A Numerical solution to the linear and nonlinear Fredholm integral equations using Legendre wavelet functions

Two methods for solving the Fredholm integral equation of the second kind in linear case, i.e. f (x) – λ ∫_a^b K (x,y)f (y)dy = g (x), and nonlinear case, i.e., f (x) = g (x) + λ ∫_a^b K (x,y)F (f (y))dy, are proposed. In order to solve the linear equation, the kernel K (x,y) as well as the functions f and g are initially approximated through Legendre wavelet functions. This leads to a system of linear equations its solution culminates in a solution to the Fredholm integral equation. In nonlinear case only K (x,y) is approximated by Legendre wavelet base functions. This leads to a separable kernel and makes it possible to employ a number of earlier methods in solving nonlinear Fredholm integral equation with separable kernels. Another feature of the proposed method is that it finds the solution as a function instead of specific solution points, what is done by the majority of the existing methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parasupersymmetry of Non-Linear and Isotropic Oscillator on Constant Curvature

In this paper we discuss the one-dimensional non-linear harmonic oscillator and isotropic oscillator in positive curvature. We will see that in the special case two different oscillator have a same behavior. Finally, by using the parasupersymmetry algebra we obtain the partner potential and superpotential. These superpotential lead us to obtain the corresponding superchares.     

CVD Growth of Carbon Nanotubes and Nanofibers: Big Length and Constant Diameter

Summary: We report mass production of carbon nanotubes (CNTs) and carbon nanofibers (CNFs) with relatively high length and aspect ratio. We synthesized carbon nanomaterials by chemical vapor deposition (CVD) of methane as the feeding gas on Fe/Mo nanoparticles that use alumina-aerogel support. Alumina-aerogel-supported Fe/Mo catalyst was prepared using sol-gel. Drying step performed using rotary evaporation and freeze-drying. CVD was performed using a quartz tube furnace. Samples were analyzed using scanning electron microscopy (SEM), Transmission electron microscopy (TEM) and Raman spectroscopy.