Boundary and Initial Conditions of PDE and ODE

Objective : To know the importance of the initial and boundary condition in the pde and ode

Classification of PDE's

How to implement Boundary conditions in cfd.

Boundary and Initial conditions provide the unique solution for the considered physics.

Initial value condition : Represent a constraint to the DE at a particular point (at any time).

Differential Initial value is an equation which specifies how the system evolves with time for the given

initial condition of the problem.

ex:

`frac{dy}{dx} = cos(x)`

y(0) = -1 (Initial Condition of the Dependent variable)

Solving , => `y = sinx + c` (generalised solution for the given ODE)

Using the initial condition (x = 0 , y = -1) , => c = -1 final solution is `y = sinx -1`

Note :

1. Initial values need not to be a values of y only , may be the derivative of the dependent variable.

2. Initial value problem does not always have a unique solution , possible for may solutions (or) No solution.

Boundary condition is a prescription os some combinations of values of the unknown solution and its

derivatives at more than one point.

Boundary Value Problem , considering the additional constraint Boundary condition. Solution satisfies

Boundary values . Boundary condition specifies a relationship between the values of the solution at

two or more locations in the considered intervals.

How many Boundary conditions need to specify for the given pde ?

Depends on the pde , Generally Number of Boundary conditions are given as the order of the derivative of the dependent variable in that direction.

`frac{del^2T}{delx^2} + frac{del^2T}{dely^2} = gamma*frac{delT}{delt}`

Along x-direction => order = 2 ; so 2 BC required.

Along y-direction => order = 2 ; so 2 BC required.

Since transient term present in the pde ; Order = 1 => 1 IC required.

So , To get an Unique solution , we required 4 BC +1 BC.

How to implement the Boundary conditions ?

Boundary condition are implemented based on the model and type of flow physics.

Types of BC :

1. Dirichlet : Prescribed value .

2. Neumann : In the form of derivative (represent the flux)

3. Cauchy. : Dirichlet + Neumann simultaneously.

4. Robin. : Derivative of dependent variable as a function of the dependent variable.

Classification of PDE's :

There are 3 Types of PDE :

1. Elliptic.

2. Parabolic.

3. Hyperbolic.

Classification of pde is very important before considering the Numerical scheme. whether it is elliptical , parabolic , hyperbolic (Number of characteristics lines).

Generally fluid mechanics governing equations are second order partial differential equations.

General pde :

`A*frac{del^2phi}{delx^2} + B*frac{del^2phi}{delxdely} +c*frac{del^2phi}{dely^2}+d*frac{delphi}{delx}+e*frac{delphi}{dely}+F = 0`

Whether the characteristic lines exist or not , depends on the higher order terms

`frac{dely}{delx} = frac{-b +-(b^2-4*a*c)^(1/2)}{2*a}`

if ;

`(b^2-4*a*c) < 0` => No characteristic , means No discontinuity in the solution , if exists gets nullified. (elliptic)

There are Boundary Value problems

ex ;

Laplace Equation.

`(b^2-4*a*c) = 0` => 1 real characteristic , time Marching problems

`(b^2-4*a*c) > 0` => 2 real characteristics

The Numerical scheme should be selected in a way that can accept the discontuinities correclty in the domain where it exists.