Classification of Second-Order Equations

Second-order eqnations are classified into different types, depending on the coefhcients of the second-order derivatives. The anxiliary conditions to determine a unique solution depend on the type of eqnation. Anxiliary conditions specified incorrectly with the type of equation lead to an ill-posed problem that may not reflect physical reality. 

The general second-order partial differential equation in -variables is written as 

For hyperbolic and parabolic equations, the conditions on (Xi,X2.x ) are prescribed on the bonndary of the region, and the conditions on t must be prescribed at f = 0. These equations are often referred to as evolution equations. These equations describe the time evolution of a process from a given initial configuration. 

As a consequence of the evolutionary nature of the hyperbolic and parabolic equations, boundary conditions are specified on the boundary of the spatial variables, and initial conditions are specified on the time variable, t. 

Elhptic equations describe steady-state or equilibrium processes. For such equations, all aux-ihary conditions must be prescribed on the boundary of the region of interest. Initial conditions given to elliptic equations lead to ill-posed problems. Solutions to these ill-posed problems exhibit sensitivity to the initial data. Small changes to the initial data cause large changes in the solution. 

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