Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in ${{\mathbb R}^n}$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L² estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.     

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

The first integral method to some complex nonlinear partial differential equations

In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.     

Singularities of multivalued solutions of nonlinear differential equations,and nonlinear phenomena

The purpose of this paper is to use the geometrical theory of nonlinear partial differential equations and the theory of singularities of maps in order to obtain the general scheme for constructing shock waves from multivalued solutions, given by smooth integral manifolds. This scheme is illustrated by some examples from gas dynamics, mechanics, acoustics and thermodynamics.     

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

The existence of multiple positive solutions of systems of singular Hammerstein integral equations is studied, where the nonlinearities involved are allowed to have singularities in their second variables and satisfy weaker conditions involving the first eigenvalues of the corresponding linear Hammerstein integral operators. Such systems contain some mathematical models arising in science and engineering. Applications are given to the existence of multiple positive radial solutions of systems of semilinear singular elliptic equations in annuli on which, to the best of our knowledge, there has been little study.     

Related solution operators in mathematical physics

Transmutation operators are derived relating many of the frequently encountered linear partial differential equations in mathematical physics. The setting for this study is vector-valued distributions. Examples are given showing how fundamental solutions are derived for both homogeneous and nonhomogeneous partial differential equations.     

Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

Eigenvalue intervals and the existence of finitely many positive eigenfunctions for semi-positone Hammerstein integral equations are obtained. The positive characteristic values and their upper and lower bounds of the corresponding linear Hammerstein integral operators are studied. Applications of the results are given to third-order differential equations with three-point boundary conditions.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

Smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions

The smoothness of generalized solutions for higher-order elliptic equations with nonlocal boundary conditions is studied in plane domains. Necessary and sufficient conditions upon the right-hand side of the problem and nonlocal operators under which the generalized solutions possess an appropriate smoothness are established.     

Numerical solutions of nonlinear evolution equations using variational iteration method

The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces

In this paper we give the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable and uniformly smooth Banach spaces.     

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h^1/4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.     

Some properties of the Cauchy-type integral for the time-harmonic Maxwell equations

We study the analog of the Cauchy-type integral for the theory of time-harmonic electromagnetic fields in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given pair of vector fields from such a surface up to a solution of the time-harmonic Maxwell equations in a domain. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between time-harmonic solutions of the Maxwell equations and some versions of quaternionic analysis.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

Additive Schwarz algorithms for the p version of the Galerkin boundary element method

Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators. Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000