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Second-order Equation (second-order + equation)
Selected AbstractsAn integral formulation procedure for the solutions to Helmholtz's equation in spherically symmetric mediaMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2010Giacomo Caviglia Abstract Starting from Helmholtz's equation in inhomogeneous media, the associated radial second-order equation is investigated through a Volterra integral equation. First the integral equation is considered in a sphere. Boundedness, uniqueness and existence of the (regular) solution are established and the series form of the solution is provided. An estimate is determined for the error arising when the series is truncated. Next the analogous problem is considered for a spherical layer. Again, boundedness, uniqueness and existence of two base solutions are established and error estimates are determined. The procedure proves more effective in the sphere. Copyright © 2009 John Wiley & Sons, Ltd. [source] Equilibrium and kinetic study for the removal of malachite green using activated carbon prepared from Borassus flabellofer male flowerASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING, Issue 3 2010P. E. Jagadeesh Babu Abstract Activated carbon was prepared from dried Borassus flabellofer male flower and batch adsorption experiments were conducted to study its potential to remove malachite green (MG) dye. The process was further optimized by studying the operating variables like initial pH of the stock solution, activation temperature, initial dye concentration, adsorbent loading and contact time. The optimized pH and activation temperatures were found to be 7.55 and 450 °C respectively, where further analysis was made using these optimal variables. Linear, Freundlich and Langmuir isotherms were studied and it was found that the Langmuir isotherms have the highest correlation coefficients compared to the others. Further, the sorption kinetics were analysed using pseudo-first-order and pseudo-second-order kinetic models. The data showed that the second-order equation was the more appropriate, which indicate that the intra-particle diffusion is the rate limiting factor. Copyright © 2009 Curtin University of Technology and John Wiley & Sons, Ltd. [source] Symmetry group classification of ordinary differential equations: Survey of some resultsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2007F. M. Mahomed Abstract After the initial seminal works of Sophus Lie on ordinary differential equations, several important results on point symmetry group analysis of ordinary differential equations have been obtained. In this review, we present the salient features of point symmetry group classification of scalar ordinary differential equations: linear nth-order, second-order equations as well as related results. The main focus here is the contributions of Peter Leach, in this area, in whose honour this paper is written on the occasion of his 65th birthday celebrations. Copyright © 2007 John Wiley & Sons, Ltd. [source] Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs,COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2007Eitan Tadmor We prove in this paper new velocity-averaging results for second-order multidimensional equations of the general form ,(,x, v)f(x, v) = g(x, v) where ,(,x, v) := a(v) · ,x , , · b(v),x. These results quantify the Sobolev regularity of the averages, ,vf(x, v),(v)dv, in terms of the nondegeneracy of the set {v: |,(i,, v)| , ,} and the mere integrability of the data, (f, g) , (L, L). Velocity averaging is then used to study the regularizing effect in quasi-linear second-order equations, ,(,x, ,), = S(,), which use their underlying kinetic formulations, ,(,x, v),, = gS. In particular, we improve previous regularity statements for nonlinear conservation laws, and we derive completely new regularity results for convection-diffusion and elliptic equations driven by degenerate, nonisotropic diffusion. © 2007 Wiley Periodicals, Inc. [source] | ||||||||||