By John P. Boyd
Thoroughly revised textual content specializes in use of spectral how to clear up boundary worth, eigenvalue, and time-dependent difficulties, but in addition covers Hermite, Laguerre, rational Chebyshev, sinc, and round harmonic services, in addition to cardinal services, linear eigenvalue difficulties, matrix-solving tools, coordinate modifications, equipment for unbounded periods, round and cylindrical geometry, and lots more and plenty extra. 7 Appendices. word list. Bibliography. Index. Over a hundred and sixty textual content figures.
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Additional info for Chebyshev and Fourier spectral methods
The motive for these definitions is that the Fourier and Chebyshev series of so-called “entire functions” [functions without singularities anywhere in the complex plane except at ∞] have “supergeometric” convergence. For the expansion of functions with poles or branch points which are a finite distance off the expansion interval — the usual case — geometric convergence is normal. Both entire functions and functions with singularities at finite x (but off the expansion interval) have r = 1, so the exponential index of convergence cannot discriminate between them even though these are quite distinct classes of functions.
Better yet, we shall show in later chapters that the accuracy of pseudospectral methods is only a little bit poorer than that of the non-interpolating kingdom – too little to outweigh the much greater simplicity and computational efficiency of the pseudospectral algorithms. Consequently, we shall emphasize pseudospectral methods in this book. Nonetheless, the justification for the pseudospectral kingdom is derived from that for the non-interpolating methods, and the latter are still superior to interpolation for specialized but important applications.
The coefficients of the two forms are related by c0 cn = a0 , n=0 (an − ibn )/2, = (an + ibn )/2, n>0 n<0 Often, it is unnecessary to use the full Fourier series. In particular, if f (x) is known to have the property of being symmetric about x = 0, which means that f (x) = f (−x) for all x, then all the sine coefficients are zero. The series with only the constant and the cosine terms is known as a “Fourier cosine series”. ) If f (x) = −f (−x) for all x, then f (x) is said to be antisymmetric about x = 0 and all the an = 0.