B+H, sin f
cos B (L-x)
X(Bmx)
sin B (L-x)
cos Bmx
*cos Bmx
cos x
sin x
sin ßx
0 < x <L
X=0
sin Bmx
X = 0
"For this particular case Bo-0 is also an eigenvalue corresponding to X = 1.
2 [10² + H²) (L
2-
B²+H}
L(B²+ H²) + H₂
0
B²+ H₂
2.
LB² + H²) + H₂
2
1/N(B)
8² + H²
2
L(B²+ H₂) + H₂
2
L+
for for 8,0
NIJ
B² + H²
L(B²+ H₂) + H₂
2
L
H₂
² + H²
+ H₁
Eigenvalues 's are
Positive Roots of
tan BL-
_B_(H₁ + H₂)
B²-H₂H₂
Plan BL=H₁
Bm cot BML=H₁
Pm tan PL=H₂
sin BL=0ª
cos BL=0
Bm cot BmL=-H₂
cos BL=0
sin AL=0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b656b52-bb2e-489d-9bef-8a32efc9339f%2Fea9e6cbd-0806-4135-b116-598d0efef2bd%2Fp7gcgj_processed.png&w=3840&q=75)
Transcribed Image Text:5
6
TABLE 2-2 The Solution X(), the Norm NB) and the Eigenvalues of the Differential Equation
Subject to the Boundary Conditions Shown in the Table Below
Boundary
Condition
at x=0
No.
1
2
7
3
4
dX
-=+H₁X=0
dx
9
-
dX
dx
dx
dX
dx
dX
ö
dX
-=0
X=0
X=0
+ H₂X=0
+H₁X=0
dX
dx
dx
dX
i
X=0.
dX
dx
A
dX
dx
dX
Boundary
Condition
at x = L
dx
+H₂X=0
= 0
X=0
d²X(x)
dx²
+ H₂X=0
0
+ H₂X=0
=0
+8²x(x)=0 in
Bcos B+H, sin f
cos B (L-x)
X(Bmx)
sin B (L-x)
cos Bmx
*cos Bmx
cos x
sin x
sin ßx
0 < x <L
X=0
sin Bmx
X = 0
"For this particular case Bo-0 is also an eigenvalue corresponding to X = 1.
2 [10² + H²) (L
2-
B²+H}
L(B²+ H²) + H₂
0
B²+ H₂
2.
LB² + H²) + H₂
2
1/N(B)
8² + H²
2
L(B²+ H₂) + H₂
2
L+
for for 8,0
NIJ
B² + H²
L(B²+ H₂) + H₂
2
L
H₂
² + H²
+ H₁
Eigenvalues 's are
Positive Roots of
tan BL-
_B_(H₁ + H₂)
B²-H₂H₂
Plan BL=H₁
Bm cot BML=H₁
Pm tan PL=H₂
sin BL=0ª
cos BL=0
Bm cot BmL=-H₂
cos BL=0
sin AL=0
![1a. Consider the
problem 2-15 from Ozisik (1993). Starting with the
heat diffusion equation, Eq. 1-11a in Ozisik (1993), show how it can be reduced to:
a²T ²T
+
əx² Əy²
in [0 < x <a; 0 <y<b], subjected to the following boundary conditions:
ƏT
ƏT
-= 0 at x = 0;
ax
T = f(x) at y = 0;
əx
0
+ HT = 0 at x = a;
ƏT
+ HT = 0 at y = b
ду
All assumptions must be clearly stated and justified, and all steps clearly explained.
= 2
1b. Using separation of variables, show how the eigenfunction from Table 2-2 in Ozisik
(1993) was derived.
X(Bm, x) = cospmx
Each step of your formulation must be clearly explained, and all assumptions justified.
1c. Show how the eigen condition from Table 2-2 in Ozisik (1993) was derived.
Pm tan Bma = H
Each step of your formulation must be clearly explained, and all assumptions justified.
1d. Show how the normalization integral from Table 2-2 in Ozisik (1993) was derived.
1
B + H²
N(Pm)
a(B² + H²) + H
Each step of your formulation must be clearly explained, and all assumptions justified.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b656b52-bb2e-489d-9bef-8a32efc9339f%2Fea9e6cbd-0806-4135-b116-598d0efef2bd%2F9o1xhp7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1a. Consider the
problem 2-15 from Ozisik (1993). Starting with the
heat diffusion equation, Eq. 1-11a in Ozisik (1993), show how it can be reduced to:
a²T ²T
+
əx² Əy²
in [0 < x <a; 0 <y<b], subjected to the following boundary conditions:
ƏT
ƏT
-= 0 at x = 0;
ax
T = f(x) at y = 0;
əx
0
+ HT = 0 at x = a;
ƏT
+ HT = 0 at y = b
ду
All assumptions must be clearly stated and justified, and all steps clearly explained.
= 2
1b. Using separation of variables, show how the eigenfunction from Table 2-2 in Ozisik
(1993) was derived.
X(Bm, x) = cospmx
Each step of your formulation must be clearly explained, and all assumptions justified.
1c. Show how the eigen condition from Table 2-2 in Ozisik (1993) was derived.
Pm tan Bma = H
Each step of your formulation must be clearly explained, and all assumptions justified.
1d. Show how the normalization integral from Table 2-2 in Ozisik (1993) was derived.
1
B + H²
N(Pm)
a(B² + H²) + H
Each step of your formulation must be clearly explained, and all assumptions justified.
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