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A physical phenomenon can normally be described by a mathematical problem. Consider a simple elliptic mathematical problem [85] of finding the unknown function φ(x) that satisfies the governing differential equation (DE) of the phenomenon
in the one-dimensional domain Ω=(0,L) subject to a set of boundary conditions at the boundary points x=0 and x=L as follows
| \[\begin{align} \varphi(0)=0\quad \text{at} \quad x=0,\nonumber\\ \left(a(x)\frac{d\varphi}{dx}\right)\Bigg\vert _{x=L}=\beta_\text{l} \quad \text{at} \quad x=L, \end{align}\] | (4.2) |
where x is the independent variable, a(x), b(x) are the material or physical properties of the system, f(x) is a given source function, and βl is the boundary load which is given by the problem. For simplicity, the known functions a(x), b(x), and f(x)) will be called a,b, and f. The first boundary condition in equation (4.2), associated with the DE, is commonly called a Dirichlet boundary condition which requires that a solution needs to take a specific value at the boundary of the domain, while the second boundary condition term is the Neumann boundary condition which specifies the value that the derivative of a solution can take at the boundary of the domain.
The mathematical problem described above is stated in its strong form. It consists of the DE governing the system dynamics, the associated boundary conditions, and the initial conditions of the problem.