Parabolic And Elliptic Equation With Constant Coefficients

Parabolic and Elliptic Equation with Constant Coefficients:

Parabolic:

The only solutions of main equations is;

Again η is choosen judiciously but in such a way that the Jacobian of the transformation is not zero. Can A be zero in this case? In the parabolic case A = 0 implies B = 0. Therefore the original equations is;

Which is canonical forms

Elliptic:

Now we have complex conjucate functions ξ, η

The canonical form is similar to the above equations.

Examples:

The charecristics equation is

and the transformation is

The solution in terms of x,t in the above equation we get;