The numerical solution of elliptic partial differential equations by novel block iterative methods
Sojoodi-HaghighiReza
2012
Partial differential equations occur in a variety of forms in
many different branches of Mathematical Physics. These equations
can be classified according to various criteria, which may include
such formal aspects as the number of dependent or independent
variables, the order of the derivatives involved and the degree of
non-linearity of the equations. These equations can also be
categorised according to the methods which are employed for solving
the partial differential equations, or according to particular
properties which their solutions may possess.
In pure analysis, to solve partial differential equations we
may use methods such as transformation or separation of variables.
However, these methods can only be applied to very special classes
of problems [WEINBERGER, 1965].
In practice, we employ numerical methods for the solution of
such systems and although the types of methods used in numerical
analysis of differential equations do not generally correspond with
those used in mathematical analysis, both depend upon particular
properties of the solution.
This thesis is concerned with the numerical solution of certain
types of partial differential equations and therefore, with practical
problems which can be treated by certain numerical methods. In
practice, the numerical methods for solving the differential problems
depend upon the nature of other auxiliary conditions, such as boundary
or initial conditions. Certain types of auxiliary conditions are
suitable only for certain corresponding types of differential equations,
and in general, physical problems suggest auxiliary conditions which
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are suitable for the differential equations involved in the problem.
If, the auxiliary conditions are specified in such a way that
there exists one and only one solution (uniqueness) for the
differential problem, and in addition, a small change in these
given auxiliary conditions result in a small change in the solution
(stability), then the problem is said to be well-posed. Since
numerical methods are by nature approximate processes, however,
these methods rarely produce exact solutions for a given problem.
However, it can be shown [see STEPHENSON, 1970], that if the
differential problem is well-posed then the solution of this problem
is expected to be accurate...