Non linear partial differential equation

 

Introduction:

A partial differential equation is called non linear partial differential equation if

1.      Depended variables and all derivatives occurs in more than one degree

2.      The product of depended variables and derivatives occurs

Example:1

According to above example condition 1 is not fulfilled. That’s why this equation is non linear

Example:2

According to above example condition 2 is not fulfilled. That’s why this equation is non linear

 

Explanation:

The majority of this present reality occurrence can be displayed by non linear partial differential equation. In current science non linear waves fact are one of the most  talking  field of science and engineering. For example: solid state physics (SSP), plasma physical science (PPS), liquid machines, strong state physical science, chemical reaction and soon

Identity with Functional Dependence:

Let   G=G (t , y) and also

G(t , y)=U(t , s) and w(t,x) = T(t , x ) are variables

So the second order differential equation becomes

 

Where v=v(t , x) and g=g(z) are function

Now Theorem 1 is.

 

Ø  g′(T) represented the derivative of  g(T) with respect to T.

Ø  ∇v∣2 donated  the squared norm of the gradient of v

Ø   Where ΔT and Δv are the Laplacians of T and v respectively

Ø  Φ is an arbitrary function dependent on its arguments as specified