An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh
A function in
Definition 1.2 Hadamard's criteria; compare with the three criteria in theory
A problem consisting of a PDE associated with boundary and/or initial conditions is called well‐posed if it fulfills the following three criteria:
1 Existence The problem has a solution.
2 Uniqueness There is no more than one solution.
3 Stability A small change in the equation or in the side (initial and/or boundary) conditions gives rise to a small change in the solution.
If one or more of the conditions abovementioned does not hold, then we say that the problem is ill‐posed. The fundamental theoretical question of PDEs is whether the problem consisting of the equation and its associated side conditions is well‐posed. However, in certain engineering applications, we might encounter problems that are ill‐posed. In practice, such problems are unsolvable. Therefore, when we face an ill‐posed problem, the first step should be to modify it appropriately in order to render it well‐posed.
Definition 1.3
An equation is called linear if in (1.3.1),
Thus, for example, the equation
(1.3.4)
(1.3.5)
Here
1.4 Differential Operators, Superposition
Differential and integral operators are examples of mappings between function classes as
Definition 1.4
An operator
(1.4.1)
where
(1.4.2)
i.e.
For instance the integral operator
A linear partial differential operator
(1.4.3)
where
The term linear in the phrase linear partial differential operator refers to the following fundamental property: if