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The orthogonal polynomials have their basic recurrence
relation, namely: (1) coefficients recurrence relation (2)
differential recurrence relation. Using the above mentioned
recurrence relation, the conversion matrix of the expansion
coefficients,the operational matrix of product, the operational
matrix of integration and the operational matrix of derivatives
were derived. In this study the conversion matrix of the
expansion coefficients, and the operational matrix of
derivatives are applied to the linear two dimensional second-
order partial differential equations. In this study, there are
three steps for the numerical procedure. The first one is to
put the partial differential equations of two dimensions and
boundary conditions on the setting points to get the ordinary
differential equations of one dimension. The second one is to
use Tau Method to solve the ordinary differentail equations of
one dimension to obtain the associated algebraic equations
formed by the unknown values of the one dimensional expansion
coefficients on the setting points. At last, the Block
Iterative Method is used to solve the associated algebraic
matrixs,therefore the one-dimensional expansion coefficients on
the setting points can be found. The research points of this
study are to augment the integration and the application of the
orthogonal polynomials being applied to the linear second-order
partial differential equations of two dimensions. In the three
types of partial differential equations, the elliptic
equations, the parabolic equations, the hyperbolic equations,
three examples of each type were used to illustrate the
applicability of the orthogonal polynomials applying to the
partial differential equations.
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