Deciphering singularities by discrete methods
Yves Tourigny,Michael Grinfeld +1 more
TL;DR: This work considers the problem of estimating numerically the parameters of singularities of solutions of differential equations and proposes a novel approach which is based on discretizing the governing equation and "timestepping" in the complex domain.
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Abstract: 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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Blowup in diffusion equations: a survey
Catherine Bandle,Hermann Brunner +1 more
TL;DR: In this paper, the authors dealt with quasilinear reaction-diffusion equations for which a solution local in time exists and the problem of determining whether the solution ceases to exist for some finite time is studied.
288
Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces
TL;DR: The numerical pole field solver is extended to enable the computation of the multivalued Painleve transcendents, which are the solutions to the third, fifth and sixthpainleve equations, on their Riemann surfaces.
12
A case study of methods of series summation: Kelvin-Helmholtz instability of finite amplitude
TL;DR: In this article, the singularities of the solution of the Birkhoff-Rott equation that governs the evolution of a planar periodic vortex sheet were computed for a flat sheet subject initially to a sinusoidal disturbance of amplitude a. The series was then summed by using various generalisations of the Pade method.
6
Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations
TL;DR: In this article, a composite modified Filon-Clenshaw-Curtis rule is applied on high-frequency oscillatory integrals with algebraic (or logarithmic) endpoint singularities, where the integral outside the oscillation subinterval is non-oscillatory.
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TL;DR: The Euler Method and its Generalizations Analysis of Runge-Kutta Methods General Linear Methods Bibliography.
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The role of critical exponents in blowup theorems
TL;DR: In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation u_t =Delta u + u^p in $R^N with nonnegative initial values.
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