Boundary and Initial Conditions

Partial differential equations are equations which involve partial derivatives of the quantities depending on space and time. The different classes of partial differential equations, based on the value of the discriminant (B² – AC) are:

1. Elliptic – (B² – AC < 0)
2. Parabolic – (B² – AC = 0)
3. Hyperbolic – (B² – AC > 0)

where A, B and C are the coefficients in the partial differential equation :

A u_xx + 2B u_xy + C u_yy + lower order terms = 0

BOUNDARY CONDITION – The boundary conditions w.r.t. the Partial Differential Equations are the set of limitations that specify the behavior of the equation at the spatial boundary of the domain.

INITIAL CONDITION – The initial conditions w.r.t. the Partial Differential Equations are the value of the equation at the initial time on the domain.

The broad classification of the types of boundary conditions are:-

1. Dirichlet boundary condition – where the value of the dependent variable is specified at the boundary
2. Neumann boundary condition – where the value of the normal derivative of the dependent variable is specified on the boundary
3. Cauchy boundary condition – where both the value of the dependent variable and its normal derivative are specified on the boundary
4. Robin boundary condition – where the value of the linear combination of the dependent variable and its normal derivative is specified on the boundary

Each class of the partial differential requires a different type of boundary condition in order to have the stable solution:

1. Elliptic partial differential equation – This type of equation require either Dirichlet or Neumann boundary conditions on a closed boundary surrounding the domain of interest. Other boundary conditions are either insufficient to determine a unique solution, overly restrictive for a solution to exist or lead to instabilities.
2. Parabolic partial differential equation – This type of equation require Cauchy boundary conditions on open surface. Other boundary conditions are either insufficient to determine a unique solution or overly restrictive for a solution to exist.
3. Hyperbolic partial differential equation – This type of equation require either Dirichlet or Neumann boundary conditions on open surface. Other boundary conditions are overly restrictive for a solution to exist.