quadratic spline collocation methods for two point boundary value problems a new collocation method based on quadratic splines is presented for second order two point boundary value problems first o h4 approximations to the first and second derivative of a function are derived using a quadratic spline interpolant of u then these approximations are used to define an o h4 perturbation of the given boundary value problem second the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal o h3 j global estimate for the jth derivative of the error is derived further o h4 j error bounds for the jth derivative are obtained for certain superconvergence points it should be observed that standard collocation at midpoints gives o h2 j bounds results from numerical experiments are reported that verify the theoretical behaviour of the method

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

optimal stopping time problem for stochastic navier stokes equations and infinite dimensional variational inequalities we prove the existence of a solution for the obstacle problem associated with the kolmogorov operator corresponding to the stopping time problem for stochastic navier stokes equations in 2 d 2005 elsevier ltd all rights reserved

matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions abstract bialecki b and g fairweather matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions journal of computational and applied mathematics 46 1993 369 386 in recent years several matrix decomposition algorithms have been developed for the efficient solution of the linear algebraic systems arising when finite difference finite element galerkin and spectral methods are applied to separable elliptic boundary value problems in a rectangle the success of these methods depends on knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two point boundary value problems in one space dimension the primary purpose of this paper is to provide an overview of matrix decomposition algorithms and show how they can be expressed in terms of a unifying framework particular emphasis is placed on algorithms formulated recently by the authors for solving the linear systems arising in orthogonal spline collocation that is spline collocation at gauss points all of the methods discussed in this paper are modular and possess a great deal of natural parallelism

basis spline collocation method for solving the schrodinger equation in axillary symmetric systems we use basis spline collocation to solve the schrodinger equation on a lattice for axially symmetric systems with and without spin the focus of the present work is on systems which have continuum states weakly bound states or strong spin orbit coupling since these are the most difficult to solve on the lattice a brief overview of the basis spline collocation method is included which concentrates primarily on those aspects of the theory which are relevant to its application in cylindrical coordinates to demonstrate the method we solve several model problems selected from the fields of atomic and nuclear physics

on the existence of the compact global attractor for semilinear reaction diffusion systems on rn we show that a class of reaction diffusion systems on rngenerates an asymptotically compact semiflow on the banach space of bounded uniformly continuous functions if such a semiflow is dissipative then a unique non empty compact minimal attractor is known to exist we apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on rnthat contain appropriate weight functions we also state conditions which guarantee that the attractor has finite hausdorff dimension

matrix decomposition algorithms for modified spline collocation for helmholtz problems we consider the solution of various boundary value problems for the helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal fourth order approximations for the solution of the collocation equations we formulate matrix decomposition algorithms fast direct methods which employ fast fourier transforms and require o n2 log n operations on an n n uniform partition of the unit square a computational study confirms the published analysis for the dirichlet problem and indicates that similar results hold for neumann mixed and periodic boundary conditions the numerical results also exhibit superconvergence phenomena not reported in earlier studies

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

generalized solutions of hjb equations applied to stochastic control on hilbert space abstract in this paper we prove the existence of borel measurable optimal control laws for infinite dimensional nonstationary stochastic control problems on hilbert space this is based on some recent results on existence of generalized solutions of hjb equations on hilbert space l2 h where is a suitable reference measure on h for illustration some examples are presented

modified nodal cubic spline collocation for poisson s equation we present a new modified nodal cubic spline collocation scheme for solving the dirichlet problem for poisson s equation on the unit square we prove existence and uniqueness of a solution of the scheme and show how the solution can be computed on an n 1 times n 1 uniform partition of the square with cost o n 2 mbox log n using a direct fast fourier transform method using two comparison functions we derive an optimal fourth order error bound in the continuous maximum norm we compare our scheme with other modified nodal cubic spline collocation schemes in particular the one proposed by houstis vavalis and rice in siam j numer anal 25 1988 pp 54 74 we believe that our paper gives the first correct convergence analysis of a modified nodal cubic spline collocation for solving partial differential equations

attractors of partial differential evolution equations in an unbounded domain there is a large number of papers in which attractors of parabolic reaction diffusion equations in bounded domains are investigated in this paper these equations are considered in the whole unbounded space and a theory of attractors of such equations is built while investigating these equations specific difficulties arise connected with the noncompactness of operators with the continuity of their spectra etc therefore some new conditions on nonlinear terms arise in this paper weighted spaces are widely applied an important feature of this problem is worth mentioning namely properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ

convergence of o h 4 cubic spline collocation methods for elliptic partial differential equations this paper presents a new class of collocation methods using cubic splines for solving elliptic partial differential equations pdes the error bounds obtained for these methods are optimal the methods are formulated and a convergence analysis is carried out for a broad class of elliptic pdes experimental results confirm the optimal convergence and indicate that these methods are computationally more efficient than methods based on either collocation with hermite cubics or on the galerkin method with cubic splines various direct and iterative methods have been applied for the solution of cubic spline collocation methods

optimal superconvergent one step nodal cubic spline collocation methods we formulate new optimal fourth order one step nodal cubic spline collocation methods for the solution of various elliptic boundary value problems in the unit square these methods are constructed so that the respective collocation equations can be solved using matrix decomposition algorithms mdas mdas are fast direct methods which employ fast fourier transforms and require o n2 log n operations on an n times n uniform partition of the unit square the results of numerical experiments exhibit expected global optimal orders of convergence as well as desired superconvergence phenomena

ergodic control of semilinear stochastic equations and the hamilton jacobi equation abstract in this paper we consider optimal control of stochastic semilinear equations with lipschitz continuous drift and cylindrical noise we show existence and uniqueness up to an additive constant of solutions to the stationary hamilton jacobi equation associated with the cost functional given by the asymptotic average per unit time cost as a consequence we find the optimizing controls given in the feedback form to obtain these results we prove also some new results on the transition semigroups of semilinear diffusion acting in the spaces of continuous function with the weighted sup norms and on the optimal control of semilinear diffusions for the discounted cost functional

second order hamilton jacobi equations in infinite dimensions some second order hamilton jacobi equations connected to stochastic optimal control problems for infinite dimensional systems driven by a white noise are studied a direct method to prove existence and uniqueness of mild solutions is developed then this solution is identified as the value function of the related stochastic control problem and a feedback formula for optimal controls is derived

attractors for partly dissipative reaction diffusion systems in rn in this paper we study the asymptotic behavior of solutions for the partly dissipative reaction diffusion equations in n we prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in 2 n 2 n l l 2000 academic press

basis spline collocation method for the lattice solution of boundary value problems abstract we study a particular utilization of the basis spline collocation method bscm for the lattice solution of boundary value problems we demonstrate the implementation of a general set of boundary conditions among the selected problems are the schrodinger equation in radial coordinates the poisson and the generalized helmholtz equations in radial and three dimensional cartesian coordinates

stationary hamilton jacobi equations in hilbert spaces and applications to a stochastic optimal control problem we study an infinite horizon stochastic control problem associated with a class of stochastic reaction diffusion systems with coefficients having polynomial growth the hamiltonian is assumed to be only locally lipschitz continuous so that the quadratic case can be covered we prove that the value function v corresponding to the control problem is given by the solution of the stationary hamilton jacobi equation associated with the state system to this purpose we write the hamilton jacobi equation in integral form and by using the smoothing properties of the transition semigroup relative to the state system and the theory of m dissipative operators we show that it admits a unique solution moreover the value function v is obtained as the limit of minima for some approximating control problems which admit unique optimal controls and states

second order hamilton jacobi equations in hilbert spaces and stochastic boundary control the paper is concerned with fully nonlinear second order hamilton jacobi bellman isaacs equations of elliptic type in separable hilbert spaces which have unbounded first and second order terms the viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved a stochastic optimal control problem driven by a parabolic stochastic pde with control of dirichlet type on the boundary is considered it is proved that the value function of this problem is the unique viscosity solution of the associated hamilton jacobi bellman equation

adi methods for cubic spline collocation discretizations of elliptic pde this paper presents the formulation analysis and implementation of alternating direction implicit adi methods for solving the linear system of algebraic equations that arise from the discretization of multidimensional linear elliptic partial differential equations pdes the theoretical analysis is carried out for a broad class of pde problems numerical experiments confirm the theoretically determined characteristics of the adi iterative schemes the computational effectiveness of the proposed schemes is shown through a detailed theoretical complexity analysis confirmed with our experimental data

iterative line cubic spline collocation methods for elliptic partial differential equations in several dimensions this paper presents a new class of second and fourth order line cubic spline collocation methods the lcsc methods for multidimensional linear elliptic partial differential equations with no cross derivative terms the lcsc methods approximate the differential operator along lines in each dimension independently and then combine the results into one large linear system expressed in terms of discretization stencils for the operator these methods have nonzero entries only in the coordinate directions the advantage of this approach is that the discretization is much simpler to derive and analyze further iterative methods are easily applied to the resulting linear systems especially on parallel computers the disadvantage is that the resulting linear system is k times larger in k dimensions using the simplicity of the methods iterative schemes are analyzed and formulated in order to solve the resulting lcsc linear systems in the case of helmholtz problems block jacobi extrapolated jacobi ej and

ergodic boundary point control of stochastic semilinear systems a controlled markov process in a hilbert space and an ergodic cost functional are given for a control problem that is solved where the process is a solution of a parameter dependent semilinear stochastic differential equation and the control can occur only on the boundary or at discrete points in the domain the linear term of the semilinear differential equation is the infinitesimal generator of an analytic semigroup the noise for the stochastic differential equation can be distributed boundary and point some ergodic properties of the controlled markov process are shown to be uniform in the control and the parameter the existence of an optimal control is verified to solve the ergodic control problem the optimal cost is shown to depend continuously on the system parameter

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s