Second Order PDE

Just like ODEs, Partial Differential Equation can have second order terms.

Definition §

Let $f(x,y)$ be a bivariate function. The second order partial derivatives of f with respect to x and y are defined by:

$$f_{xx}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial x^2}$$

$$f_{yy}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial y})=\frac{{\partial }^{2}f}{\partial y^2}$$

$$f_{xy}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial y\partial x}$$

$$f_{yx}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial y})=\frac{{\partial }^{2}f}{\partial x\partial y}$$

Theorem - Continuity Of Partial Derivatives §

If the second order partial derivatives of $f$ exist and are continuous then:

$$f_{xy}=f_{yx}$$