Partial Differential Equations

A partial differential Equation is characterised by at least $2$ independent variables and the occurance of their partial derivatives.
Important linear partial differential equations of the second order are:

1. One-dimensional wave equation
   
   $${\frac{\partial^2 u}{\partial t^2}\hspace{0.3cm}=\hspace{0.3cm}c^2\frac{\partial^2 u}{\partial x^2}}$$ $c^2$ is a constant
   
   $u_{tt}\hspace{0.6cm}=\hspace{0.3cm}c^2 u_{xx}$

2. One-dimensional heat equation / Standard Diffusion
   
   $${\frac{\partial u}{\partial t}\hspace{0.3cm}=\hspace{0.3cm}c^2\frac{\partial^2 u}{\partial x^2}}$$ $c^2$ is the thermal diffusivity ( constant)
   
   $${\frac{\partial u}{\partial t}\hspace{0.3cm}=\hspace{0.3cm}c^2\nabla^2u}$$ for the definition of $\nabla^2u$ see (16) on page
   
   $u_t\hspace{0.5cm}=\hspace{0.3cm}c^2 u_{xx}$

3. Two-dimensional Laplace equation
   
   $${\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\hspace{0.3cm}=\hspace{0.3cm}0}$$

4. Two-dimensional Poisson equation
   
   $${\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\hspace{0.3cm}=\hspace{0.3cm}f(x,y)}$$

5. Two-dimensional wave equation
   
   $${\frac{\partial^2 u}{\partial t^2}\hspace{0.3cm}=\hspace{0.3cm}c^2\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)}$$

6. Three-dimensional Laplace equation
   
   $${\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}\hspace{0.3cm}=\hspace{0.3cm}0}$$