What does PDE mean?

Answer 1

In mathematics, a PDE is a Partial Differential Equation.

To partially differentiate an equation, read below:

Suppose you have a function f(x,y) and suppose you want to partially differentiate it w.r.t. x then you consider y as a constant and find d/dx(f(x,y)).

Eg. - Let f(x,y)=xy+x+y then on partially differentiating f(x,y) w.r.t. x -

d/dx(f(x,y))

= d/dx(xy) + d/dx(x) + d/dx(y)

= y(d/dx(x)) + 1 + 0 (as y is constant)

= y +1

Some application(s) of partial differential equations that I know -

1. Find the centre of a conic:

Suppose you have a curve as a function of x and y, say f(x,y).

Then to find its centre -

-> Partially differentiate f(x,y) w.r.t. x. Let the equation obtained be e1.

-> Partially differentiate f(x,y) w.r.t. y. Let the equation obtained be e2.

Solve e1 and e2 to get (x,y) which is the centre of the curve.

2. To find the (conservative) force acting on an object if its Potential energy is given as a function of distance:

-> Let the potential energy function be U(x,y).

-> To find the force acting on object in x-direction, find minus(partial derivative of U(x,y) w.r.t. x).

-> Same method to find force acting on the object in y-direction.

-> Only works for conservative force.

For more information, please see the related link.