Literature review of Boundary and Initial Conditions in Differential equations

Boundary Conditions:

It specifies how a system will react at the boundary that is in space coordinate generally at x=0 & at x=L so here 0 to L is boundary and the value of function f(x) is given at these boundaries

Intial Conditions:

It specifies that how the system react at starting that is at time t=0 what is the value of f(x) as f is a function of space and time x,t

Why We Need Them:

Because without boundary and intial conditions arbitrariness in the solutions makes it almost useless (if possible) to write down the general solution.

Common types of boundary conditions used in solving the differential equations

The common types of boundary condition we use are

If there is a function f with independent variable x

(i) Dirichlet boundary condition:

At x=0 f(x)=9 and and at x=10 f(x)=5

(ii) Neumann boundary condition

At x=0 f(x)=5 and f'(x)=3

Different Classes of PDEs.

PDEs can be classified into different categories

(1) Based on order of PDE

The order of PDE is described by the highest derivative in the equation

for E.g.

First Order PDE

`((delu)/(delx))^3+(delu)/(delx)+u=0`

Third order PDE

`(del^3u)/(delx^3)+(delu)/(delx)-u=3`

(2) Based on degree of PDE

Linear : A PDE with first degree in all of its variable

E.g.

`(del^3u)/(delx^3)+(delu)/(delx)-u=3`

Non-Linear

E.g. Third Degree

`((delu)/(delx))^3+(delu)/(delx)+u=0`

Quasi-Linear

All the terms with highest order derivatives of dependent variables occur linearly, that is the coefficients of such terms are functions of only lower order derivatives of the dependent variables. However, terms with lower order derivatives can occur in any manner.

`ux_1(del^2u)/(delx_1^2)+u^2x_1x_2(del^2u)/(delx_1delx_2)+ux_2(del^2u)/(delx_2^2)=x_1((delu)/(delx_1))^2`

(3) Based on variables

Homogenous: If all the terms in the PDE are dependent and its derivative.

For eg. `(del^3u)/(delx^3)+(delu)/(delx)-u=3`

Non-Homogenous : If there is a term with an independent variable

For eg. `(del^3u)/(delx^3)+(delu)/(delx)-u=3x^2-x`

(4) Based on second order PDEs of conics section

We usually come across three-types of second order PDEs in mechanics.

Each type of PDE has certain characteristics that help determine if a particular finite element approach is appropriate to the problem being described by the PDE.

Interestingly, just knowing the type of PDE can give us insight into how smooth the solution is, how fast information propagates, and the effect of initial and boundary conditions.

`a(del^2u)/(delx_1^2)+b(del^2u)/(delx_1delx_2)+c(del^2u)/(delx_2^2)+d(delu)/(delx_1)+e(delu)/(delx_2)+fu=g`

here a,b,c,d,e,f `in x_1, x_2`

If `b^2-4ac<0` it is called elliptic

`b^2-4ac>0` it is called hyperbolic

`b^2-4ac=0` it is called parabolic

Elliptic

The equations of elasticity (without inertial terms) are elliptic PDEs.

In contrast, the solutions of elliptic PDEs are always smooth, even if the initial and boundary conditions are rough (though there may be singularities at sharp corners). In addition, boundary data at any point affect the solution at all points in the domain.

eg. `(del^2u)/(delx_1^2)+(del^2u)/(delx_1delx_2)+(del^2u)/(delx_2^2)=x_1(delu)/(delx_1)`

Hyperbolic

Hyperbolic PDEs describe wave propagation phenomena.

In hyperbolic PDEs, the smoothness of the solution depends on the smoothness of the initial and boundary conditions. For instance, if there is a jump in the data at the start or at the boundaries, then the jump will propagate as a shock in the solution. If, in addition, the PDE is nonlinear, then shocks may develop even though the initial conditions and the boundary conditions are smooth. In a system modelled with a hyperbolic PDE, information travels at a finite speed referred to as the wave speed. Information is not transmitted until the wave arrives.

eg. `(del^2u)/(delx_1^2)+3(del^2u)/(delx_1delx_2)+(del^2u)/(delx_2^2)=x_1(delu)/(delx_1)`

Parabolic

The heat conduction equation is an example of a parabolic PDE.

Parabolic PDEs are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.

eg. `(del^2u)/(delx_1^2)+2(del^2u)/(delx_1delx_2)+(del^2u)/(delx_2^2)=x_1(delu)/(delx_1)`