Recurrence relations for orthogonal polynomials for PDEs in polar and cylindrical geometries

Area of Science:

• Mathematics
• Numerical Analysis
• Orthogonal Polynomials

Background:

• Orthogonal polynomials are fundamental in approximation theory and solving differential equations.
• Standard orthogonal polynomials often lack specific boundary conditions required for certain physical problems.
• Dirichlet boundary conditions on disks lead to complex solutions in Cartesian coordinates.

Purpose of the Study:

• To introduce two novel families of orthogonal polynomials on the interval (-1,1).
• To ensure these polynomials satisfy specific boundary conditions relevant to PDEs.
• To develop efficient computational methods based on these polynomials.

Main Methods:

• Orthogonalization of linear combinations of Legendre polynomials.
• Derivation of three-term recurrence relations for the new polynomial families.
• Utilizing recurrence relations to compute recurrences for generalized Jacobi polynomials.

Main Results:

• Two distinct families of orthogonal polynomials satisfying specified boundary conditions were constructed.
• Three-term recurrence relations were successfully derived for both families.
• An efficient method for computing recurrences of generalized Jacobi polynomials was established.

Conclusions:

• The introduced orthogonal polynomials provide a powerful tool for problems with specific boundary conditions.
• The derived recurrence relations enable efficient numerical computations.
• This work extends the applicability of orthogonal polynomials in solving PDEs.