Question

use the laplace transform method to solve the boundary value problem

∂u/∂t=3(∂u/∂x), u(x,0)=4e^-2x

Answer 1

Appliing Laplase transworm with respect to t we have

U(x,p)p-u(x,0)=$3\cdot U(x,p)'_x$

or

$\frac{d}{dx}U(x,p)=U(x,p)\cdot \frac{p}{3}-\frac{4}{3}\cdot e^{-2x}$

homogeneous equation has the general solution

$U(x,p)=Ce^\frac{px}{3}$

Let C=C(x).

Then

$C'(x)e^\frac{px}{3}=-\frac{4}{3}\cdot e^{-2x};$

C'(x)=$-\frac{4}{3}\cdot e^{(-2-\frac{p}{3})x}$

$C(x)=\frac{4}{6+p}e^{(-2-\frac{p}{3})x}$

U(x,p)=$\frac{4}{6+p}e^{-2x}$

Using table of Laplase transform we have

$\frac{4}{p+6}\doteqdot 4\cdot e^{-6t}$

Finalyy we have

u(x,t)=$4\cdot e^{-(6t+2x)}$