Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

A dynamic thermoviscoelastic problem: An existence and uniqueness result

This work deals with the variational analysis of a dynamic problem which models the temperature evolution in a thermoviscoelastic body. The variational problem is formulated as a coupled system of evolutionary nonlinear variational equations. Then, the existence of a unique weak solution is proved using Banach fixed-point arguments and results on time-dependent families of subgradients.     

A new modification of the variational iteration method for van der Pol equations

In this paper, we provide a new modification of the variational iteration method (MVIM) for solving van der Pol equations. The modification couples the classical variational iteration method with He’s polynomials, where the He’s polynomials are applied to the approximate solution and the initial condition to eliminate secular terms. For the large ?, the numerical results demonstrate that the modification method get an accurate approximate period than the other presented methods.     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Application of He's variational iteration method to nonlinear Jaulent–Miodek equations and comparing it with ADM

Instead of finding a small parameter for solving nonlinear problems through perturbation method, a new analytical method called He's variational iteration method (VIM) is introduced to be applied to solve nonlinear Jaulent–Miodek, coupled KdV and coupled MKdV equations in this article. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers can be identified optimally via the variational theory. The results are compared with exact solutions.     

Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system

We consider a structural acoustic wave equation with nonlinear acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with boundary conditions on the interface. We prove wellposedness in the Hadamard sense for strong and weak solutions. The main tool used in the proof is the theory of nonlinear semigroups. We present the system of partial differential equations as a suitable Cauchy problem . Though the operator A is not maximally dissipative we are able to show that it is a translate of a maximally dissipative operator. The obtained semigroup solution is shown to satisfy a suitable variational equality, thus giving weak solutions to the system of PDEs. The results obtained (i) dispel the notion that the model does not generate semigroup solutions, (ii) provide treatment of nonlinear models, and (iii) provide existence of a correct state space which is invariant under the flow-thus showing that physical model under consideration is a dynamical system. The latter is obtained by eliminating compatibility conditions which have been assumed in previous work (on the linear case).     

Investigation of the behavior of weak waves in the solution of a vector variational wave equation

We consider nonsmooth solutions of the system of Euler-Lagrange equations corresponding to a variational problem with several unknown functions of several variables and with a quadratic functional. The propagation of weak discontinuities is described by the equations of the method of singular characteristics developed by Melikyan. The onset and interaction of weak discontinuities of the solution caused by nonsmooth initial conditions are studied by numerical-analytic methods. We develop two computer programs for shock-fitting and shock-capturing computations. The approach was earlier applied by the authors to the analysis of a variational wave equation, namely, to the solution of the Euler-Lagrange equation for a variational problem with a single unknown function.     

Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid–solid interaction problem

In this paper we consider a fluid–solid interaction problem posed in the plane. We employ a mixed variational formulation in the obstacle, in which the Cauchy stress tensor and the rotation are the only unknowns. This new mixed formulation is coupled, through suitable transmission conditions on the wet interface, with a Helmholtz equation satisfied by the pressure of the fluid in the unbounded domain. We use a traditional primal variational formulation in this part of the domain and incorporate the far field information through boundary integral equations. We approximate the resulting weak formulation by a Galerkin scheme based on PEERS in the solid and on a FEM-BEM approach in the fluid part. We show that our scheme is uniquely solvable and convergent, and then provide optimal error estimates. Finally, we illustrate our analysis with some computational experiments.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations

In this article, we study a streamline diffusion‐based discontinuous Galerkin approximation for the numerical solution of a coupled nonlinear system of Schrödinger equations and extend the resulting method to a multiscale variational scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. In the weak formulation, to make the underlying bilinear form coercive, it was necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. We justify the theory by implementing an example of an application of the time‐dependent Schrödinger equation in the coupled ultrafast laser. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Existence results of semilinear differential variational inequalities without compactness

ABSTRACT

The purpose of this paper is to study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing semilinear evolution equations and generalized variational inequalities. First we prove essential properties of the solution set for generalized variational inequalities. Then without requiring any compactness condition for the evolution operator or for the nonlinear term, two existence results for mild solutions are established by applying a weak topology technique combined with a fixed point theorem.     

A-调和方程很弱解的正则性

本文证明了二阶拟线性偏微分方程很弱解的正则性．若ｕ是（１）的一个很弱解并属于一个合适的包含Ｗ１,ｐ　ｌｏｃ（ ）的空间,则ｕ属于 （ ）,即ｕ是（１）通常意义下的弱解．变分积分弱极值的同样结果被得到．     

非光滑泛函临界点理论的一个应用

本文利用局部Lipschitz泛函的临界点理论，来得到一个改进的半线性方程的Landesman Lazer型结果，问题来自于力学，称为变分不等式的特征值问题，在[3]中，D.Goeleven,D.Motreanu和P.D.Panagiotopoulos等人讨论了共振的情形，并得到了弱解的存在性的结果。但是，在他们结论中（见[3]中定理4．1），条件（H1）与（H2）却是互不相容的，本文得到了变分不等式的特征值问题在共振情形下的弱解存在性的相应结果。     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

Direct problem for a nonlinear evolutionary system

In this article, we consider the first initial boundary-value problem for an evolutionary system describing nonlinear interactions of electromagnetic and elastic waves. The system under study consists of three coupled differential equations, one of them is a hyperbolic equation (an analogue of the Lamé equations) and the other two equations form a parabolic system (an analogue of the diffusion Maxwell system). Existence and uniqueness results are established. We also prove the stability estimate of a weak solution.     

A variational formulation of nonlocalmaterials withmicrostructure based on dual macro- and micro-balances

We investigate a variational setting of nonlocal materials with microstructure and outline aspects of its numerical implementation. Thereby, the current state of the evolving microstructure is described by independent global degrees in addition to the macroscopic displacement field, so-called order parameters. Focussing on standard-dissipative materials, the constitutive response is governed by two fundamental functions for the energy storage and the dissipation. Based on these functions, a global dissipation postulate is introduced. Its exploitation constitutes a global variation formulation of nonlocal materials, which can be related to a minimization principle. Following this methodology, we end up with coupled macro- and microscopic field equations and corresponding boundary conditions. On the numerical side, we consider the weak counterpart of these coupled field equations and obtain after linearization a fully coupled system for increments of the displacement and the order parameters. Due to the underlying variational structure, this system of equations is symmetric. In order to show the capability of the proposed setting, we specify the above outlined scenario to a model problem of isotropic damage mechanics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled system of integral equations in reflexive banach spaces

We present an existence theorem for at least one weak solution for a coupled system of integral equations of Volterra type in a reflexive Banach spaces relative to the weak topology.     

Some problems for nonlinear pseudoparabolic equations in nontube domains

We prove existence and uniqueness of a weak solution to the first initial-boundary value problem for some class of quasilinear pseudoparabolic equations in nontube domains. Also, we study unique solvability in these domains for the variational inequality connected with the above class of equations.     

Helical symmetric solution of 3d Navier-Stokes equations arising from geometric shape of the boundary

In this article, we investigate three-dimensional solution with helical symmetry in a gap between two concentric rotating cylinders, inside is a helicoidal surface (screw propeller) while outside is a cylindrical surface. Establish the partial differential equations and its variational formulation satisfied by a helical solution in a helical coordinate system using tensor analysis method, we provide a computational method for the power and propulsion of the screw. The existence and uniqueness of weak helical solutions are proved.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.