Analytical Solution of the Poisson's Equation for OneDimensional Domains
General solution using the Heat Transfer example
Consider the heat transfer without convection effects along the following bar:
Remember that the conduction phenomena refers to "the transfer of thermal energy from a region of higher temperature to a region of lower temperature through direct molecular communication within a medium or between mediums in direct physical contact without a flow of the material medium".
As boundary conditions, the temperature is fixed at the beginning of the bar, and the heat flow is given at the end of the bar:
Taking a differential length and by establishing the balance of heat flows, it can be written:
 outflow inflow
Using the Fourier law[1]:
Therefore:
If an external heat source is considered, the balance of heat flows becomes:
Rewritting the boundary conditions:
 en
 en
 Generic Solution:
Contents 
The simplest case: homogeneous medium, no sources
Case 1. With Dirichlet boundary conditions
sources  medium  boundary
conditions 







To obtain the constants cte_{1} and cte_{2}:
with the result of:
that is:
Specific example:


 To modify the parameters, edit this Matlab code
Case 2. With Dirichlet and Neumann boundary conditions
sources  medium  boundary
conditions 







that is:
note that:
Specific example:


 To modify the parameters, edit this Matlab code
Heterogeneous medium, no sources
Case 3. With Dirichlet boundary conditions
sources  medium  boundary
conditions 







To obtain the constants cte_{1} and cte_{2} in both intervals:
therefore cte_{1} is the same in both intervals.
 and
with the result of:
note that if k_{1}=k_{2}, the case 3 becomes the case 1.
Specific example:

results  

unknow  gradient  flow 



 To modify the parameters, edit this Matlab code
Case 4. With Dirichlet and Neumann boundary conditions
sources  medium  boundary
conditions 







To obtain the constants cte_{1} and cte_{2} in both intervals:
with the result of:
 note that if k_{1}=k_{2}, the case 4 becomes the case 2.
Specific example:

results  

unknow  gradient  flow 



 To modify the parameters, edit this Matlab code
With a source, homogeneous medium
Case 5. Constant source over the whole domain, Dirichlet boundary conditions
sources  medium  boundary
conditions 







To obtain the constants cte_{1} and cte_{2}:
with the result of:
note that if Q = 0, the case 5 becomes the case 1.
Specific example:
parameters sources medium b.c. geometry results unknow gradient flow  To modify the parameters, edit this Matlab code
Case 6. A constant source over a part of the domain, Dirichlet boundary conditions
sources medium boundary conditions
 constant source
 homogeneous
 Dirichlet
To obtain the constants cte_{1} and cte_{2} in both intervals:
interval 1:
interval 2:
with the result of:
note that if Q_{0} = 0, the case 6 becomes the case 1.
Specific example:parameters sources medium b.c. geometry results unknow gradient flow  To modify the parameters, edit this Matlab code
Case 7. Constant source over the whole domain, Dirichlet and Neumann boundary conditions
sources medium boundary conditions
 constant sources
 homogeneous
 Dirichlet and Neumann
To obtain the constants cte_{1} and cte_{2}:
with the result of:
note that if Q = 0, the case 5 becomes the case 2.
Specific example:parameters sources medium b.c. geometry results unknow gradient flow
 To modify the parameters, edit this Matlab code