Dense Short Solution Segments from Monotonic Delayed Arguments

We construct a delay functional d on an open subset of the space $$C^1_r=C^1([-r,0],\mathbb {R})$$ C r 1 = C 1 ( [ - r , 0 ] , R ) and find $$h\in (0,r)$$ h ∈ ( 0 , r ) so that the equation $$\begin{aligned} x'(t)=-x(t-d(x_t)) \end{aligned}$$ x ′ ( t ) = - x ( t - d ( x t ) ) defines a continuous semiflow of continuously differentiable solution operators on the solution manifold $$\begin{aligned} X=\{\phi \in C^1_r:\phi '(0)=-\phi (-d(\phi ))\}, \end{aligned}$$ X = { ϕ ∈ C r 1 : ϕ ′ ( 0 ) = - ϕ ( - d ( ϕ ) ) } , and along each solution the delayed argument $$t-d(x_t)$$ t - d ( x t ) is strictly increasing, and there exists a solution whose short segments$$\begin{aligned} x_{t,short}=x(t+\cdot )\in C^2_h,\quad t\ge 0, \end{aligned}$$ x t , s h o r t = x ( t + · ) ∈ C h 2 , t ≥ 0 , are dense in an infinite-dimensional subset of the space $$C^2_h$$ C h 2 . The result supplements earlier work on complicated motion caused by state-dependent delay with oscillatory delayed arguments.

Non-Differentiable Solution of Nonlinear Biological Population Model on Cantor Sets

The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local fractional sense. Two examples have been presented showing the effectiveness of this method in solving this model on Cantor sets.

A non-differentiable solution for the local fractional telegraph equation

In this paper, we consider the linear telegraph equations with local fractional derivative. The local fractional Laplace series expansion method is used to handle the local fractional telegraph equation. The analytical solution with the non-differentiable graphs is discussed in detail. The proposed method is efficient and accurate.

The non-differentiable solution for local fractional Laplace equation in steady heat-conduction problem

In this paper, we investigate the local fractional Laplace equation in the steady heat-conduction problem. The solutions involving the non-differentiable graph are obtained by using the characteristic equation method (CEM) via local fractional derivative. The obtained results are given to present the accuracy of the technology to solve the steady heat-conduction in fractal media.

A RENORMALIZATION PROOF OF THE KAM THEOREM FOR NON-ANALYTIC PERTURBATIONS

We shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non-analytic perturbation (the latter will be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which the perturbations are analytic approximations of the original one. We shall finally show that the sequence of the approximate solutions will converge to a differentiable solution of the original problem.

Nonlinear dynamical boundary-value problem of hydrogen thermal desorption

The nonlinear boundary-value problem for the diffusion equation, which models gas interaction with solids, is considered. The model includes diffusion and the sorption/desorption processes on the surface, which leads to dynamical nonlinear boundary conditions. The boundary-value problem is reduced to an integro-differential equation of a special kind; existence and uniqueness of the classical (differentiable) solution theorems are proved. The results of numerical experiments are presented.