SPECTRALLY ACCURATE SOLUTION OF NONPERIODIC DIFFERENTIAL-EQUATIONS BYTHE FOURIER-GEGENBAUER METHOD

It is well known that the Fourier partial sum of an analytic nonperiod ic function, supported on a finite interval, converges slowly inside t he interval and exhibits O(1) spurious oscillations near the boundarie s (the Gibbs phenomenon). An effective algorithm which allows one to c ompletely overcome the Gibbs phenomenon was developed in [J. Comput. A ppl. Math., 43 (1992), pp. 81-92]. The basic concept of this approach consists of the re-expansion of the Fourier partial sums into the rapi dly convergent Gegenbauer series. In this paper we extend the Fourier- Gegenbauer (F-G) method of [J. Comput. Appl. Math., 43 (1992), pp. 81- 92] to the evaluation of the spatial derivatives of a piecewise analyt ic function. Also, we apply this method to the solution of nonperiodic boundary value problems. Although the derivatives of a discontinuous function are not in L-2, the exponential convergence of the truncated Gegenbauer series can be proved, and the rate of convergence can be es timated. The solution of differential equations is accomplished in two steps. First, a particular solution with arbitrary boundary condition s is constructed using the F-G method. This particular solution is the n corrected to satisfy the prescribed boundary conditions of the probl em by adding a proper linear combination of homogeneous solutions. For boundary layer problems the intermediate (particular) solution has st eep profiles near the boundaries. These steep regions introduce a larg e error into the final solution, which presumably has a smooth profile on the whole interval. A method which compensates for this loss of ac curacy by using the appropriately constructed homogeneous solutions is proposed.