1.2 PDE’s

Rather than giving a strict mathematical definition, let us look at an example of a PDE, the heat equation in 1 space dimension

• It is a PDE since partial derivatives are involved.
  To remind you of what that means: {∂u(x,t)\over ∂x} denotes the differentiation of u(x,t) w.r.t. x keeping t fixed,

  {∂({x}^{2}t + x{t}^{2})\over ∂x} = 2xt + {t}^{2}.

  (1.6)

• It is called linear since u and its derivatives appear linearly, i.e., once per term. No functions of u are allowed. Terms like {u}^{2}, \mathop{sin}\nolimits (u), u{∂u\over ∂x}, etc., break this rule, and lead to non-linear equations. These are interesting and important in their own right, but outside the scope of this course.
• Equation (1.5) above is also homogeneous (which just means that every term involves either u or one of its derivatives, there is no term that does not contain u). The equation

  {{∂}^{2}u(x,t)\over ∂{x}^{2}} = {1\over k} {∂u(x,t)\over ∂t} +\mathop{ sin}\nolimits (x)

  (1.7)

  is called inhomogeneous, due to the \mathop{sin}\nolimits (x) term on the right, that is independent of u.

Finally we would like to define the order of a PDE as the power in the highest derivative, even it is a mixed derivative (w.r.t. more than one variable).

Quiz Which of these equations is linear? and which is homogeneous?

1. {{∂}^{2}u\over ∂{x}^{2}} + {x}^{2}{∂u\over ∂y} = {x}^{2} + {y}^{2}.

   (1.12)

2. {y}^{2}{{∂}^{2}u\over ∂{x}^{2}} + u{∂u\over ∂x} + {x}^{2}{{∂}^{2}u\over ∂{y}^{2}} = 0.

   (1.13)

3. { \left ({∂u\over ∂x}\right )}^{2} + {{∂}^{2}u\over ∂{y}^{2}} = 0.

   (1.14)

What is the order of the following equations?

1. {{∂}^{3}u\over ∂{x}^{3}} + {{∂}^{2}u\over ∂{y}^{2}} = 0.

   (1.15)

2. {{∂}^{2}u\over ∂{x}^{2}} − 2 {{∂}^{4}u\over ∂{x}^{3}∂y} + {{∂}^{2}u\over ∂{y}^{2}} = 0.

   (1.16)