discrete time orthogonal spline collocation methods for schrodinger equations in two space variables two discrete time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time dependent schrodinger equation in two space variables these are crank nicolson and alternating direction implicit adi schemes employing c1 piecewise polynomial spaces of arbitrary degree ge 3 in each space variable the stability of the schemes is examined and optimal order a priori l2 and h1 error estimates at each time step are derived parallel implementation of the adi scheme is discussed

mean first passage time of surface mediated diffusion in spherical domains we present an exact calculation of the mean first passage time to a target on the surface of a 2d or 3d spherical domain for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion the presented approach is based on an integral equation which can be solved analytically numerically validated approximation schemes which provide more tractable expressions of the mean first passage time are also proposed in the framework of this minimal model of surface mediated reactions we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface

split step orthogonal spline collocation methods for nonlinear schrodinger equations in one two and three dimensions abstract split step orthogonal spline collocation osc methods are proposed for one two and three dimensional nonlinear schrodinger nls equations with time dependent potentials firstly the nls equation is split into two nonlinear equations and one or more one dimensional linear equations commonly the nonlinear subproblems could be integrated directly and accurately but it fails when the time dependent potential cannot be integrated exactly in this case we propose three approximations by using quadrature formulae but the split order is not reduced discrete time osc schemes are applied for the linear subproblems in numerical experiments many tests are carried out to prove the reliability and efficiency of the split step osc ssosc methods solitons in one two and three dimensions are well simulated and conservative properties and convergence rates are demonstrated we also apply the ways of solving the nonlinear subproblems to the split step finite difference ssfd methods and the time splitting spectral tssp methods and the approximate ways still work well finally we apply the ssosc methods to solve some problems of bose einstein condensates

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

asymptotically linear cooperative elliptic system existence and multiplicity abstract in this paper we consider an asymptotically linear cooperative elliptic system at resonance by a penalization technique and by the use of morse theory we prove firstly the existence of a non zero solution if the difference between the morse index at zero and the morse index at infinity is big enough later on we will prove the existence of n 1 pairs of non trivial solutions where n is roughly speaking the difference between the morse index at zero and the morse index at infinity

a nonoverlapping domain decomposition method for orthogonal spline collocation problems a nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the dirichlet boundary value problem for poisson s equation on an l shaped region we prove existence and uniqueness of the collocation solution and derive optimal order hs norm error bounds for s 0 1 2 the collocation solution on two interfaces is computed using the preconditioned conjugate gradient method and the collocation solution on three squares is computed by a matrix decomposition method that uses fast fourier transforms the total cost of the algorithm is o n2 log n where the number of unknowns in the collocation solution is o n2

an orthogonal spline collocation alternating direction implicit crank nicolson method for linear parabolic problems on rectangles we formulate an alternating direction implicit crank nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous dirichlet boundary condition orthogonal spline collocation with piecewise hermite bicubics is used for spatial discretization we show that for sufficiently small time stepsize the scheme is stable and of optimal order accuracy in time and the h1 norm in space we also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms

the mean first rotation time of a planar polymer we estimate the mean first time called the mean rotation time mrt for a planar random polymer to wind around a point this polymer is modeled as a collection of n rods each of them being parameterized by a brownian angle we are led to study the sum of i i d imaginary exponentials with one dimensional brownian motions as arguments we find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function finally we obtain an asymptotic formula for the mrt whose leading order term depends on the square root of n and interestingly depends weakly on the mean initial configuration our analytical results are confirmed by brownian simulations our analytical results are confirmed by brownian simulations

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems fast direct methods are presented for the solution of linear systems arising in high order tensor product orthogonal spline collocation applied to separable second order linear elliptic partial

analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables optimal order error estimates are derived for the continuous time collocation method and a discrete time collocation method the crank nicolson collocation method for approximating the solution of a semilinear parabolic initial boundary value problem in r times 0 t where r is the unit square in each case at each time level the approximation is a c 1 piecewise polynomial of degree r geqq 3 defined by collocation at gauss points in r optimal order error estimates are also derived for a family of discrete time collocation methods for a semilinear second order hyperbolic initial boundary value problem in r times 0 t

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

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

narrow escape problem with a mixed trap and the effect of orientation we consider the mean first passage time mfpt of a two dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections high order asymptotic formulas for the mfpt and the fundamental eigenvalue of the laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture we obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a green s function and a certain alignment vector we find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle in the scenario where the trap undergoes prescribed motion in the domain the mfpt is seen to be particularly sensitive to the orientation of the trap

a local min max orthogonal method for finding multiple solutions to noncooperative elliptic systems a local min max orthogonal method together with its mathematical justification is developed in this paper to solve noncooperative elliptic systems for multiple solutions in an order first it is discovered that a noncooperative system has the nature of a zero sum game a new local characterization for multiple unstable solutions is then established under which a stable method for multiple solutions is developed numerical experiments for two types of noncooperative systems are carried out to illustrate the new characterization and method several important properties for the method are explored or verified multiple numerical solutions are found and presented with their profiles and contour plots as a new bifurcation phenomenon multiple asymmetric positive solutions to the second type of noncooperative systems are discovered numerically but are still open for mathematical verification

analytical solutions for an escape problem in a disc with an arbitrary distribution of exit holes along its boundary we analytically construct solutions for the mean first passage time and splitting probabilities for the escape problem of a particle moving with continuous brownian motion in a confining planar disc with an arbitrary distribution i e of any number size and spacing of exit holes absorbing sections along its boundary the governing equations for these quantities are poisson s equation with a non zero constant forcing term and laplace s equation respectively and both are subject to a mixture of homogeneous neumann and dirichlet boundary conditions our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map using special transcendental functions that are defined in terms of an associated schottky group they are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over no slip no shear surfaces as well as results from potential theory all of which were themselves derived using the same theory of schottky groups they are exact up to the determination of a finite set of mapping parameters which is performed numerically their evaluation also requires the numerical inversion of the parameterising conformal map computations for a series of illustrative examples are also presented

on optimal order error estimates for the nonlinear schro dinger equation implicit runge kutta methods in time are used in conjunction with the galerkin method in space to generate stable and accurate approximations to solutions of the nonlinear cubic schrodinger equation the temporal component of the discretization error is shown to decrease at the classical rates in some important special cases

fast direct solvers for piecewise hermite bicubic orthogonal spline collocation equations fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise hermite bicubics is employed for the approximate solution of poisson s

orthogonal spline collocation for nonlinear dirichlet problems we study the orthogonal spline collocation osc solution of a homogeneous dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation the approximate solution is sought in the space of hermite bicubic splines we prove local existence and uniqueness of the osc solution obtain optimal order h1 and h2 error estimates and prove the quadratic convergence of newton s method for solving the osc problem

asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture abstract let x t be a positive recurrent diffusion process corresponding to an operator l on a domain d r d with oblique reflection at d if d r d for each x d we define a volume preserving norm that depends on the diffusion matrix a x we calculate the asymptotic behavior as e 0 of the expected hitting time of the e ball centered at x and of the principal eigenvalue for l in the exterior domain formed by deleting the ball with the oblique derivative boundary condition at d and the dirichlet boundary condition on the boundary of the ball this operator is non self adjoint in general the behavior is described in terms of the invariant probability density at x and det a x in the case of normally reflected brownian motion the results become isoperimetric type equalities

unconditional stability and convergence of crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system the paper is concerned with the unconditional stability and optimal l 2 error estimates of linearized crank nicolson galerkin fems for a nonlinear schrodinger helmholtz system in mathbb r d d 2 3 by introducing a corresponding time discrete system we separate the error into two parts i e the temporal error and the spatial error since the latter is tau independent the uniform boundedness of numerical solutions in l infty norm follows an inverse inequality immediately without any restrictions on time stepsize then optimal error estimates are obtained in a routine way numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results

orthogonal spline collocation methods for the subdiffusion equation we develop two kinds of numerical schemes to efficiently solve the subdiffusion equation which is used to describe anomalous subdiffusive transport processes the time fractional derivative is first discretized by l1 approximation and the grunwald letnikov approximation respectively then we use the orthogonal spline collocation method to approximate the two semi discretized subdiffusion equations the stability and convergence of time semi discretization and full discretization schemes are both established strictly for the two schemes both of them are unconditionally stable numerically the convergent orders in space including the solution and its first derivative are four for the hermite cubic spline approximation and theoretically we get that at least the solution itself has a fourth order convergent rate extensive numerical results are presented to show the convergent order and robustness of the numerical schemes

discrete time orthogonal spline collocation methods for the nonlinear schrodinger equation with wave operator in this paper discrete time orthogonal spline collocation schemes are proposed for the nonlinear schrodinger equation with wave operator these schemes are constructed by using orthogonal spline collocation approaches combined with finite difference methods the conservative property the convergence and the stability of these methods are theoretically analyzed and also verified by extensive numerical experiments in addition some interesting phenomena which require further theoretical analysis are discussed numerically

narrow escape part iii non smooth domains and riemann surfaces we consider the narrow escape problem in two dimensional riemannian manifolds with a metric g with corners and cusps in an annulus and on a sphere specifically we calculate the mean time it takes a brownian particle diffusing in a domain to reach an absorbing window when the ratio varepsilon frac partial omega a g partial omega g between the absorbing window and the otherwise reflecting boundary is small if the boundary is smooth as in the cases of the annulus and the sphere the leading term in the expansion is the same as that given in part i of the present series of papers however when it is not smooth the leading order term is different if the absorbing window is located at a corner of angle then e tau frac omega g alpha d log frac 1 varepsilon o 1 if near a cusp then e tau grows algebraically rather than logarithmically thus in the domain bounded between two tangent circles the expected lifetime is e tau frac omega d 1 1 d frac 1 varepsilon o 1 where d 1 is the ratio of the radii for the smooth boundary case we calculate the next term of the expansion for the annulus and the sphere it can also be evaluated for domains that can be mapped conformally onto an annulus this term is needed in real life applications such as trafficking of receptors on neuronal spines because log frac 1 varepsilon is not necessarily large even when varepsilon frac partial omega a g partial omega g is small in these two problems there are additional parameters that can be small such as the ratio of the radii of the annulus the contributions of these parameters to the expansion of the mean escape time are also logarithmic in the case of the annulus the mean escape time is e tau frac omega g pi d log frac 1 varepsilon frac 12 log frac 1 delta o 1

galerkin alternating direction methods for nonrectangular regions using patch approximations galerkin alternating direction procedures are considered for the nonlinear parabolic equation u t sum i j 1 n frac partial partial xi i left tilde a ij xi u frac partial u partial xi j right sum i 1 n tilde b i xi u frac partial u partial xi i f xi t u which is given on a nonrectangular curved region we use isoparametric elements and an approximation to the jacobian of the isoparametric map which is based on patches of finite elements with these procedures a multidimensional problem can be solved as a series of one dimensional problems optimal order l 2 and h 1 error estimates as well as optimal order work estimates are derived

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems on evolving domains we consider the approximate solution of nonlinear reaction diffusion systems on evolving domains that arise in a variety of areas including biology chemistry ecology and physics by mapping a fixed domain onto the evolving domain at each time level we generalize to evolving domains the adi extrapolated crank nicolson orthogonal spline collocation technique developed in 8 9 for fixed domains the new method is tested on the schnakenberg model and we demonstrate numerically that it preserves the second order accuracy in time and optimal accuracy in space for piecewise hermite cubics in various norms moreover the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known

diffusion escape through a cluster of small absorbing windows we study the first eigenvalue of the laplace equation in a bounded domain in r d d 2 3 with mixed neumann dirichlet zaremba boundary conditions the neumann condition is imposed on most of the boundary and the dirichlet boundary consists of a cluster of small windows when the windows are well separated the first eigenvalue is asymptotically the sum of eigenvalues of mixed problems with a single dirichlet window however when two or more dirichlet windows cluster tightly together they interact nonlinearly we compare our asymptotic approximation of the eigenvalue to the escape rate of simulated brownian particles through the small windows

a mixed volume element with upwind multistep mixed volume element and convergence analysis for numerical simulation of nuclear waste contaminant disposal abstract in this paper the authors discuss a numerical simulation problem of three dimensional contamination treatment from nuclear waste the mathematical model is defined by a nonlinear system of an initial boundary problem of four partial differential equations an elliptic equation for the fluid pressure three convection diffusion equations for the brine radionuclides and heat the pressure appears within the concentration equations and heat conduction equation and the darcy velocity controls the concentrations and the temperature the pressure is solved by the conservative mixed volume element method and the order of accuracy is improved by the darcy velocity the concentrations and temperature are treated by a combination of a multistep method the upwind approximation and a mixed volume element the partial derivatives to time variable are approximated by a multistep method a mixed volume element and an upwind scheme are used to discretize the diffusion and the convection respectively the composite method can solve the convection dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy the mixed volume element has the local conservation of mass and energy and it can obtain the concentrations and temperature and their adjoint vector functions simultaneously the conservation nature plays an important role in numerical simulation of underground fluid by the technique of a priori estimates of differential equations we derive an optimal second order result in l 2 norm numerical examples are given to show the effectiveness and practicability of our approach and the composite method is testified as a powerful tool for solving the challenging benchmark problem

discrete time orthogonal spline collocation methods for vibration problems discrete time orthogonal spline collocation schemes are formulated and analyzed for vibration problems involving various boundary conditions each problem is written as a schrodinger type system which is then approximated by crank nicolson and or alternating direction implicit orthogonal spline collocation schemes these schemes are shown to be second order accurate in time and of optimal order accuracy in space in the hm norm m 1 2

an adi extrapolated crank nicolson orthogonal spline collocation method for nonlinear reaction diffusion systems an alternating direction implicit adi orthogonal spline collocation osc method is described for the approximate solution of a class of nonlinear reaction diffusion systems its efficacy is demonstrated on the solution of well known examples of such systems specifically the brusselator gray scott gierer meinhardt and schnakenberg models and comparisons are made with other numerical techniques considered in the literature the new adi method is based on an extrapolated crank nicolson osc method and is algebraically linear it is efficient requiring at each time level only o n operations where n is the number of unknowns moreover it is shown to produce approximations which are of optimal global accuracy in various norms and to possess superconvergence properties

exact mean exit time for surface mediated diffusion we present an exact expression for the mean exit time through the cap of a confining sphere for particles alternating phases of surface and of bulk diffusion the present approach is based on an integral equation which can be solved analytically in contrast to the statement of berezhkovskii and barzykin j chem phys 136 54115 2012 we show that the mean exit time can be optimized with respect to the desorption rate under analytically determined criteria

discontinuous galerkin finite element approximation of nondivergence form elliptic equations with cordes coefficients nondivergence form elliptic equations with discontinuous coefficients do not generally possess a weak formulation thus presenting an obstacle to their numerical solution by classical finite element methods we propose a new hp version discontinuous galerkin finite element method for a class of these problems which satisfy the cordes condition it is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size h and suboptimal with respect to the polynomial degree p by only half an order numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under hp refinement for problems with discontinuous coefficients and nonsmooth solutions

hybrid asymptotic numerical approach for estimating first passage time densities of the two dimensional narrow capture problem a hybrid asymptotic numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two dimensional domain with a reflecting boundary as motivation for this study we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean this implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined to solve the underlying diffusion equation the method of laplace transforms is used to obtain an elliptic problem of modified helmholtz type in the limit of vanishing trap sizes each trap is represented as a dirac point source that permits the solution of the transform equation to be represented as a superposition of helmholtz green s functions using this solution we construct asymptotic short time solutions of the first passage time density which captures peaks associated with rapid capture by the absorbing traps when numerical evaluation of the helmholtz green s function is employed followed by numerical inversion of the laplace transform the method reproduces the density for larger times we demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time dependent solution of the diffusion equation and discrete particle simulations in particular we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged the hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes

orthogonal spline collocation method for the fourth order diffusion system abstract the fourth order diffusion systems depict the wave and photon propagation in intense laser beams and play an important role in the phase separation in binary mixture in this paper by using orthogonal spline collocation osc method in spatial direction and classical l1 approximation in temporal direction a fully discrete scheme is established for a class of fourth order fractional reaction diffusion equations for the original unknown u and auxiliary variable v u the full discrete unconditional stabilities based on a priori analysis are derived by virtue of properties of osc moreover the convergence rates in l 2 norm for unknown u are strictly investigated at the same time the optimal error estimates in h 1 norm for unknown u and in l 2 norm for variable v are also derived respectively for further verifying the theoretical analysis some numerical examples are provided

analytical and numerical results for an escape problem a particle moves with brownian motion in a unit disc with reflection from the boundaries except for a portion called a window or gate in which it is absorbed the main problems are to determine the first hitting time and spatial distribution a closed formula for the mean first hitting time is discovered and proven for a gate of any size also given is the probability density of the location where a particle hits if initially the particle is at the center or uniformly distributed numerical simulations of the stochastic process with finite step size and a sufficient number of sample paths are compared with the exact solution to the brownian motion the limit of zero step size providing an empirical formula for the difference histograms of first hitting times are also generated

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

optimal error analysis of cranknicolson schemes for a coupled nonlinear schrdinger system in 3d the paper is concerned with the time step condition of the commonly used semi implicit cranknicolson finite difference schemes for a coupled nonlinear schrdinger system in three dimensional space we present the optimal l2 error estimate without any restriction on time step while all previous works require certain time step conditions our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate inequality of the error function numerical examples for both two dimensional and three dimensional models are investigated and numerical results illustrate our theoretical analysis

solutions for resonant elliptic systems with nonodd or odd nonlinearities we consider by minimax techniques a class of cooperative elliptic systems at resonance if the nonlinear term is nonodd we establish some new existence theorems about nontrivial solutions if the nonlinearity is odd we obtain existence of infinitely many solutions a topic about which very little is known

a preconditioned conjugate gradient method for nonselfadjoint or indefinite orthogonal spline collocation problems we study the computation of the orthogonal spline collocation solution of a linear dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form lu sum a ij x u x ix j sum b i x u x i c x u we apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator and prove that the resulting algorithm has a convergence rate independent of the partition step size we solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm on a uniform n n partition the cost of the algorithm for computing the collocation solution within a tolerance epsilon is o n 2 ln n ln epsilon

convergence analysis of orthogonal spline collocation for elliptic boundary value problems existence uniqueness and optimal order h2 h1 and l2 error bounds are established for the orthogonal spline collocation solution of a dirichlet boundary value problem on the unit square the linear elliptic nonself adjoint partial differential equation is given in nondivergence form the approximate solution which is a tensor product of continuously differentiable piecewise polynomials of degree r geq 3 is determined by satisfying the partial differential equation at the nodes of a composite gauss quadrature

the narrow escape problem the narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain the problem is equivalent to solving the mixed dirichlet neumann boundary value problem for the poisson equation with small dirichlet and large neumann parts the mixed boundary value problem which goes back to lord rayleigh originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the neumann problem in domains with bottlenecks we review here recent developments in the non standard asymptotics of the problem which are based on several ingredients a better resolution of the singularity of neumann s function resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks and the breakup of composite domains into simpler components the new methodology applies to two and higher dimensional problems selected applications are r

first passage quantities of brownian motion in a bounded domain with multiple targets a unified approach in this paper we introduce a general computation method to systematically determine the mean first passage time mfpt the global mean first passage time gmfpt and splitting probabilities for a continuous brownian motion in a confined 2d or 3d domain with multiple absorbing targets in the bulk or on the boundary this method is applied to spherically symmetric domains in the limit of small sized targets and asymptotic expansions of the mpft gmfpt and splitting probabilities are obtained in four distinct cases 3d domains with targets in the bulk 3d domains with targets on the boundary 2d domains with targets on the bulk and 2d domains with targets on the boundary this approach gives a unified description of existing exact results which were obtained using specific technics and also yields new results in particular for n targets splitting probabilities

efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles piecewise hermite bicubic orthogonal spline collocation laplace modified and alternating direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed the schemes are shown to be unconditionally stable and of optimal order accuracy in the h 1 and discrete maximum norms for space and time respectively implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms

maximum norm error bound of a linearized difference scheme for a coupled nonlinear schrodinger equations in this article a linearized conservative difference scheme for a coupled nonlinear schrodinger equations is studied the discrete energy method and an useful technique are used to analyze the difference scheme it is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm numerical experiments are presented to support the theoretical results

an alternating direction implicit backward differentiation orthogonal spline collocation method for linear variable coefficient parabolic equations we present a new fully discrete method to solve a linear variable coefficient parabolic equation on a rectangular region the scheme is an alternating direction implicit method which uses the third order backward differentiation formula for the time discretization and piecewise hermite bicubic orthogonal spline collocation for the spatial discretization the l 2 norm stability and convergence analysis is carried out for the heat equation the stability analysis reveals that the scheme is stable with respect to the right hand sides but it is unstable with respect to the initial conditions we explain using a damping property of the scheme and round off error analysis why this instability is harmless we also show how a different choice of initial values leads to optimal convergence orders numerical results demonstrate the third order accuracy in time and optimal order accuracy in space using maximum and sobolev norms respectively

narrow escape problem for the unit sphere homogenization limit optimal arrangements of large numbers of traps and the n 2 conjecture a narrow escape problem is considered to calculate the mean first passage time mfpt needed for a brownian particle to leave a unit sphere through one of its n small boundary windows traps a procedure is established to calculate optimal arrangements of n 1 equal small boundary traps that minimize the asymptotic mfpt based on observed characteristics of such arrangements a remarkable property is discovered that is the sum of squared pairwise distances between optimally arranged n traps on a unit sphere is integer equal to n 2 it is observed for 2 n 1004 with high precision it is conjectured that this is the case for such optimal arrangements for all n a dilute trap limit of homogenization theory when n can be used to replace the strongly heterogeneous dirichlet neumann mfpt problem with a spherically symmetric robin problem for which an exact solution is readily found parameters of the robin homogenization problem are computed that capture the first four terms of the asymptotic mfpt close agreement of asymptotic and homogenization mfpt values is demonstrated the homogenization approach provides a radically faster way to estimate the mfpt since it is given by a simple formula and does not involve expensive global optimization to determine locations of n 1 boundary traps

orthogonal spline collocation methods for schr o dinger type equations in one space variable we examine the use of orthogonal spline collocation for the semi discreti za tion of the cubic schr o dinger equation and the two dimensional parabolic equation of tappert in each case an optimal order l 2 estimate of the error in the semidiscrete approximation is derived for the cubic schr o dinger equation we present the results of numerical experiments in which the integration in time is performed using a routine from a software library

h 1 norm error bounds for piecewise hermite bicubic orthogonal spline collocation schemes for elliptic boundary value problems two piecewise hermite bicubic orthogonal spline collocation schemes are considered for the approximate solution of elliptic nonhomogeneous dirichlet boundary value problems on rectangles in the first scheme the nonhomogeneous dirichlet boundary condition is approximated by means of the piecewise hermite cubic interpolant while the piecewise cubic interpolant at the boundary gauss points is used in the second scheme the piecewise hermite bicubic interpolant of the exact solution of the boundary value problem is employed as a comparison function to show that the h 1 norm of the error for each scheme is o h 3

adi spectral collocation methods for parabolic problems we discuss the crank nicolson and laplace modified alternating direction implicit legendre and chebyshev spectral collocation methods for a linear variable coefficient parabolic initial boundary value problem on a rectangular domain with the solution subject to non zero dirichlet boundary conditions the discretization of the problems by the above methods yields matrices which possess banded structures this along with the use of fast fourier transforms makes the cost of one step of each of the chebyshev spectral collocation methods proportional except for a logarithmic term to the number of the unknowns we present the convergence analysis for the legendre spectral collocation methods in the special case of the heat equation using numerical tests we demonstrate the second order accuracy in time of the chebyshev spectral collocation methods for general linear variable coefficient parabolic problems

existence and multiplicity of rotating periodic solutions for resonant hamiltonian systems abstract in the present paper we consider a class of resonant hamiltonian systems x j h x t x in r 2 n we will use saddle point reduction morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions i e x t t q x t for any t r with t 0 and q an symplectic orthogonal matrix in the case q k i 2 n for any positive integer k such a rotating periodic solution is just a quasi periodic solution moreover if h is even in x we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

asymptotically linear system of three equations near resonance abstract this paper deals with the asymptotically linear system u 1 1 u 3 f 1 x u 1 u 2 u 3 in u 2 2 u 2 f 2 x u 1 u 2 u 3 in u 3 3 u 1 f 3 x u 1 u 2 u 3 in u 1 u 2 u 3 0 on where i 0 for i 1 2 3 with 2 1 3 is a real parameter and r n is a bounded domain with smooth boundary the linear part of the system has two simple eigenvalues with nonnegative eigenfunctions each with at least one zero component we provide sufficient conditions which guarantee bifurcation from infinity of positive solutions from both one or none of the two simple eigenvalues under additional assumptions on the nonlinear perturbations we determine the direction of bifurcation as well we use bifurcation theory to establish our results

an asymptotically linear non cooperative elliptic system with lack of compactness in this paper we consider an asymptotically linear non cooperative elliptic system at resonance the functional associated to the system is strongly indefinite and so the morse index of each critical point is infinite to overcome this difficulty we will use morse theory for strongly indefinite functionals developed by abbondandolo we shall prove that if the system has a non resonant solution the morse index of which is different from the morse index at infinity then there exists another solution

alternating direction multistep methods for parabolic problems iterative stabilization efficient multistep procedures for time stepping galerkin methods for parabolic partial differential equations are presented and analyzed the procedures involve using an alternating direction preconditioned iterative method for approximately solving the linear equations arising at each timestep in a discrete galerkin method optimal order convergence rates are obtained for the iterative methods work estimates of almost optimal order are obtained

orthogonal spline collocation methods for biharmonic problems orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square particular attention is given to the dirichlet biharmonic problem which is solved using capacitance matrix techniques

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

a symmetric characteristic fve method with second order accuracy for nonlinear convection diffusion problems the finite volume element fve methods used currently are essentially low order and unsymmetric in this paper by biquadratic elements and multistep methods we construct a second order fve scheme for nonlinear convection diffusion problem on nonuniform rectangular meshes to overcome the numerical oscillation we discretize the problem along its characteristic direction the choice of alternating direction strategy is critical in this paper which guarantees the high efficiency and symmetry of the discrete scheme optimal order error estimates in h 1 norm are derived and a numerical example is given at the end to confirm the usefulness of the method

galerkin alternating direction methods for a kind of nonlinear hyperbolic equations on nonrectangular regions patch approximation and an approximation to the jacobian of the isoparametric map are applied to discuss a three level and a four level galerkin alternating direction procedures for a kind of nonlinear hyperbolic equation which is given on a nonrectangular curved region with these procedures multi dimensional problems can be solved as a series of one dimensional problems and the l 2 norm error estimates are derived

an alternating direction galerkin method for a class of second order hyperbolic equations in two space variables a new alternating direction implicit adi galerkin method is devised and analyzed for solving a certain class of second order hyperbolic initial boundary value problems in two space variables this class includes the wave equation in cartesian coordinates polar coordinates and cylindrical coordinates with radial symmetry optimal a priori h 0 1 and l 2 error estimates are derived

an alternating direction implicit orthogonal spline collocation scheme for nonlinear parabolic problems on rectangular polygons we consider a nonlinear parabolic initial boundary value problem on a rectangular polygon with the solution satisfying robin boundary conditions with variable coefficients an approximation to the solution at the desired time value is obtained using an alternating direction implicit extrapolated crank nicolson scheme in which orthogonal spline collocation with piecewise polynomials of an arbitrary degree is used for spatial discretization at each time level the scheme determines the intermediate solution along horizontal lines and the approximate solution along vertical lines passing through the gauss points only at the last time level is the approximate solution along vertical lines converted into the approximate solution defined on the entire rectangular polygon this property of our approach leads to its efficient implementation and its applicability to rectangular polygons

narrow escape part ii the circular disk we consider brownian motion in a circular disk whose boundary partial omega is reflecting except for a small arc partial omega a which is absorbing as varepsilon partial omega a partial omega decreases to zero the mean time to absorption in partial omega a denoted e tau becomes infinite the narrow escape problem is to find an asymptotic expansion of e tau for varepsilon ll1 we find the first two terms in the expansion and an estimate of the error the results are extended in a straightforward manner to planar domains and two dimensional riemannian manifolds that can be mapped conformally onto the disk our results improve the previously derived expansion for a general domain e tau frac omega d pi big log frac 1 varepsilon o 1 big d is the diffusion coefficient in the case of a circular disk we find that the mean first passage time from the center of the disk is e tau boldmath x 0 bf 0 frac r 2 d big log frac 1 varepsilon log 2 frac 1 4 o varepsilon big the second term in the expansion is needed in real life applications such as trafficking of receptors on neuronal spines because log frac 1 varepsilon is not necessarily large even when e is small we also find the singular behavior of the probability flux profile into partial omega a at the endpoints of partial omega a and find the value of the flux near the center of the window