# Step by step analytical solution of temperature distribution

1. Mar 20, 2012

### Fr34k

1. The problem statement, all variables and given/known data
Lets say we have a 2D rectangular plate with a point heat source and some boundary conditions. I would like to solve and understand step by step the solution to this second order differential equation. Lets say dimensions of rectangular are a and b. 2 opposite sides are at a fixed temperature T1 and the other 2 are insulated. The initial temperature of the "body" is lets say T0. Our source is a δ source somewhere on the rectangular (x,y) and starts at t0. We ignore the heat dissipation due to convection and radiation. How would the temperature distribution look like T(x,y,t)? We also know the k-thermal conductivity, ρ density and c thermal diffusivity.

2. Relevant equations
$\nabla^2T-\frac{1}{k}\frac{\delta T}{\delta t}+\frac{q(x,y,t)}{c}=0$

And please don't just tell/write the solution. I would like to understand the method of solving these type of problems. I have been reading lots of books on differential equations with simpler problems or similar problems but I can not put 1 and 1 together to solve this exact situation.

2. Mar 20, 2012

### sunjin09

The general solution I think is obtained by taking the Fourier series/transform of all variables x,y,z,t, converting into algebraic equation, solving for it and taking the inverse FS/T. You'll probably need to know the FS/T of the delta function source. The inverse FS/T is a multi-dimensional singular summation/integral where if the geometry is simple enough, you might be able to evaluate in closed form, using certain techniques. The solution obtained this way is nothing but the system's Green's function

3. Apr 11, 2012

### Fr34k

Done it some time ago but I'll post what I did if someone else will have same problems.
The essential thing is to break the partial equation down into sum of components each solving a part of the whole.
one for boundary conditions + one laplace for stationary solution + one time dependant solution
The only trick here is in the stationary solution, where you have to center the system to get nicer solutions.
The rest are just known solutions to partial differential equations.