Question

Find the bounded solution for the following PDE using Laplace transform.

Answer 1

@mukushla

Answer 2

taking laplace wrt t ?

Answer 3

yes

Answer 4

after applying laplace we have \[sU-u(x,0)=U_{xx}\]\[sU-0=U_{xx}\]\[U_{xx}-sU=0\]solve this for \(U\)

Answer 5

i did this and got
\[\Large U(x,s)=c_{1}(s)e^{\sqrt{s}x}+c_{2}(s)e^{-\sqrt{s}x}\]

Answer 6

how to proceed then .

Answer 7

sorry i was out

Answer 8

solution is bounded so \(c_1=0\)

i'll be back in 5 min

Answer 9

oh what did i say ... thats wrong...

till here we have\[ U(x,s)=c_{1}e^{\sqrt{s}x}+c_{2}e^{-\sqrt{s}x}\]

Answer 10

i think you are right c1(s) should be zero.

Answer 11

lets find an acceptable reasoning for this

we know that\[\lim_{t \rightarrow 0} u(x,t)=\lim_{s \rightarrow \infty} sU(x,s)=0\]

Answer 12

so yes...\(c_1=0\)

Answer 13

so\[U(x,s)=ce^{-x\sqrt{s}}\]from laplace table\[u(x,t)=c\frac{x}{2\sqrt{\pi t^3}}e^{-\frac{x^2}{t}}\]

Answer 14

but i think this is not right !!!

Answer 15

i am thinking the same :P