Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives

In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.     

Resonant control of solitons

It is shown that the effect of “scattering on resonance” can be used to control envelope solitons in the driven nonlinear Schrödinger equation. The control occurs by the frequency modulated driving with multiple crossing of the resonant frequency of the soliton.     

Boundary problems for fractional differential equations

In this paper, the existence of solutions of fractional differential equations with nonlinear boundary conditions is investigated. The monotone iterative method combined with lower and upper solutions is applied. Fractional differential inequalities are also discussed. Two examples are added to illustrate the results.     

Interbasis expansions for isotropic harmonic oscillator

The exact solutions of the isotropic harmonic oscillator are reviewed in Cartesian, cylindrical polar and spherical coordinates. The problem of interbasis expansions of the eigenfunctions is solved completely. The explicit expansion coefficients of the basis for given coordinates in terms of other two coordinates are presented for lower excited states. Such a property is occurred only for those degenerated states for given principal quantum number n.     

Simultaneous Determination of Acetaminophen and Phenobarbital in Suppositories by Partial Least-Squares Spectrophotometric Calibration

Abstract

The use of partial least-squares spectrophotometric calibration for the simultaneous determination of suppositories in a multicomponent formulation is presented. This method is applied to the deternination of acetaminophen and phenobarbital in suppository preparations. The results show that these components in a molar ratio of about 61:1 in suppositories have been determined simultaneously with high precision.     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Equilibrium analysis for a mass-conserving model in presence of cavitation

We study the existence of equilibrium positions for the load problem in Lubrication Theory. The problem consists of two surfaces in relative motion separated by a small distance filled by a lubricant. The system is described by the modified Reynolds equation (Elrod–Adams model) which describes the behavior of the lubricant and an extra integral equation given the balance of forces. The balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom.     

Properties of ε-value functions for a two person differential game

We consider the problem of approximate minimax for the Bolza problem of optimal control. Starting from the method of dynamic programming (Bellman) we define the ?-value function to be the approximation for the value function being a solution to the Hamilton–Jacobi equation.     

变形谐振子势的能谱方程

讨论了3种变形谐振子势:左右两边不同参数的谐振子势、左边方形势加右边谐振子势和谐振子势中间加δ势中的能量本征态函数.这些函数都可以由厄米函数表示.由波函数及其一次导数在原点的衔接条件,得到了能谱方程.     

The use of Sinc‐collocation method for solving Falkner–Skan boundary‐layer equation

The MHD Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity on the semi‐infinite domain. The proposed approach is equipped by the orthogonal Sinc functions that have perfect properties. This method solves the problem on the semi‐infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, the governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by the Sinc‐collocation method. It is shown that the Sinc‐collocation method converges to the solution at an exponential rate. Copyright © 2010 John Wiley & Sons, Ltd.