5. Solve the PDE method. ду Эх ду = 0, y(x,0) = e−², y(0,t) = e-21 using Laplace transform Ət
5. Solve the PDE method. ду Эх ду = 0, y(x,0) = e−², y(0,t) = e-21 using Laplace transform Ət
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
**5. Solve the PDE \( \frac{\partial y}{\partial x} - \frac{\partial y}{\partial t} = 0 \) with initial conditions \( y(x, 0) = e^{-2x} \) and \( y(0, t) = e^{-2t} \) using the Laplace transform method.**
**Solution Outline:**
To solve the partial differential equation (PDE) given by
\[ \frac{\partial y}{\partial x} - \frac{\partial y}{\partial t} = 0, \]
subject to the initial conditions:
\[ y(x, 0) = e^{-2x}, \]
\[ y(0, t) = e^{-2t}, \]
we can employ the Laplace transform method. Here is the step-by-step process:
1. **Apply the Laplace Transform**:
- Take the Laplace transform of both sides with respect to \( t \).
2. **Solve the Transformed Equation**:
- Use the properties of the Laplace transform to simplify and solve the transformed equation.
3. **Apply Inverse Laplace Transform**:
- Calculate the inverse Laplace transform to obtain the solution in the time domain.
4. **Verify the Solution**:
- Check that the obtained solution satisfies the original PDE and the given initial conditions.
Detailed steps and calculations will follow to solve this problem entirely using the steps outlined above.
---
Note: The Laplace transform method is a powerful technique for solving linear PDEs, especially when initial or boundary conditions are given. The transform converts the PDE into an ordinary differential equation (ODE), which is typically easier to solve. After solving the ODE, the inverse Laplace transform is applied to obtain the solution to the original PDE.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a780900-9b69-4fe9-9f43-185066935266%2F770b938c-3950-4c25-a49d-51cd54222040%2Fg6ra5ts_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**5. Solve the PDE \( \frac{\partial y}{\partial x} - \frac{\partial y}{\partial t} = 0 \) with initial conditions \( y(x, 0) = e^{-2x} \) and \( y(0, t) = e^{-2t} \) using the Laplace transform method.**
**Solution Outline:**
To solve the partial differential equation (PDE) given by
\[ \frac{\partial y}{\partial x} - \frac{\partial y}{\partial t} = 0, \]
subject to the initial conditions:
\[ y(x, 0) = e^{-2x}, \]
\[ y(0, t) = e^{-2t}, \]
we can employ the Laplace transform method. Here is the step-by-step process:
1. **Apply the Laplace Transform**:
- Take the Laplace transform of both sides with respect to \( t \).
2. **Solve the Transformed Equation**:
- Use the properties of the Laplace transform to simplify and solve the transformed equation.
3. **Apply Inverse Laplace Transform**:
- Calculate the inverse Laplace transform to obtain the solution in the time domain.
4. **Verify the Solution**:
- Check that the obtained solution satisfies the original PDE and the given initial conditions.
Detailed steps and calculations will follow to solve this problem entirely using the steps outlined above.
---
Note: The Laplace transform method is a powerful technique for solving linear PDEs, especially when initial or boundary conditions are given. The transform converts the PDE into an ordinary differential equation (ODE), which is typically easier to solve. After solving the ODE, the inverse Laplace transform is applied to obtain the solution to the original PDE.
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