1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1+x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).
1. Our goal is to find a real-valued function u(x, t) solving 1 (1) ut +2ux = 0, 1+x² (This is called the transport equation.) We will do this in stages. (a) Suppose v = v(x, t) satisfies (2) Ut = 0, Compute v(x, t) for all x, t. u(x, 0) v(x,0) = - 1 1+x² (b) Suppose u satisfies the equations in (1) and define v(x, t) := u(x + 2t, t). Show that v satisfies the equations in (2). (c) Combine (a) and (b) to deduce an explicit function u satisfying (1).
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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![### Solving the Transport Equation
Our goal is to find a real-valued function \( u(x, t) \) solving:
1. **Equations:**
\[
u_t + 2u_x = 0, \quad u(x, 0) = \frac{1}{1 + x^2}
\]
(This is called the **transport equation**.) We will do this in stages.
2. **Stage (a):**
- Suppose \( v = v(x, t) \) satisfies:
\[
v_t = 0, \quad v(x, 0) = \frac{1}{1 + x^2}
\]
- Compute \( v(x, t) \) for all \( x, t \).
3. **Stage (b):**
- Suppose \( u \) satisfies the equations in (1) and define \( v(x, t) := u(x + 2t, t) \). Show that \( v \) satisfies the equations in (2).
4. **Stage (c):**
- Combine (a) and (b) to deduce an explicit function \( u \) satisfying (1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac048e26-edc8-46ba-8829-c16c07a7a6b4%2Fa3a004ba-1349-45d2-be12-aba835868ef2%2Fr6iosh7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving the Transport Equation
Our goal is to find a real-valued function \( u(x, t) \) solving:
1. **Equations:**
\[
u_t + 2u_x = 0, \quad u(x, 0) = \frac{1}{1 + x^2}
\]
(This is called the **transport equation**.) We will do this in stages.
2. **Stage (a):**
- Suppose \( v = v(x, t) \) satisfies:
\[
v_t = 0, \quad v(x, 0) = \frac{1}{1 + x^2}
\]
- Compute \( v(x, t) \) for all \( x, t \).
3. **Stage (b):**
- Suppose \( u \) satisfies the equations in (1) and define \( v(x, t) := u(x + 2t, t) \). Show that \( v \) satisfies the equations in (2).
4. **Stage (c):**
- Combine (a) and (b) to deduce an explicit function \( u \) satisfying (1).
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