How would i do this question for PDE, I think it just involves knowing general soln. but I need to know how to do any these without doing integrals to practice for midterm. Thanks ### Problem Statement**b.** Given the initial condition \( u(x,0) = \sin(x) + 0.5 \sin(3x) \), \( x \in [0, 2\pi] \), for a rod with ends fixed at zero temperature, write down the solution \( u(x,t) \) of the heat equation \( u_t = 3u_{xx} \) without doing any integration.### ExplanationThis problem involves solving a heat equation with given initial conditions. The function \( u(x, t) \) represents the temperature distribution along a rod at position \( x \) and time \( t \).- **Initial Condition:** The initial temperature distribution along the rod is defined by \( u(x, 0) = \sin(x) + 0.5 \sin(3x) \).- **Domain:** \( x \) ranges from 0 to \( 2\pi \).- **Boundary Conditions:** Both ends of the rod are kept at zero temperature.- **Heat Equation:** The evolution of temperature is governed by the partial differential equation \( u_t = 3u_{xx} \). Here, \( u_t \) represents the time derivative, and \( u_{xx} \) represents the second spatial derivative.### ApproachFor such problems, the solution can often be written in terms of eigenfunctions that satisfy the boundary conditions. Since the initial condition is given as a linear combination of sine functions, the solution will typically take the form:\[ u(x, t) = a_1 e^{-k_1 t} \sin(x) + a_2 e^{-k_2 t} \sin(3x) \]**where:**- \( k_1 \) and \( k_2 \) are constants determined by the heat equation.- \( a_1 \) and \( a_2 \) are coefficients derived from the initial condition.In this case, since the equation is \( u_t = 3u_{xx} \), each term of the form \( \sin(nx) \) evolves over time as \( e^{-3n^2t} \). Thus, the solution can be directly written as:\[ u(x, t) = e^{-3t} \sin(x) + 0.5 e^{-27t} \sin(3x) \]This solution satisfies both the initial and boundary conditions

Name: How would i do this question for PDE, I think it just involves knowing
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Description: How would i do this question for PDE, I think it just involves knowing general soln. but I need to know how to do any these without doing integrals to practice for midterm. Thanks ### Problem Statement**b.** Given the initial condition \( u(x,0) = \s

The heat equation is ut=3uxx Consider the solution as ux, t=XxTt where X (0)=X2π=0. Substitute u in…

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Given the initial condition u(x,0) = sin(x)+0.5sin(3x), xe[0,27], for a rod with ends fixed at zero temperature, write down the solution u(x,t) of the heat equation u, = 3u without doing any integration. xx

Given the initial condition u(x,0) = sin(x)+0.5sin(3x), xe[0,27], for a rod with ends fixed at zero temperature, write down the solution u(x,t) of the heat equation u, = 3u without doing any integration. xx

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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