Literature review of Boundary and Initial Conditions
Boundary and initial conditions in Differential equations:
The solutions for the ordinary differential equations are usually not unique. The Partial differential equations (PDE) are specified with a set of initial and boundary conditions. The boundary condition is the representation of the function that exhibits the behaviour of the boundary area of definition. The boundary conditions helps us to estimate the actual extend or boundary area of the particular PDE. The initial condition is as like as boundary condition but it explains the time and direction of the PDE. Let us consider the differential equation which represent the characteristic of a particle under the influence of a constant force.
`(d^2x)/dt^2 = a` which leads to
`dx/dt = at + v0` and
`x=1/2(at^2)+v0t+x0`
from the above equations, the v0 is obtained as a result of the partial difference of x with respect to t at t=0 and the x0 is obtained in the condition of x at t=0. These are linear initial conditions. There is an important difference in the arbitrariness of integration constants in ordinary differential equations and partial differential equations. The solution of ordinary differential equation is constant but the solution of partial differential equation contain arbitrary functions.
Consider another example, the heat transfer in a solid material. The heat transfer actually takes place based on a differential equation. The initial condition explains the starting parameter of the heat transfer in that material. The parameter that exhibits the nature of the heat when it enters the solid material is known as initial condition. The initial condition helps us to examine the nature of the heat transfer occurs in the material. The boundary condition is one which explains the rate at which the heat transfer occurs. It also tells us the distance parameter of the heat transfer. This initial and boundary conditions helps us to solve the PDEs in an effective way.
Types of boundary conditions used in solving the differential equations:
- Dirichlet boundary condition
- Neumann boundary condition
- Robin boundary condition
- Mixed boundary condition
- Cauchy boundary condition
Different classes of PDEs:
- Linear and non linear PDE:
The linear PDE is one which has first degree in all its variables and derivatives.
- Homogeneous PDE:
Let us consider an operator with components of x1,x2,x3,t. When the sum of partial derivative of all the components is equal to 0. Then the PDE is known as homogeneous PDE.
- Elliptical, Hyperbolic and Parabolic PDE:
The elliptical PDE are the PDEs that represents the equation of elasticity. The elliptical PDEs are always smooth though the intial and boundary conditions are rough.
The hyperbolic PDE represents the wave propagation phenomena. The hyperbolic PDE's smoothness is depended upon the smoothness of initial and boundary conditions.
The parabolic PDEs are time dependent. The heat conduction equation is a parabolic PDE.
These three PDEs can be identified by examine the various conditions of `b^2-4ac` value. If `b^2-4ac<0`, it will be an elliptical PDE. If `b^2-4ac >0`, it will be an hyperbolic PDE. If `b^2-4ac=0`, it will be an parabolic PDE.
Reference:
https://en.wikiversity.org/wiki/Partial_differential_equations