Take k to be a positive real number. Suppose that u is the solution to the diffusion equation Ut = kupr on the closed interval [0, l]. (4.1) Solve the differential equation given that u(t, 0) = u(t, l) = 0 and u(0, x) = sin (2) + 3 sin (). (4.2) Solve the differential equation given that 7rx Uz(t,0) = u#(t, l) = 0 and u(0, x) = 1 + 3 cos (T) + 2 cos (T).
Take k to be a positive real number. Suppose that u is the solution to the diffusion equation Ut = kupr on the closed interval [0, l]. (4.1) Solve the differential equation given that u(t, 0) = u(t, l) = 0 and u(0, x) = sin (2) + 3 sin (). (4.2) Solve the differential equation given that 7rx Uz(t,0) = u#(t, l) = 0 and u(0, x) = 1 + 3 cos (T) + 2 cos (T).
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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![**Diffusion Equation Problem Set**
Consider a positive real number \( k \). Assume that \( u \) is the solution to the diffusion equation given by:
\[ u_t = ku_{xx} \]
on the closed interval \([0, \ell]\).
**Problem 4.1**
Solve the differential equation given that:
\[ u(t, 0) = u(t, \ell) = 0 \quad \text{and} \quad u(0, x) = \sin\left(\frac{2\pi x}{\ell}\right) + 3 \sin\left(\frac{5\pi x}{\ell}\right). \]
---
**Problem 4.2**
Solve the differential equation given that:
\[ u_x(t, 0) = u_x(t, \ell) = 0 \quad \text{and} \quad u(0, x) = 1 + 3 \cos\left(\frac{6\pi x}{\ell}\right) + 2 \cos\left(\frac{7\pi x}{\ell}\right). \]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e487533-50a5-4edd-80c5-edc489931218%2F0219f099-b9c2-41a9-a6c1-9b207d728314%2F9pvmtli.png&w=3840&q=75)
Transcribed Image Text:**Diffusion Equation Problem Set**
Consider a positive real number \( k \). Assume that \( u \) is the solution to the diffusion equation given by:
\[ u_t = ku_{xx} \]
on the closed interval \([0, \ell]\).
**Problem 4.1**
Solve the differential equation given that:
\[ u(t, 0) = u(t, \ell) = 0 \quad \text{and} \quad u(0, x) = \sin\left(\frac{2\pi x}{\ell}\right) + 3 \sin\left(\frac{5\pi x}{\ell}\right). \]
---
**Problem 4.2**
Solve the differential equation given that:
\[ u_x(t, 0) = u_x(t, \ell) = 0 \quad \text{and} \quad u(0, x) = 1 + 3 \cos\left(\frac{6\pi x}{\ell}\right) + 2 \cos\left(\frac{7\pi x}{\ell}\right). \]
---
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