**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:\[ u_x(t, 0) = u_x(t, \ell) = 0 \quad \text{and} \quad u(0, x) = 1 + 3 \cos\left(\frac{6\pi x}{\ell}\right) + 2 \cos\left(\frac{7\pi x}{\ell}\right). \]---

Answered: Image /qna-images/answer/0219f099-b9c2-41a9-a6c1-9b207d728314.jpg

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**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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