https://file.aiquickdraw.com/mathbot/file/1728526859998o7ec75jz.jpg

The problem given involves solving a boundary value problem (BVP) using the finite difference method. Let’s break down the steps required for the question part (a):

### Given Equation:
The governing equation for the temperature distribution \( u(x) \) along the tapered conical cooling fin is:
\[
\frac{d^2u}{dx^2} + \left( \frac{2}{x} \right) \frac{du}{dx} - p u = 0
\]
with the boundary conditions:
\[
u(0) = 0, \quad u(1) = 1
\]

### Step-by-Step Procedure:
We need to **manually derive the finite-difference formula** for this differential equation using a step size \( dx = 0.2 \).

#### 1. **Discretize the Domain:**
The range of \( x \) is from \( x = 0 \) to \( x = 1 \). With \( dx = 0.2 \), the grid points are:
\[
x_0 = 0, x_1 = 0.2, x_2 = 0.4, x_3 = 0.6, x_4 = 0.8, x_5 = 1
\]
Thus, \( u_0 = u(0) = 0 \) and \( u_5 = u(1) = 1 \).

#### 2. **Finite Difference Approximations:**
The second derivative and first derivative terms can be approximated using finite difference formulas.

- The second derivative \( \frac{d^2u}{dx^2} \) at point \( x_i \) can be approximated as:
\[
\frac{d^2u}{dx^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{(dx)^2}
\]
- The first derivative \( \frac{du}{dx} \) at point \( x_i \) can be approximated as:
\[
\frac{du}{dx} \approx \frac{u_{i+1} - u_{i-1}}{2dx}
\]

#### 3. **Substitute into the Governing Equation:**
Substitute the finite difference approximations into the governing equation. For an arbitrary point \( x_i \):
\[
\frac{u_{i+1} - 2u_i + u_{i-1}}{(dx)^2} + \left( \frac{2}{x_i} \right) \frac{u_{i+1} - u_{i-1}}{2dx} - p u_i = 0
\]

#### 4. **Simplify the Equation:**
Multiply through by \( (dx)^2 \) to clear the denominators:
\[
u_{i+1} - 2u_i + u_{i-1} + \left( \frac{2 dx}{x_i} \right) (u_{i+1} - u_{i-1}) - p (dx)^2 u_i = 0
\]
This is the finite difference equation at each point \( x_i \).

#### 5. **Apply Boundary Conditions:**
At \( x_0 = 0 \), we have \( u_0 = 0 \), and at \( x_5 = 1 \), we have \( u_5 = 1 \). This allows us to set up a system of equations for the internal grid points \( u_1, u_2, u_3, u_4 \).

#### 6. **Finite Difference Formula at Each Point:**
Now that the finite-difference formula is derived, you can compute the values of \( u_1, u_2, u_3, u_4 \) using a matrix form and solving the system of linear equations.

Do you want me to continue with the calculation for the specific intervals, or explain more details?

### Here are 5 related questions you might be interested in:
1. How do you handle non-uniform grids in finite difference methods?
2. What is the significance of the parameter \( p \) in this equation?
3. Can this method be extended to 2D or 3D heat transfer problems?
4. How does the accuracy of finite difference methods depend on the step size \( dx \)?
5. How can one check the stability of a finite difference scheme?

**Tip:** When using finite difference methods, smaller step sizes improve accuracy but at the cost of more computational effort.

The temperature distribution in a tapered conical cooling fin is described by the following differential equation: d²u/dx² + (2/x)(du/dx) - p*u = 0 with boundary conditions u(0) = 0 and u(1) = 1. Derive the finite-difference formula for this equation with dx = 0.2.

This problem involves solving a boundary value problem using the finite-difference method for a tapered conical cooling fin. The given equation includes a second-order differential equation with boundary conditions and requires discretization with dx = 0.2.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation