How Can Greens Function Be Used to Solve the Heat Equation with Delta Functions?

[maggie56]

Hi, i am trying to find greens function for the heat equation with a=1, i.e du/dt - d2u/dx2 = f(x,t) for 0<x<infinty i also have the conditions, u(x,0)=0 and u(0,t)=0
When i have found my greens function i have to allow f(x,t) = delta(x-1) delta(t-1) and obtain a solution using the greens function

i have tried to do it using two different methods, one using Fourier transforms and one using separation of variables, but both times i get unstuck when it comes to doing the integrals for 0 to infinity with delta functions because that is zero? I don't understand how i can work it so that i can simplify by getting rid of the delta functions, can i find the integral of negatie infinity to positive infinity and halve it or anything like that?
Any help would really be appreciated
Thanks

 

[lippyka]

The Greens function for the heat equation with a=1 isG(x,t;x',t') = (1/2√π)*∫0∞ exp(-s²/4)*exp(-s*|x-x'|/2) dsThen in order to solve the equation with f(x,t) = delta(x-1) delta(t-1):u(x,t) = (1/2√π)*∫0∞ exp(-s²/4)*exp(-s*|x-1|/2)*delta(t-1) dsNote that since the delta function vanishes everywhere except at t=1, the integral over t has been replaced by 1. Integrating x from 0 to ∞, we obtain:u(x,t) = (1/2√π)*∫0∞ exp(-s²/4)*(1-exp(-s/2)) ds which can be evaluated using integration by parts. Finally, the solution is given by u(x,t) = (1/2√π)*[exp(-1/4)(1-exp(-|x-1|/2))], where 0<x<∞ and t=1.