Boundary and Initial Condition Of Differential Equations

Differential Equation: It is an equation with a function and one or more of its derivative.

Mainly, there are two types of differential equations, which are-

1. Ordinary differential equation

2. Partial differential equation

Ordinary differential equation: A differential equation that involves an unknown function of one variable and its derivative.

for example: `5dy/dx +3y= 6`

Partial differential equation: It is a Differential equation that contains unknown multi-variable function and their partial derivatives.

For example: `2(delu)/(delx) -3(delu)/(dely)+1= 0`

Initial value condition: An initial condition is an extra bit of information about a DE that tells you the value of the function at a particular point. DE(s) with initial conditions are commonly called Intial Value Problems.

For example:

`dy/dx=cos x`

`y(0)=-1`

On solving this DE we will get `y=sin x +c`.

After getting the above general equation and using the initial condition which is taking x=0 and y=-1, we will get c= -1.

The final particular solution is `y= sin x -1`.

Important points for initial value problem:

1. the initial values doesn't have only the values of y, higher order equation might have initial values for y as well as its derivatives.

2. an initial value problem doesn't always have a unique solution. it's possible that there may be multiple solutions or even no solutions.

Boundary value condition in a differential eqn is a set of additional constraints called boundary condition. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

It is opposed to initial value problems in which only the conditions on one extreme of the interval are known whereas in BVP the conditions on both extremes of the interval are known.

WHY DO WE NEED INITIAL AND BOUNDARY CONDITIONS:

Boundary value problems are extremely important as they model a vast amount of phenomena and applications, from solid mechanics to heat transfer, from fluid mechanics to acoustic diffusion. They arise naturally in every problem based on a differential equation to be solved in space, while initial value problems usually refer to problems to be solved in time.

Common types of boundary conditions used in solving the differential equations:

Both ordinary and partial DE need boundary conditions to be solved. Different types of BE can be imposed on the boundary of the domain. the choice of the boundary condition is fundamental for the resolution of the computational problem. There are mainly five types of boundary conditions:

1. Dirichlet BC: this condition specifies the value that the unknown function needs to take on along the boundary of the domain.

for eg: this condition mainly used in the Laplace equation.

2. Neumann BC: this condition when imposed on ODE or PDE, it specifies the values that the derivative of a solution is going to take on the boundary of the domain

3. Robin BC: this BC consists of a linear combination of the values of the field and its derivative of the boundary. this condition is also known as impedance condition.

4. Mixed BC: it consists of applying different types of BC in different parts of the domain. it is important that BC must be applied on the whole boundary.

5. Cauchy BC: it is the condition on both the unknown field and its derivative. it differs from Robin condition because it implies the imposition of two constraints while robin condition implies only one constraint on the linear combination of unknown function and its derivative.

Classification of Partial differential equation:

1. First order PDE

when we talk about the first order PDE, then the equation has only the frist derivative of the unknown function having ' m' variable(suppose), It can be expressed as follows:-

`F(x1, ..., xm, u, (delu)/(delx1), ..., (delu)/(delxm))=0`

2. Linear and Non-Linear PDE:

If the dependent variable and all its partial derivatives occur linearly in any PDE, then such an equation is called linear PDE.

Otherwise a non-linear PDE.

For example:

`(del^2u)/(delx^2) + (del^2u)/(dely^2)=0`

Here the power of each derivative is 1, so this is linear PDE

3. Quasi-Linear PDE:

A PDE is said to be Quasi-linear if all the terms with the highest order derivative of dependent variables occur linearly, that is the coefficient of those terms are functions of only lower order derivatives of the dependent variables. However, terms with lower-order derivatives can occur in any manner.

for example:

`ux(del^2u)/(delx^2) + u^2xy(del^2u)/(delxdely) + uy(del^2u)/(dely^2) + ((delu)/(delx))^2 + ((delu)/(dely))^2 +u^3 =0`

4. Homogeneous PDE:

If all the terms of a PDE contain the dependent variable or its partial derivatives the such a PDE is called non-homogeneous or homogeneous PDE(s) otherwise.

eg:-

`(del^2u)/(delx^2) + (del^2u)/(dely^2) =0`

The above equation is a Homogeneous equation