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A solution of a PDE of type (1.3.1)–(1.3.3) is a function that identically satisfies the corresponding PDE, and the associated initial and boundary conditions, in some region of the variables , or (and ). Note that a solution of an equation of order has to be times differentiable. A function in that satisfies a PDE of order is called a classical (or strong) solution of the PDE. We sometimes also have to deal with solutions that are not classical. Such solutions are called weak solutions. In this note, in the variational formulation for FEMs, we actually deal with weak solutions. For a more thorough discussion on weak solutions, see Chapter 2 or any textbook in distribution theory.
Definition 1.2Hadamard'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), is a linear function of the unknown function and its derivatives.
Thus, for example, the equation is a linear equation, while is a nonlinear equation. The nonlinear equations are often further classified into subclasses according to the type of their nonlinearity. Generally, the nonlinearity is more pronounced when it appears in higher‐order derivatives. For example, the following equations are both nonlinear
(1.3.4)
(1.3.5)
Here denotes the norm of the gradient of . While (1.3.5) is nonlinear, it is still linear as a function of the highest‐order derivative (here and ). Such a nonlinearity is called quasilinear. On the other hand, in (1.3.4), the nonlinearity is only in the unknown solution . Such equations are called semilinear.
1.4 Differential Operators, Superposition
Differential and integral operators are examples of mappings between function classes as where . We denote by the operation of a mapping (operator) on a function .
Definition 1.4
An operator that satisfies
(1.4.1)
where and are functions, is called a linear operator. We may generalize (1.4.1) as
(1.4.2)
i.e. maps any linear combination of 's to corresponding linear combination of 's.
For instance the integral operator defined on the space of continuous functions on defines a linear operator from into , which satisfies both (1.4.1) and (1.4.2).
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