Integral representation of a solution to the Stokes–Darcy problem

With methods of potential theory, we develop a representation of a solution of the coupled Stokes–Darcy model in a Lipschitz domain for boundary data in H^?1/2. Copyright © 2014 John Wiley & Sons, Ltd.     

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4⁺ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12 : 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4⁺ T cells. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergence of FFT‐based homogenization for strongly heterogeneous media

The FFT‐based homogenization method of Moulinec–Suquet has recently attracted attention because of its wide range of applicability and short computational time. In this article, we deduce an optimal a priori error estimate for the homogenization method of Moulinec–Suquet, which can be interpreted as a spectral collocation method. Such methods are well‐known to converge for sufficiently smooth coefficients. We extend this result to rough coefficients. More precisely, we prove convergence of the fields involved for Riemann‐integrable coercive coefficients without the need for an a priori regularization. We show that our L² estimates are optimal and extend to mildly nonlinear situations and L^p estimates for p in the vicinity of 2. The results carry over to the case of scalar elliptic and curl ? curl‐type equations, encountered, for instance, in stationary electromagnetism. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

Q‐symmetry and conditional Q‐symmetries for Drinfel'd–Sokolov–Wilson system

We study in this paper the Q‐symmetry and conditional Q‐symmetries of Drinfel'd–Sokolov–Wilson equations. The solutions which we obtain in this paper take the form of convergent power series with easily computable components. Copyright © 2014 John Wiley & Sons, Ltd.     

A deductive approach to the solution of the problem of optimal pairs trading from the viewpoint of stochastic control with time‐dependent parameters

In a fairly recent paper (2008 American Control Conference, June 11‐13, 1035‐1039), the problem of dealing with trading in optimal pairs was treated from the viewpoint of stochastic control. The analysis of the subsequent nonlinear evolution partial differential equation was based upon a succession of Ansätze, which can lead to a solution of the terminal‐value problem. Through an application of the Lie Theory of Continuous Groups to this equation, we show that the Ansätze are based upon the underlying symmetries of the equation (their (14)). We solve the problem in a more general context by allowing the parameters to be explicitly time dependent. The extension means thatmore realistic problems are amenable to the samemode of solution. Copyright © 2014 JohnWiley & Sons, Ltd.     

Existence of global weak solutions for Navier‐Stokes‐Poisson equations with quantum effect and convergence to incompressible Navier‐Stokes equations

In this paper, we consider a three dimensional quantum Navier‐Stokes‐Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier‐Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

On existence and multiplicity of similarity solutions to a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow for decelerated flows

Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. Copyright © 2015 John Wiley & Sons, Ltd.     

Fredholm boundary value problems for perturbed systems of dynamic equations on time scales

This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo‐inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.