Blowup in diffusion equations: a survey

Applied and computational complex analysis. Vol. 1. Power series, integration, conformal mapping, location of zeros: Henrici P. xv + 682 pp., John Wiley and Sons, Inc., New York — London, 1974☆

Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces

A case study of methods of series summation: Kelvin-Helmholtz instability of finite amplitude

Highly oscillatory integrals, having amplitudes with algebraic (or logarithmic) endpoint singularities, are considered. An integral of this kind is first transformed into a regular oscillatory integral over an unbounded interval. After applying the method of finite sections, a composite modified Filon–Clenshaw–Curtis rule, recently developed by the author, is applied on it. By this strategy the original integral can be computed in a more stable manner, while the convergence orders of the composite Filon–Clenshaw–Curtis rule are preserved. By introducing the concept of an oscillation subinterval, we propose algorithms, which employ composite Filon–Clenshaw–Curtis rules on rather small intervals. The integral outside the oscillation subinterval is non-oscillatory, so it can be computed by traditional quadrature rules for regular integrals, e.g. the Gaussian ones. We present several numerical examples, which illustrate the accuracy of the algorithms.

Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations

We consider the problem of estimating numerically the parameters of singularities of solutions of differential equations. We propose a novel approach which is based on discretizing the governing equation and "timestepping" in the complex domain. Some applications to ordinary and partial differential equations are discussed.

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Deciphering singularities by discrete methods

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Deciphering singularities by discrete methods