| 15 | | The solution to this equation needs to simultaneously 1) satisfy this equation at all points within a bounding region and 2) satisfy the boundary conditions on that region. Thus, this equation + solution can be thought of as an instantaneous system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed. Indeed, the numerical methods for solving hyperbolic equations compared to elliptic equations are much different. Apart from their classification as either boundary value or initial value problems, they can be thought of as "time-evolution" or "static" problems respectively, from a computational point of view (see Fortran Numerical Recipes, Press et al, Vol. 2, Chapt. 19 - Partial Differential Equations). The following figure from that book illustrates this concept: |
| | 15 | The solution to this equation needs to simultaneously 1) satisfy this equation at all points within a bounding region and 2) satisfy the boundary conditions on that region. This then lends an "instantaneous" feel to the system, much different than the wave-like solutions of hyperbolic equations which travel with finite speed. |
| | 16 | |
| | 17 | In fact, apart from their classification as either a boundary value problem for the elliptic equation, or initial value problem for the hyperbolic, they can be considered of either a "time-evolution" or "static" nature. This is a more helpful classification in terms of designing numerical methods to solve these different types of equations (see Fortran Numerical Recipes, Press et al, Vol. 2, Chapt. 19 - Partial Differential Equations). The following figure from that book illustrates this concept: |
