Abstract
Integration operational matrix methods based on Zernike polynomials are used
to determine approximate solutions of a class of non-homogeneous partial
differential equations (PDEs) of first and second order. Due to the nature of
the Zernike polynomials being described in the unit disk, this method is
particularly effective in solving PDEs over a circular region. Further, the
proposed method can solve PDEs with discontinuous Dirichlet and Neumann
boundary conditions, and as these discontinuous functions cannot be defined at
some of the Chebyshev or Gauss-Lobatto points, the much acclaimed
pseudo-spectral methods are not directly applicable to such problems. Solving
such PDEs is also a new application of Zernike polynomials as so far the main
application of these polynomials seem to have been in the study of optical
aberrations of circularly symmetric optical systems. In the present method, the
given PDE is converted to a system of linear equations of the form Ax = b which
may be solved by both l1 and l2 minimization methods among which the l1 method
is found to be more accurate. Finally, in the expansion of a function in terms
of Zernike polynomials, the rate of decay of the coefficients is given for
certain classes of functions.