This note deals with the initial shift problem that arises from discrete-time iterative learning control. A unified learning scheme is considered for a class of nonlinear systems with well-defined relative degree, which adopts the error data with anticipation in time and provides wider freedom for the updating law formation. The sufficient convergence condition is derived to enable the system t...

We study the positivity and regularity of solutions to the fractional porous medium equations ut + (−∆) u = 0 in (0,∞)× Ω, for m > 1 and s ∈ (0, 1) and with Dirichlet boundary data u = 0 in (0,∞)× (R \ Ω), and nonnegative initial condition u(0, ·) = u0 ≥ 0. Our first result is a quantitative lower bound for solutions which holds for all positive times t > 0. As a consequence, we find a global H...

We solve the Cauchy problem for the Korteweg–de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the mutual spectral bands either coincide or are disjoint.

We consider a reaction-diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is asymptotically stable. As in [25], [24], and [14], we show that the sequence of traveling waves is itself asymptotically stable: as t → ∞, the solution approaches the concatenated wave pattern, with di...

Results concerning the global existence and uniqueness of mild solutions for a class of first-order abstract stochastic integro-differential equations with variable delay in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t are establish...

In this paper, we consider the nonlinear viscoelastic equation utt −Δu+ t ∫ 0 g(t − τ )Δu(τ) dτ + ut |ut |m−2 = u|u|p−2 with initial conditions and Dirichlet boundary conditions. For nonincreasing positive functions g and for p >m, we prove that there are solutions with positive initial energy that blow up in finite time. © 2005 Elsevier Inc. All rights reserved.

We investigate the effect of friction on domain formation in disoriented chiral condensate. Including a friction term, we solve the equation of motion of the linear sigma model fields, in the Hartree approximation. With boostinvariance and cylinderical symmetry, irrespective of friction, on average, we donot find any indication of domain like formation with quenched initial condition. However, ...

We deene the continuum up to order isomorphism (and hence homeomorphism) as the nal coalgebra of the functor X !, ordinal product with !. This makes an attractive analogy with the deenition of the ordinal ! itself as the initial algebra of the functor 1; X, prepend unity, with both deenitions made in the category of posets. The variants 1; (X !), X o !, and 1; (X o !) yield respectively Cantor ...

In this paper, we applied the homotopy analysis method (HAM) to construct the analytical solutions of the space fractional diffusion equations. The derivatives are defined in the Jumarie’s fractional derivative sense. The explicit solutions of the equations have been presented in the closed form by using initial conditions. Two typical examples have been discussed. The results reveal that the m...

In this paper, we present mainly two logical systems that clarify pragmatic aspects of the process of explanation. The first concerns a proof theory that leads to the derivation of possible initial conditions from an explanandum and a given theory. The second logic concerns the derivation of questions in view of the verification of some possible initial condition, or of one out of several possi...