Question

8.3.1. Solve the initial value problem for the heat equation with time-dependent sources $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + Q(x, t)$ $u(x, 0) = f(x)$ subject to the following boundary conditions: (a) $u(0, t) = 0$, $\frac{\partial u}{\partial x}(L, t) = 0$ (b) $u(0, t) = 0$, $u(L, t) + 2\frac{\partial u}{\partial x}(L, t) = 0$ *(c) $u(0, t) = A(t)$, $\frac{\partial u}{\partial x}(L, t) = 0$

          8.3.1. Solve the initial value problem for the heat equation with time-dependent sources
$\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + Q(x, t)$
$u(x, 0) = f(x)$
subject to the following boundary conditions:
(a) $u(0, t) = 0$,
$\frac{\partial u}{\partial x}(L, t) = 0$
(b) $u(0, t) = 0$,
$u(L, t) + 2\frac{\partial u}{\partial x}(L, t) = 0$
*(c) $u(0, t) = A(t)$,
$\frac{\partial u}{\partial x}(L, t) = 0$

8.3.1. Solve the initial value problem for the heat equation with time-dependent sources
(∂ u)/(∂ t) = k (∂^2 u)/(∂ x^2) + Q(x, t)
u(x, 0) = f(x)
subject to the following boundary conditions:
(a) u(0, t) = 0,
(∂ u)/(∂ x)(L, t) = 0
(b) u(0, t) = 0,
u(L, t) + 2(∂ u)/(∂ x)(L, t) = 0
*(c) u(0, t) = A(t),
(∂ u)/(∂ x)(L, t) = 0

Added by Jose B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/13/2023

Step 1

Step 1: Write down the heat equation with the given boundary conditions for part (c): ∂u/∂t = k(∂^2u/∂x^2) + Q(x,t) u(x,0) = f(x) u(0,t) = A(t) (∂u/∂x)(L,t) = 0 Show more…

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8.3.1. Solve the initial value problem for the heat equation with time-dependent sources (delu)/(delt)=k( heta ^(2)u)/(delx^(2))+Q(x,t) u(x,0)=f(x) subject to the following boundary conditions: (a) u(0,t)=0, (delu)/(delx)(L,t)=0 (b) u(0,t)=0, u(L,t)+2(delu)/(delx)(L,t)=0 ^(**)(C)u(0,t)=A(t), (delu)/(delx)(L,t)=0 Part C only please!!! 8.3.1. Solve the initial value problem for the heat eguation with time-dependent sources =kg+Q(x,i ux,0)=fx) subject to the following boundary conditions: au0,t=0, n(L,1)=0 (b)u(0,t)=0 u(L,t)+2gn(L,t)=0 *(c) u(0,t)=A(t) gh(L,t)=0