Introduction To Partial Differential Equations

Mathematics is thuniverse of knowledge. From starting of universe it is a secret of all objects of this universe. Every ending has a starting. Differential Equations is the part of  universe of Mathematics.
             Differential equations are divided into two parts those are Ordinary Differential Equations and Partial Differential Equations.
For example : dy/dx + xy² = x²
          It is an ordinary differential equation.
                           ∂u/∂t + (∂u/∂x)² = 4
          It is a partial differential equation.
     
        I have already discussed the theorems related to ordinary differential equations. The theorems and equations are all important to the solutions of ordinary differential equations.

 The geometrical structure for ordinary  differential equations is given by the following;

        Let us now discuss about partial differential equation and the theorems and equations related to it .
   Definition : A partial differential equation is a relation between the independent variables , the dependent variable and its partial derivatives .

    For example : x ∂θ/∂x + y ∂θ/∂y + θ =0

Geometrical Interpretation of partial differential equations can be shown using the graph given below.

The slope in the graph defines partial differential equation geometrically.

     This article may useful.

Reference:

1.     Partial differential equations for scientists and engineers by Dover Books On Mathematics.

2.    Introduction to partial differentials by Arihant.