a second order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions abstract a difference scheme is derived for a class of nonlocal parabolic equations with natural boundary conditions by the method of reduction of order it is shown that the scheme is uniquely solvable and unconditionally convergent with the convergence rate of order o h 2 2 a numerical example with some comparisons is presented

weak 1 1 bounds for oscillatory singular integrals pour un polynome p r n r n r et un noyau de calderon zygmund k l operateur t f x pv exp ip x y k x y f y dy est de type faible sur l 1 avec une borne dependant seulement de k cz et le degre de p

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

second order schemes for a boundary value problem with neumann s boundary conditions a new second order finite difference scheme based on the 3 3 alternating direction implicit method and a new second order finite difference technique based on the 5 5 implicit formula are discussed for solving a nonlocal boundary value problem for the two dimensional diffusion equation with neumann s boundary conditions while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of the one dimensional case using a suitable transformation the solution of this problem is equivalent to the solution of two other problems the former which is a one dimensional nonlocal boundary value problem giving the value of through using the unconditionally stable standard implicit 3 1 backward time centred space denoted btcs scheme using this result the second problem will be changed to a classical two dimensional diffusion equation with neumann s boundary conditions which will be solved numerically by using the unconditionally stable alternating direction implicit 3 3 technique or the fully implicit finite difference scheme the results of a numerical example are given and computation times are presented error estimates derived in the maximum norm are also tabulated

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

asymptotic behavior of an initial boundary value problem for the vlasov poisson fokker planck system the asymptotic behavior for the vlasov poisson fokker planck system in bounded domains is analyzed in this paper boundary conditions defined by a scattering kernel are considered it is proven that the distribution of particles tends for large time to a maxwellian determined by the solution of the poisson boltzmann equation with dirichlet boundary condition in the proof of the main result the conservation law of mass and the balance of energy and entropy identities are rigorously derived an important argument in the proof is to use a lyapunov type functional related to these physical quantities

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

diffusion subject to the specification of mass abstract existence uniqueness and continuous dependence upon the data are demonstrated for the solution u u x t of the diffusion equation ut uxx s x t 0 u x 0 x 0 ux 1 t g t 0 0 b 0 x dx m 0 a numerical procedure is discussed and results of numerical experiments are presented

on the initial boundary value problem for the boltzmann equation a theorem on the traces of the solutions of initial boundary value problems for the boltzmann equation is proved this result makes it possible to extend a recent theorem of existence proved by hamdache to more realistic situations

the boltzmann equation with weakly inhomogeneous data in bounded domain this paper is concerned with the boltzmann equation with specular reflection boundary condition we construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum the result extends the case of cauchy problem considered by arkeryd esposito pulvirenti 4 to the specular reflection boundary value problem in bounded domain

harmonic analysis on nilpotent groups and singular integrals i oscillatory integrals abstract this paper is devoted to the study of the operator given by where k is a standard calderon zygmund kernel and p is a real polynomial on r n r n we show t is bounded on lp to itself when 1

initial boundary value problems for the boltzmann equation global existence of weak solutions for an open set of 3 bounded or not we consider initial boundary value problems for the boltzmann equation for general gas surface interaction laws and for hard potentials we prove a global existence result for weak solutions the proof uses the regularization of the collision operator and the renormalization method for the regularized problem by using weak compactness in l1 and averaged stability ofq f f we prove the existence of weak solutions of our problem

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

the boltzmann equation with large amplitude initial data in bounded domains abstract the paper is devoted to constructing the global solutions around global maxwellians to the initial boundary value problem on the boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries we allow a class of non negative initial data which have arbitrary large amplitude and even contain vacuum the result shows that the oscillation of solutions away from global maxwellians becomes small after some positive time provided that they are initially close to each other in l 2 this yields the disappearance of any initial vacuum and the exponential convergence of large amplitude solutions to equilibrium in large time the isothermal diffuse reflection boundary condition plays a vital role in the analysis the most key ingredients in our strategy of the proof include i l x v 2 l x l v 1 l x v estimates along a bootstrap argument ii pointwise estimates on the upper bound of the gain term by the product of l norm and l 2 norm iii an iterative procedure on the nonlinear term

uniform estimates for oscillatory integral operators abstract in this paper we prove the uniform l p boundedness of oscillatory singular integral operators with smooth phase functions which satisfy a certain finite type condition these phase functions include the class of real analytic functions the uniform boundedness does not hold if the finite type condition is removed we also obtained estimates for related operators including an optimal l 2 estimate for operators with singular kernels whose phase functions have non singular hessians