We consider the following partial integro-diierential equations which generalize the classical phase eld equations with a non-conserved order parameter and describe the process of phase transitions where memory eeects are present, 8 < : u t + 2 t = R t 0 a 1 (t ?) u()dd 2 t = R t 0 a 2 (t ?) 2 + f() + u ]()dd; where is a small parameter. The functions u and… (More)
Uniform finite difference methods are constructed via nonstandard finite difference methods for the numerical solution of singularly perturbed quasilinear initial value problem for delay differential equations. A numerical method is constructed for this problem which involves the appropriate Bakhvalov meshes on each time subinterval. The method is shown to… (More)
We obtain the maximum principles for the first-order neutral functional differential equation Mxt ≡ x t − Sx are linear continuous operators, A and B are positive operators, C 0,ω is the space of continuous functions, and L ∞ 0,ω is the space of essentially bounded functions defined on 0, ω. New tests on positivity of the Cauchy function and its derivative… (More)
For the delay differential equations ¨ x(t) + a(t) ˙ x(g(t)) + b(t)x(h(t)) = 0, g(t) ≤ t, h(t) ≤ t, and ¨ x(t) + a(t) ˙ x(t) + b(t)x(t) + a 1 (t) ˙ x(g(t)) + b 1 (t)x(h(t)) = 0 explicit exponential stability conditions are obtained.
Sturm's type theorems on separation of zeros of solutions are proved for the second order linear differential equations with delayed argument.
In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet… (More)
We consider the following partial integro-differential equation (Allen–Cahn equation with memory): 2 φ t = t 0 a(t − t)[ 2 ∆φ + f (φ) + h](t) dt , where is a small parameter, h a constant, f (φ) the negative derivative of a double well potential and the kernel a is a piecewise continuous, differentiable at the origin, scalar-valued function on (0, ∞). The… (More)
A boundary value problem is considered for an N-th order functional diierential equation with impulses. It is reduced to the same boundary value problem for another equation of the same order without impulses. The reduction is based on constructing of an isomorphism between the space of the functions which are piece-wise absolutely continuous up to the (N… (More)
Recommended by Alexander Domoshnitsky We establish some new nonlinear Gronwall-Bellman-Ou-Iang type integral inequalities with two variables. These inequalities generalize former results and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of differential equations. Example of applying these… (More)
Copyright q 2010 B.-Y. Long and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For p ∈ R, the generalized logarithmic mean L p a, b, arithmetic mean Aa, b, and geometric mean Ga, b… (More)