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: Ꭷ2Ꭲ Ꭷ2Ꭲ + ax² ay² in [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: Ꭷ2Ꭲ Ꭷ2Ꭲ + ax² ay² in [0
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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This is a multiple-part question, I just need help with part C, Table 2.2 is provided and you can refer to above parts for equations and boundary equations.
![## Educational Content on Heat Diffusion Problem
### Problem Overview
Consider the problem 2-15 from Ozisik (1993). This involves reducing the heat diffusion equation, as provided in Equation 1-11a in Ozisik (1993), to the form:
\[
\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0
\]
This equation applies in the domain \([0 < x < a; 0 < y < b]\), subjected to the following boundary conditions:
- \(\frac{\partial T}{\partial x} = 0\) at \(x = 0\)
- \(\frac{\partial T}{\partial x} + HT = 0\) at \(x = a\)
- \(T = f(x)\) at \(y = 0\)
- \(\frac{\partial T}{\partial y} + HT = 0\) at \(y = b\)
### Steps and Justifications
All assumptions must be clearly stated and justified, with each step of the reduction process thoroughly explained.
### Problem 1b: Separation of Variables
Using the method of separation of variables, demonstrate how the eigenfunction from Table 2-2 in Ozisik (1993) was derived:
\[
X(\beta_m, x) = \cos \beta_m x
\]
Each step in the formulation should be clearly explained, and all assumptions must be adequately justified.
### Problem 1c: Eigen Condition
Illustrate how the eigen condition from Table 2-2 in Ozisik (1993) was derived:
\[
\beta_m \tan \beta_m a = H
\]
As with previous sections, each step of your formulation must be clearly explained, with all assumptions justified.
### Diagrams and Graphs
While there are no specific diagrams or graphs in this text, it may be beneficial to include visual aids such as:
- Graphical representation of the domain \([0 < x < a; 0 < y < b]\)
- Illustrations of the boundary conditions
- Graphs demonstrating the solution obtained through separation of variables
Visual aids can enhance understanding by providing intuitive insight into the problem structure and solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b656b52-bb2e-489d-9bef-8a32efc9339f%2F3d2123ee-51ef-4286-a136-fd029b2e1adf%2Fowietv8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Educational Content on Heat Diffusion Problem
### Problem Overview
Consider the problem 2-15 from Ozisik (1993). This involves reducing the heat diffusion equation, as provided in Equation 1-11a in Ozisik (1993), to the form:
\[
\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0
\]
This equation applies in the domain \([0 < x < a; 0 < y < b]\), subjected to the following boundary conditions:
- \(\frac{\partial T}{\partial x} = 0\) at \(x = 0\)
- \(\frac{\partial T}{\partial x} + HT = 0\) at \(x = a\)
- \(T = f(x)\) at \(y = 0\)
- \(\frac{\partial T}{\partial y} + HT = 0\) at \(y = b\)
### Steps and Justifications
All assumptions must be clearly stated and justified, with each step of the reduction process thoroughly explained.
### Problem 1b: Separation of Variables
Using the method of separation of variables, demonstrate how the eigenfunction from Table 2-2 in Ozisik (1993) was derived:
\[
X(\beta_m, x) = \cos \beta_m x
\]
Each step in the formulation should be clearly explained, and all assumptions must be adequately justified.
### Problem 1c: Eigen Condition
Illustrate how the eigen condition from Table 2-2 in Ozisik (1993) was derived:
\[
\beta_m \tan \beta_m a = H
\]
As with previous sections, each step of your formulation must be clearly explained, with all assumptions justified.
### Diagrams and Graphs
While there are no specific diagrams or graphs in this text, it may be beneficial to include visual aids such as:
- Graphical representation of the domain \([0 < x < a; 0 < y < b]\)
- Illustrations of the boundary conditions
- Graphs demonstrating the solution obtained through separation of variables
Visual aids can enhance understanding by providing intuitive insight into the problem structure and solution.
![**TABLE 2-2**: The Solution \( X(\beta_m, x) \), the Norm \( N(\beta_m) \), and the Eigenvalues \( \beta_m \) of the Differential Equation
**Differential Equation:**
\[
\frac{d^2 X(x)}{dx^2} + \beta^2 X(x) = 0 \quad \text{in} \quad 0 < x < L
\]
This equation is accompanied by a diagram indicating a boundary from \( x = 0 \) to \( x = L \).
**Boundary Conditions and Solutions:**
The table provides solutions for \( X(\beta_m, x) \), norms \( 1/N(\beta_m) \), and eigenvalues \( \beta_m \) subject to different boundary conditions:
| No. | Boundary Condition at \( x = 0 \) | Boundary Condition at \( x = L \) | Solution \( X(\beta_m, x) \) | Norm \( 1/N(\beta_m) \) | Eigenvalues \( \beta_m^2 \) are Positive Roots of |
|-----|----------------------------------------|---------------------------------------|--------------------------------------|---------------------------------|-------------------------------------------------|
| 1 | \(-\frac{dX}{dx} + H_1 X = 0\) | \(\frac{dX}{dx} + H_2 X = 0\) | \(\beta_m \cos \beta_m x + H_1 \sin \beta_m x\) | \(2 \left[ (\beta_m^2 + H_1^2) \left( L + \frac{H_2}{\beta_m^2 + H_2^2} \right) + H_1 \right]^{-1} \) | \(\tan \beta_m L = \frac{\beta_m (H_1 + H_2)}{\beta_m^2 - H_1 H_2}\) |
| 2 | \(-\frac{dX}{dx} = 0\) | \(\frac{dX}{dx} = 0\) | \(\cos \beta_m (L-x)\) | \(\frac{2}{L(\beta_m^2 + H_2^2) + H_1}\) | \( \beta_m \tan \beta_m L = H_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b656b52-bb2e-489d-9bef-8a32efc9339f%2F3d2123ee-51ef-4286-a136-fd029b2e1adf%2Ftjina9_processed.png&w=3840&q=75)
Transcribed Image Text:**TABLE 2-2**: The Solution \( X(\beta_m, x) \), the Norm \( N(\beta_m) \), and the Eigenvalues \( \beta_m \) of the Differential Equation
**Differential Equation:**
\[
\frac{d^2 X(x)}{dx^2} + \beta^2 X(x) = 0 \quad \text{in} \quad 0 < x < L
\]
This equation is accompanied by a diagram indicating a boundary from \( x = 0 \) to \( x = L \).
**Boundary Conditions and Solutions:**
The table provides solutions for \( X(\beta_m, x) \), norms \( 1/N(\beta_m) \), and eigenvalues \( \beta_m \) subject to different boundary conditions:
| No. | Boundary Condition at \( x = 0 \) | Boundary Condition at \( x = L \) | Solution \( X(\beta_m, x) \) | Norm \( 1/N(\beta_m) \) | Eigenvalues \( \beta_m^2 \) are Positive Roots of |
|-----|----------------------------------------|---------------------------------------|--------------------------------------|---------------------------------|-------------------------------------------------|
| 1 | \(-\frac{dX}{dx} + H_1 X = 0\) | \(\frac{dX}{dx} + H_2 X = 0\) | \(\beta_m \cos \beta_m x + H_1 \sin \beta_m x\) | \(2 \left[ (\beta_m^2 + H_1^2) \left( L + \frac{H_2}{\beta_m^2 + H_2^2} \right) + H_1 \right]^{-1} \) | \(\tan \beta_m L = \frac{\beta_m (H_1 + H_2)}{\beta_m^2 - H_1 H_2}\) |
| 2 | \(-\frac{dX}{dx} = 0\) | \(\frac{dX}{dx} = 0\) | \(\cos \beta_m (L-x)\) | \(\frac{2}{L(\beta_m^2 + H_2^2) + H_1}\) | \( \beta_m \tan \beta_m L = H_
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