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Abstract. A collocation procedure is developed for the initial value problem u (t) = f (t, u(t)), u(0) = 0, using the globally defined sinc basis functions. It is shown that this sinc procedure converges to the solution at an exponential rate, i.e., O(M 2 exp(−κ √ M )) where κ > 0 and 2M basis functions are used in the expansion. Problems on the domains R = (−∞, ∞) and R + = (0, ∞) are used to illustrate the implementation and accuracy of the procedure.

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In this paper three methods are derived for approximating /, given its Laplace transform g on (0, oo), i.e., J f(t) exp(-st) dt = g(s). Assuming that g e L2(0, oo), the first method is based on a Sine-like rational approximation of g, the second on a Sine solution of the integral equation J f(t) exp(-st) dt = g(s) via standard regularization, and the third method is based on first converting J f(t) exp(-st) dt = g(s) to a convolution integral over R, and then finding a Sine approximation to / via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.

Abstract. The Sinc-Galerkin method developed in [5], when applied to the second-order selfadjoint boundary value problem, gives rise to a nonsymmetric coefficient matrix. The technique in [5] is based on weighting the Galerkin inner products in such a way that the method will handle boundary value problems with regular singular points. In particular, the method does an accurate job of handling problems with singular solutions (the first or a higher derivative of the solution is unbounded at one or both of the boundary points). Using n function evaluations, the method of [5] converges at the rate exp(-icjñ), where k is independent of n. In this paper it is shown that, by changing the weight function used in the Galerkin inner products, the coefficient matrix can be made symmetric. This symmetric method is applicable to a slightly more restrictive set of boundary value problems than the method of [5], The present method, however, still handles a wide class of singular problems and also has the same exp(-Ky7i ) convergence rate.

This error is of almost no practical consequence, but should be fixed for logical correctness. With EPS = 10-4 , the correction produces changes on the order of 10-6 in PROB and BOUND.The author is indebted to James McNicol of the Scottish Crop Research Institute and Mahesh Parmar of University of Oxford for bringing the fatal error to his attention. The first two errors were introduced after the final submission but before page proofs, so that none of the results stated in the description of the algorithm are affected by them. As mentioned above, the third error produces very small changes. The author apologizes for any inconvenience these oversights have caused to users.To assure users that no further errors were introduced in page proofs, the author has typed the entire program (except for comments) verbatim from the Applied Statistics listing. After correcting the above errors, the newly typed program produces identical answers to the final submission for all of the test cases.

A fully-Galerkin approach to the coefficient recovery (parameter identification) problem for a linear parabolic partial differential equation is introduced. The forward problem is discretised with a sinc basis in the temporal domain and a finite element basis in the spatial domain. Tikhonov regularisation is applied to deal with the ill-posedness of the inverse problem. In the solution of the resulting nonlinear optimisation problem. advantage is taken of the diagonalisation solution procedure used for the discretised forward problem. An example with noisy data is included.