
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 31st, 2017, 05:34 PM  #1 
Newbie Joined: Aug 2017 From: USA Posts: 1 Thanks: 0  Analytical Solution
I am trying to derive an analytical solution for a heat transfer problem. I have a conduction term and a heat sink term that is proportional to temperature. The conduction term alone would leave me with the Laplace equation, T''=0, When I add the heat sink term and the constants, I believe the equation I am trying to solve is aT''bT=0. I am doing this in 1D only. I know the boundary conditions for T and T' at x=0 and x=1. How do I solve this problem? Do I use a Laplace transform?

August 31st, 2017, 06:18 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,155 Thanks: 1422 
Are a and b nonzero constants that have the same sign?

September 12th, 2017, 04:21 PM  #3  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,824 Thanks: 752  Quote:
If a= 0 s^2= b/a is impossible. The equation in that case is bT= 0 so that T= 0. If a and b are the same sign then b/a is positive so $\displaystyle s= \pm\sqrt{b/a}$ is real and the general solution is $\displaystyle Ae^{\sqrt{b/a}x}+ Be^{\sqrt{b/a}x}$. When x= 0, that is $\displaystyle A+ B$ and when x= 1, that is $\displaystyle Ae^{b/a}+ Be^{b/a}$ set those equal to the given values and you have two linear equations to solve for A and B. If a and b are of opposite sign then b/a is negative so $\displaystyle s= \pm\sqrt{b/a}$ is complex. If we write the those complex numbers as $\displaystyle \alpha\pm \beta$ then the general solution to the differential equation is $\displaystyle e^{\alpha x}(A \cos(\beta x)+ B\sin(\beta x)$. When x= 0, that is $\displaystyle A \cos(\beta)$ and when x= 1 $\displaystyle e^{\alpha}(A \cos(\beta)+ B \sin(\beta)$. Again, set those equal to the given values and solve for A and B. Last edited by skipjack; September 12th, 2017 at 04:29 PM.  

Tags 
analytical, solution 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Poisson Equation Analytical Solution  mojojojo17  Differential Equations  0  February 16th, 2016 02:09 AM 
Is there an analytical solution for a regression of this form?  animalfarm  Advanced Statistics  2  May 8th, 2014 02:15 AM 
analytical solution to BVP with fu'n and 2nd dr'tve of fu'n  munkifisht  Calculus  2  June 13th, 2012 01:02 PM 
is there an analytical solution to (x+a)^g(xa)^g=p*a*x  sciacallojo2  Economics  7  August 9th, 2011 12:37 PM 
poisson pde analytical solution  jangolobow  Calculus  1  April 2nd, 2010 09:51 AM 