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On Fractional Helmholtz Equations

Samuel, M., Thomas, Anitha (2010)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 33E12, 33C60, 35R11In this paper we derive an analytic solution for the fractional Helmholtz equation in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Leffler function. In all three cases the solutions are represented also in terms of Fox's H-function.

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ - ( a + b Ω | u | ² d x ) Δ u = u + λ u q - 1 / | x | β in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

On local motion of a compressible barotropic viscous fluid bounded by a free surface

W. Zajączkowski (1992)

Banach Center Publications

We consider the motion of a viscous compressible barotropic fluid in ℝ³ bounded by a free surface which is under constant exterior pressure, both with surface tension and without it. In the first case we prove local existence of solutions in anisotropic Hilbert spaces with noninteger derivatives. In the case without surface tension the anisotropic Sobolev spaces with integration exponent p > 3 are used to omit the coefficients which are increasing functions of 1/T, where T is the existence time....

On mild solutions of gradient systems in Hilbert spaces

Andrzej Rozkosz (2013)

Open Mathematics

We consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.

On nonoscillation of canonical or noncanonical disconjugate functional equations

Bhagat Singh (2000)

Czechoslovak Mathematical Journal

Qualitative comparison of the nonoscillatory behavior of the equations L n y ( t ) + H ( t , y ( t ) ) = 0 and L n y ( t ) + H ( t , y ( g ( t ) ) ) = 0 is sought by way of finding different nonoscillation criteria for the above equations. L n is a disconjugate operator of the form L n = 1 p n ( t ) d d t 1 p n - 1 ( t ) d d t ... d d t · p 0 ( t ) . Both canonical and noncanonical forms of L n have been studied.

On non-overdetermined inverse scattering at zero energy in three dimensions

Roman G. Novikov (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We develop the ¯ -approach to inverse scattering at zero energy in dimensions d 3 of [Beals, Coifman 1985], [Henkin, Novikov 1987] and [Novikov 2002]. As a result we give, in particular, uniqueness theorem, precise reconstruction procedure, stability estimate and approximate reconstruction for the problem of finding a sufficiently small potential v in the Schrödinger equation from a fixed non-overdetermined (“backscattering” type) restriction h | Γ of the Faddeev generalized scattering amplitude h in the...

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