chebyshev fourier spectral methods in bipolar coordinates bipolar coordinates provide an efficient cartography for a variety of geometries the exterior of two disks or cylinders a half plane containing a disk an eccentric annulus with a small disk offset from the center of an outer boundary that is a large circle and so on a pseudospectral method that employs a tensor product basis of fourier functions in the cyclic coordinate and chebyshev polynomials in the quasi radial coordinate gives easy to program spectral accuracy we show however that as the inner disk becomes more and more offset from the center of the outer boundary circle the grid is increasingly non uniform and the rate of exponential convergence increasingly slow one dimensional coordinate mappings significantly reduce the non uniformity in spite of this non uniformity the chebyshev fourier method is quite effective in an idealized model of the wind driven ocean circulation resolving both internal and boundary layers bipolar coordinates are also a good starting point for solving problems in a domain which is not one of the bipolar compatible domains listed above but is a sufficiently small perturbation of such this is illustrated by applying boundary collocation with bipolar harmonics to solve laplace s equation in a perturbed eccentric annulus in which the disk shaped island has been replaced by an island bounded by an ellipse similarly a perturbed bipolar domain can be mapped to an eccentric annulus by a smooth change of coordinates

implicit spectrally accurate method for moving boundary problems using immersed boundary conditions concept a fully implicit spectral algorithm for the analysis of moving boundary problem is described the algorithm is based on the concept of immersed boundary conditions ibc i e the computational domain is fixed while the time dependent physical domain is submerged inside the computational domain and is described in the context of the diffusion type problems the physical conditions along the edges of the physical domain are treated as internal constraints the method eliminates the need for adaptive grid generation that follows evolution of the physical domain and provides sharp resolution of the location of the boundary various tests confirm the spectral accuracy in space and the first and second order accuracy in time the computational cost advantage of the ibc method as compared with the more traditional algorithm based on the mapping concept is demonstrated

approximate controllability of fractional stochastic differential equations driven by mixed fractional brownian motion via resolvent operators abstractin this manuscript we investigate the approximate controllability of a class of fractional stochastic differential equations driven by mixed fractional brownian motion in hilbert space with hurst parameter h 12 1 the main results are obtained by employing fractional calculus analytic resolvent operators and with the help of schaefer s fixed point theorem under the assumption that the corresponding linear system is approximately controllable finally in order to validate the obtained theoretical results an illustrative example is provided

on complex singularities of solutions of the equation ℋux u up 0 it has been proved recently by bona and li yi bona j l and li yi a 1997 j math pure appl 76 377 that solitary wave solutions of a certain class of nonlinear nonlocal equations can be extended into the complex domain in this paper we present an approach for the study of singularities of such extensions this approach is applied to the localized solutions of the equation hux u u p 0 p 3 4 5 where h is the hilbert transform the location of the closest to real axis singularity z z0 was found numerically the analysis of this type of singularity shows that in the vicinity of z z0 these complex extensions of solutions cannot be represented by the series 1 z z0 n 0 an z z0 n i e they do not correspond to some power of a meromorphic function

on controllability of linear stochastic systems we discuss several concepts of controllability for partially observable stochastic systems complete controllability approximate controllability controllability and s controllability and show that complete and approximate controllability notions are equivalent and in turn are equivalent to the controllability for linear stochastic systems controlled by gaussian processes we derive necessary and sufficient conditions for these concepts of controllability these criteria reduce to the well known rank condition

controllability of fractional integro differential evolution equations with nonlocal conditions this paper concerns the controllability for a class of fractional integro differential evolution equations with nonlocal initial conditions by using the fractional calculus measure of noncompactness and the monch fixed point theorem we obtain a controllability result for the nonlocal cauchy problem of the fractional integro differential evolution equations involving noncompact semigroups and the nonlocal functions without lipschitz continuity an example is given to illustrate the effectiveness of the abstract results

a direct spectral method for determination of flows over corrugated boundaries a completely spectral algorithm for analysis of flows over corrugated boundaries is proposed the algorithm treats the flow problem as an internal rather then a boundary value problem where the flow conditions are specified along a line in the interior of the computational domain the method eliminates the need for a coordinate generation and or premapping required for regularization of the computational domain in standard implementations of spectral discretizations various tests confirm the spectral accuracy of the algorithm

controllability for a class of integro differential evolution equations involving non local initial conditions abstractthis paper deals with the exact controllability for a class of first order integro differential evolution equations involving non local initial conditions by using sadovskii fixed point theorem exact controllability results are obtained without assuming the compactness and lipschitz conditions for non local functions an example is given to illustrate the effectiveness of the abstract results

nonlocal cauchy problems governed by compact operator families abstract we study the existence of mild solutions for the nonlocal cauchy problem u t au t f t u t t t 0 t 0 t u t 0 g u u 0 and give new criterions which extend some existing results in this area

approximate controllability of fractional order semilinear systems with bounded delay abstract in this paper sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order the existence and uniqueness of mild solution of the system is also proved the results are obtained by using contraction principle and the schauder fixed point theorem some examples are given to illustrate the theory

spectral radial basis functions for full sphere computations the singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function there are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry one choice is methods based on standard chebyshev polynomials although these may be efficiently computed using fast transforms differentiability to all orders of the obtained solution at the origin is not guaranteed the second is the so called one sided jacobi polynomials that explicitly satisfy the required behavioural conditions in this paper we compare these two approaches in their accuracy differentiability and computational speed we find that the most accurate and concise representation is in terms of one sided jacobi polynomials however due to the lack of a competitive fast transform chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently although not infinitely differentiable solutions provided they are adequately converged

existence and uniqueness of periodic solution for a class of semilinear evolution equations this paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a hilbert space h u t au t f t u t 0 t u 0 u where a d a h h is a positive definite self adjoint operator 0 and f 0 h h satisfy caratheodory condition we present some spectral conditions for the nonlinearity f t u to guarantee the existence and uniqueness these spectral conditions are the generalization for nonresonance condition of the self adjoint elliptic boundary value problems

controllability for a class of fractional neutral integro differential equations with unbounded delay abstract in this paper we consider a class of fractional neutral integro differential equations with unbounded delay in banach spaces this paper deals with the exact controllability for fractional integro differential neutral control systems first we establish a new set of sufficient conditions for the exact controllability of nonlinear fractional systems by using a fixed point analysis approach further we extend the result to study the exact controllability concept with nonlocal conditions examples are provided to illustrate our main results

manifold regularized correlation object tracking in this paper we propose a manifold regularized correlation tracking method with augmented samples to make better use of the unlabeled data and the manifold structure of the sample space a manifold regularization based correlation filter is introduced which aims to assign similar labels to neighbor samples meanwhile the regression model is learned by exploiting the block circulant structure of matrices resulting from the augmented translated samples over multiple base samples cropped from both target and nontarget regions thus the final classifier in our method is trained with positive negative and unlabeled base samples which is a semisupervised learning framework a block optimization strategy is further introduced to learn a manifold regularization based correlation filter for efficient online tracking experiments on two public tracking data sets demonstrate the superior performance of our tracker compared with the state of the art tracking approaches

gridless spectral algorithm for stokes flow with moving boundaries a gridless spectrally accurate algorithm for the stokes flow with moving boundaries is presented the algorithm uses fixed computational domain with boundaries of the flow domain moving inside the computational domain the spatial discretization is based on the fourier expansions in the streamwise direction and the chebyshev expansions in the transverse direction temporal discretization uses one and two steps implicit formulations the boundary conditions on the moving boundaries are imposed using the immersed boundary conditions concept numerical tests confirm the spectral accuracy in space and theoretically predicted accuracy in time different variants of the solution procedure are presented and their relative advantages are discussed

numerical and perturbative computations of solitary waves of the benjamin ono equation with higher order nonlinearity using christov rational basis functions computation of solitons of the cubically nonlinear benjamin ono equation is challenging first the equation contains the hilbert transform a nonlocal integral operator second its solitary waves decay only as o 1 x 2 to solve the integro differential equation for waves traveling at a phase speed c we introduced the artificial homotopy h u x x c u 1 d u 2 du 3 0 d 0 1 and solved it in two ways the first was continuation in the homotopy parameter d marching from the known benjamin ono soliton for d 0 to the cubically nonlinear soliton at d 1 the second strategy was to bypass continuation by numerically computing perturbation series in d and forming pade approximants to obtain a very accurate approximation at d 1 to further minimize computations we derived an elementary theorem to reduce the two parameter soliton family to a parameter free function the soliton symmetric about the origin with unit phase speed solitons for higher order benjamin ono equations are also computed and compared to their korteweg devries counterparts all computations applied the pseudospectral method with a basis of rational orthogonal functions invented by christov which are eigenfunctions of the hilbert transform

existence and asymptotic stability of periodic solution for evolution equations with delays abstract in this paper we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a hilbert space h u t a u t f t u t u t 1 u t n t r where a d a h h is a positive definite selfadjoint operator f r h n 1 h is a nonlinear mapping which is periodic in t and 1 2 n are positive constants we present essential conditions on the nonlinearity f to guarantee that the equation has periodic solutions or an asymptotically stable periodic solution the discussion is based on analytic semigroups theory and an integral inequality with delays

semilinear integrodifferential equations with nonlocal initial conditions abstract of concern is the cauchy problem for semilinear integrodifferential equations with nonlocal initial conditions under general and natural hypotheses we establish some new theorems about the existence and uniqueness of solutions for the cauchy problem as a consequence we unify and extend the corresponding theorems given previously for the cauchy problem for differential equations or integrodifferential equations with nonlocal initial conditions moreover we present two examples one of which comes from heat conduction in materials with memory to show that the existing results are not applicable to them in contrast with ours

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

controllability for a class of fractional order neutral evolution control systems abstract in this paper we consider a class of fractional neutral control systems governed by abstract nonlinear fractional neutral differential equations this paper deals with the exact controllability for fractional differential neutral control systems first we establish a new set of sufficient conditions for the controllability of nonlinear fractional systems by using a fixed point analysis approach further we extend the result to study the controllability concept with nonlocal conditions in particular the controllability of nonlinear systems is established under the natural assumption that the associated linear control system is exactly controllable

a computational method of solving free boundary problems in vortex dynamics abstract a general method for computing two dimensional and axisymmetric three dimensional solutions of the steady euler equations is developed the method is based upon variational principles which are formulated entirely in terms of the natural conserved quantities for the unsteady euler equations the vorticity fields considered have a free boundary and may be defined either in a domain with boundary or in free space moreover the common restriction to the case of constant vorticity regions is not imposed the global convergence of the resulting iterative solver is established the numerical implementation of the method is discussed and several computational examples are given

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

spectrally accurate algorithm for moving boundary problems for the navier stokes equations a spectral algorithm based on the immersed boundary conditions ibc concept is developed for simulations of viscous flows with moving boundaries the algorithm uses a fixed computational domain with flow domain immersed inside the computational domain boundary conditions along the edges of the time dependent flow domain enter the algorithm in the form of internal constraints spectral spatial discretization uses fourier expansions in the stream wise direction and chebyshev expansions in the normal to the wall direction up to fourth order implicit temporal discretization methods have been implemented it has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed the new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements