The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domainΩand if solutions of the quasilinear problem are inΩ, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

We show the normal hyperbolicity property for the equilibria of the evolution equation∂m(r,t)/∂t=-m(r,t)+g(βJ*m(r,t)+βh), h,β≥0,and using the normal hyperbolicity property we prove the continuity (upper semicontinuity and lower semicontinuity) of the global attractors of the flow generated by this equation, with respect to functional parameterJ.

The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers.

International Journal of Differential Equations publishes research on differential equations, and related integral equations, from all scientists who use differential equations as tools within their own discipline.

We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434, Brisbane, QLD 4001, Australia 2 Department of Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA 3Department of Mathematics, The University of Jordan, Amman 11942, Jordan 4 School of Mathematics, Cardiff University, Cardiff CF2 4YH, UK 5Department of Engineering Mechanics, Hohai University, Xikang Road No. 1, Nanjing, Jiangsu 210098, China

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

We employ the Heat Balance Integral Method (HBIM) to solve a number of thermal and phase change problems which occur in a multitude of industrial contexts. As part of our analysis, we propose a new error measure for the HBIM that combines the least-squares error with a boundary immobilisation method. We describe how to determine this error for three basic thermal problems and show how it can be used to determine an optimal heat balance formulation. We then show how the HBIM, with the new error measure, may be used to approximate the solution of an aircraft deicing problem. Finally we apply the new method to two industrially important phase change problems.

Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws and error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.