1 The Approximate Solution Curve Tracing (ASCT) Method 1.1 The correct solution space (CSS) Figure 1: The CSS represents the correct solution .space which all of the depth values on the illuminated object are included in. LB is the lower bound of the CSS. All of the depth values on the illuminated object will be not "deeper" than it. UB is the upper boumt of the CSS. All of the depth values on ...

In this article, we prove that the initial value problem associated with the Korteweg-de Vries equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ 2θ ≥ 2 and the initial value problem associated with the nonlinear Schrödinger equation is well-posed in weighted Sobolev spaces X s,θ, for s ≥ θ ≥ 1. Persistence property has been proved by approximation of the solutions and using a pri...

We give a construction of a divergence-free vector field u0 ∈ H s ∩ B ∞,∞ , for all s < 1/2, such that any Leray-Hopf solution to the Navier-Stokes equation starting from u0 is discontinuous at t = 0 in the metric of B ∞,∞ . For the Euler equation a similar result is proved in all Besov spaces B r,∞ where s > 0 if r > 2, and s > n(2/r − 1) if 1 ≤ r ≤ 2.

We make remarks on Fernández Guasti’s paper [J. Phys. A: Math. Gen. 39 (2006) 11825-11832] by pointing out some mistakes Fernández Guasti derived therein. PACS number: 45.20.Dd, 02.60.Cb, 45.05.+x Fernández Guasti just showed that a time-dependent oscillator equation could be solved numerically for various trajectories in amplitude and phase variables [1]. His solutions exhibit a finite time-de...

By making use of asymptotic properties of nonoscillatory solutions the oscillation behavior of solutions for the integro-dynamic equation x(t) = e(t) − ∫ t 0 k(t, s)f(s, x(s))∆s, t ≥ 0 and the integral equation x(t) = e(t) − ∫ t 0 k(t, s)f(s, x(s))∆s, t ≥ 0 on time-scales are investigated. Easily verifiable sufficient conditions are established for the oscillation of all solutions. The results ...

In an isometric tetanus in frog's sartorius muscle tension approaches the plateau exponentially with rate constant alpha. alpha a depends on sarcomere length, s, and temperature, T, according to the Arrhenius equation See PDF for Equation for temperatures between 1 and 20 degrees C and for sarcomere lengths 2.0-2.8 microm. The energy of activation, E, does not vary significantly with s; E = 13....

We propose an alternative non-monotonic modal formalism called non-monotonic modal logic of belief. It is based on replacing the classical fixpoint equation E = ThS(A ∪ {Mφ : E 6`S ¬φ}) with the belief fixpoint equation E = ThS(A ∪ {Mφ : E 6`S ¬Mφ}). The solutions of the belief fixpoint equation, called belief S-expansions, are tightly related to the logic of belief KD45. We show interpretation...

This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation u, + u2(, = u,, the general nonlinear equation u, + u%, = u, is considered and expanded in powers of 6. The coefficients of the S series to sixth order in powers of S is determined and Pad& summation is us...

We study a kinetic model for chemotaxis introduced by Othmer, Dunbar, and Alt [22], which was motivated by earlier results of Alt, presented in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser, it was rigorously shown that, in three dimensions, this kinetic model leads to the classical KellerSegel model as its drift-diffusion limit when the equation of the chemo-attractant i...

in this paper, we use the continuous legendre wavelets on the interval [0,1] constructed by razzaghi m. and yousefi s. [6] to solve the linear second kind integral equations. we use quadrature formula for the calculation of the products of any functions, which are required in the approximation for the integral equations. then we reduced the integral equation to the solution of linear algebraic ...