# Parabolic And Elliptic Equation With Constant Coefficients

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**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;