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(which certainly is valid for any polynomial)
can be written in the operational form
and thus we deduce that
which is to be interpreted as an abbreviation of the statement
that the operators and
are equivalent when applied to any polynomial
of
degree for any n.
Further, we obtain the additional relations
Classification of second order PDE's:
The
general linear second order partial differential equation in two
independent variables is given by
where the coefficients are functions of x and y and the subscripts
denote partial derivatives with respect to the independent
variables. The above equation is called
This classification depends in general on the region of the
plane under consideration. The differential equation
for instance, is elliptic for
The well known examples of the three types are:
Heat Flow equation:
Wave equation:
Laplace equation:
The parabolic and hyperbolic type of equations are either initial value problems or initial boundary value problems whereas the elliptic type equation is always a boundary value problem. The boundary conditions can be one of the following three types:
- The Dirichlet or first boundary condition: Here,
the solution is prescribed along the boundary
- The Neumann or Second boundary condition: Here,
the derivative of the solution is specified along the boundary.
- The third or mixed boundary condition:
Here, the solution and its derivative are prescribed along the
boundary.
Next:Derivation of an exact
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