Parabolic Equations

In 1811, the French Academy chose the problem of Heat transfer for its annual prize.The prize was awarded to Joseph Fourier for two contributions:

1. For developing appropriate differential differential equation for Heat Conduction problem.
2. A novel method for solving it.

Fourier's method for solving linear PDE's is based on the technique of separation of variables. Step involved in this method are:

1. Assume that the solution of the homogeneous P.D.E can be expressed in variables separation form. e.g
   $$u(x,t)=X(x)T(t).$$
   It should satisfy the given boundary conditions. It turns out that X and T should be solution of linear O.D.E's that can be derived from the given PDE.

2. Using the superposition principle form, a general solution of the PDE in form of infinite series of product solutions.
3. Compute the coefficients of this series. (Fourier series)

Several deep ideas and technical steps are involved in this method of Separation of variables. Discussing the is beyond the scope of this course. In this course, we apply this method to several relatively simple problems without much theoretical justification.Such solutions obtained without their Mathematical Analysis are called as 'formal solutions'.