Boundary conditons and intial conditions

Boundary conditions are those condition that are given at extremes of domain of given PDE

Initial conditions are those that are given at lower limits or values of domain of given PDE

 Types of boundary conditions in PDEs:

Dirichilet boundary condition:

In this boundary condition,the value of the function is specified at a particular function.

Eg : Consider a bar being fixed at one end. Then value of temperature at fixed point is specified as 0 degrees celsius. 

Neuman boundary condition:

In this condition, the boundary condition is specified as normal derivative of the function at the surface.

Considering above eg, temperature is specified as dt/dn= 0 degree celsius.

Mixed boundary condition:

This condition is combination of neuman and dirichilet boundary condition. Here ,value of the unknown function at the surface (initial condition) and the sum of value of unknown function at surface and its normal derivative are specified as boundary conditions.

Considering above example, T = 0 degree celsius and  T + dt/dn = 5 degree celsius

Robin boundary condition:

Here sum of value of unknown function at surface (initial condition) and its normal derivative is pecified as boundary condition.

Eg : Using above example, T + dt/dn = 5 degree celsius

Cauchy boundary condition:

In this condition, value of function and its normal derivative are given as boundary conditions. That is the normal derivative of unknown function is specified along with initial condition.

Usisng above example, T = 0 degree celsius and dt/dn = 5 degree celsius.