high order difference schemes for unsteady one dimensional diffusion convection problems abstract for unsteady 1d diffusion convection problems this paper develops an extensive analysis of two level three point finite difference schemes of order 2 in time and 4 in space this general class of fds includes several schemes independently proposed by different authors one main objective is the identification of difference schemes yielding satisfactory numerical results for strongly convective problems i e when the cell reynolds number b h 2 is greater than unity the stability and the oscillatory behaviour of the schemes are carefully studied and the analyses are completed by some numerical experiments we outline some key points i the great difficulty to obtain accurate numerical results for large values of ii the possibility of virtually optimum schemes is essentially theoretical and requires in practice careful experiments iii for strongly convective problems some second order explicit schemes are almost as efficient and less costly than implicit fourth order schemes

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

high order compact finite difference scheme for option pricing in stochastic volatility models we derive a new high order compact finite difference scheme for option pricing in stochastic volatility models the scheme is fourth order accurate in space and second order accurate in time under some restrictions theoretical results like unconditional stability in the sense of von neumann are presented where the analysis becomes too involved we validate our findings by a numerical study numerical experiments for the european option pricing problem are presented we observe fourth order convergence for non smooth payoff

global stabilization of mimo minimum phase nonlinear systems without strict triangular conditions an extension of backstepping to a class of multivariable minimum phase nonlinear systems is proposed the systems are assumed to be in a special interlaced form which includes a lower triangular form as a special case the extension involves the recursive application of backstepping and augmentation

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

the standard h problem and the maximum principle the general linear case the classical relatively simple ideas of linear quadratic lq optimization are used in a time domain treatment of the standard h infty problem given the power of time domain analysis the problem can be solved in the general framework of linear time varying possibly infinite dimensional finite as well as infinite horizon systems under no structural restrictions such as block dimensions zero blocks in the d operator etc in the spirit of recent finite dimensional linear time invariant lti results the solution is given in terms of two coupled riccati equations it includes a criterion for suboptimality and a parametrization of all suboptimal compensators results pertinent to lti periodic and asymptotic systems are obtained as corollaries

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

high order adi scheme for option pricing in stochastic volatility models we propose a new high order alternating direction implicit adi finite difference scheme for the solution of initial boundary value problems of convection diffusion type with mixed derivatives and non constant coefficients as they arise from stochastic volatility models in option pricing our approach combines different high order spatial discretisations with hundsdorfer and verwer s adi time stepping method to obtain an efficient method which is fourth order accurate in space and second order accurate in time numerical experiments for the european put option pricing problem using heston s stochastic volatility model confirm the high order convergence

a second order accurate numerical method for the two dimensional fractional diffusion equation spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow finance and others in this paper we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation this numerical method combines the alternating directions implicit adi approach with a crank nicolson discretization and a richardson extrapolation to obtain an unconditionally stable second order accurate finite difference method the stability and the consistency of the method are established numerical solutions for an example super diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method

multigrid method for fractional diffusion equations the fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted grunwald formula the scheme is unconditionally stable and the coefficient matrix possesses the toeplitz like structure a multigrid method is proposed to solve the resulting system meanwhile the fast toeplitz matrix vector multiplication is utilized to lower the computational cost with only o nlogn complexity where n is the number of the grid points numerical experiments are given to demonstrate the efficiency of the method

branch and bound algorithms for the multi product assembly line balancing problem abstract this paper considers a flexible manufacturing system for several products each requiring a number of sequential nonpreemptive tasks some of which may be common to many products work stations are capable of handling any task the objective is to minimize the number of work stations necessary to manufacture all products so that tasks are performed in the proper order and so that all tasks assigned to any given station can be executed within a prescribed time frame a new branch and bound approach for this problem is presented it contains two main improvements over previously published tree search algorithms an improved lower bounding procedure and a more powerful partitioning scheme the algorithm can be used either as a heuristic in its truncated version or as an exact method in its full version tests performed on randomly generated problems confirm the effectiveness of the proposed improvements

robust stabilization with positive real uncertainty beyond the small gain theory in many applications of feedback control phase information is available concerning the plant uncertainty for example lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions closed loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators in this paper the properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real but otherwise unknown plant uncertainty these results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty one of the principal motivations for these results is to utilize phase information in guaranteeing robust stability in this these sense results go beyond the usual limitations the small gain theorem and quadratic lyapunov functions which may be conservative when phase information is available the results of this paper are based upon a riccati equation formulation of the positive real lemma and thus are in the spirit of recent riccati based approaches to bounded real h control

input output norms in general linear systems upper bounds on the induced norm of the input output operators in linear systems with l2 input and output signals are provided the main idea is to transform the original problem into a parametized set of indefinite lq optimization problems using the classical maximum principle the existence of solutions to the latter is equivalent to the solvability of a boundary value hamiltonian system or equivalenty of an indefinite riccati equation our time domain approach enables the treatment of a very general linear set up including time varying infinite dimensional finite and infinite horizon systems

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

exact and approximation algorithms for the min max k traveling salesmen problem on a tree abstract given k identical salesmen where k 2 is a constant independent of the input size the min max k traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree shaped network so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized the problem is known to be np hard even when k 2 in this paper we have developed a pseudo polynomial time exact algorithm for this problem with any constant k 2 closing a question that has remained open for a decade along with this we have further developed a 1 approximation algorithm for any 0

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

new lower bounds for the three dimensional orthogonal bin packing problem abstract in this paper we consider the three dimensional orthogonal bin packing problem which is a generalization of the well known bin packing problem we present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature the comparison is also done concerning asymptotic worst case performance ratios the new lower bounds can be more efficiently computed in polynomial time in addition we study the non oriented model which allows items to be rotated

on the reduction of certain frequency shaped linear quadratic dissipative design problems to an h sub spl infin formulation this paper concerns a concept of frequency based quadratic dissipativity and its reduction to an equivalent h sub spl infin or spl mu condition the paper is motivated by the observation that a strictly proper system cannot be strictly passive hence results on the synthesis of positive real systems using the cayley transform directly cannot be applied in most physical situations

synthesis of positive real multivariable feedback systems this is a tutorial paper describing the application of infinity norm optimal control theory to the synthesis of sector bounded and positive real multivariable feedback systems the paper gives a concise treatment of the practical aspects of the theory describing how it has been implemented via state space oriented computer algorithms included is a concise summary of the state space approach to infinity norm control developed by doyle and by safonov and verma based on the hankel model reduction theory and incorporating the multivariable hankel results of glover

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

fourth order accurate scheme for the space fractional diffusion equations because of the nonlocal properties of fractional operators higher order schemes play a more important role in discretizing fractional derivatives than classical ones the striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy nowadays there are already two types of second order discretization schemes for space fractional derivatives the first type is given and discussed in sousa and li arxiv 1109 2345v1 2011 chen and deng appl math model 38 2014 pp 3244 3259 chen deng and wu appl numer math 70 2013 pp 22 41 and the second type is a class of schemes presented in tian zhou and deng math comp to appear also available online from arxiv 1201 5949 2012 the core object of this paper is to derive a class of fourth order approximations called the weighted and shifted lubich difference operators for space fractional derivatives then we use the derived schemes t

high order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions we present a high order compact finite difference approach for a class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second order derivative terms in n spatial dimensions problems of this type arise frequently in computational fluid dynamics and computational finance we derive general conditions on the coefficients which allow us to obtain a high order compact scheme which is fourth order accurate in space and second order accurate in time moreover we perform a thorough von neumann stability analysis of the cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms and also give partial results for the general case the results suggest unconditional stability of the scheme as an application example we consider the pricing of european power put basket options in the multidimensional black scholes model for two and three underlying assets due to the low regularity of typical initial conditions we employ th

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

computing the asymptotic worst case of bin packing lower bounds abstract this paper addresses the issue of computing the asymptotic worst case of lower bounds for the bin packing problem we introduce a general result that allows to bound the asymptotic worst case performance of any lower bound for the problem and to derive for the first time the asymptotic worst case of the well known bound l 3 by martello and toth we also show that the general result allows to easily derive the asymptotic worst case of several lower bounds proposed in the literature

the three dimensional bin packing problem the problem addressed in this paper is that of orthogonally packing a given set of rectangular shaped items into the minimum number of three dimensional rectangular bins the problem is strongly np hard and extremely difficult to solve in practice lower bounds are discussed and it is proved that the asymptotic worst case performance ratio of the continuous lower bound is an exact algorithm for filling a single bin is developed leading to the definition of an exact branch and bound algorithm for the three dimensional bin packing problem which also incorporates original approximation algorithms extensive computational results involving instances with up to 90 items are presented it is shown that many instances can be solved to optimality within a reasonable time limit

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

capacitated vehicle routing on trees t v e is a tree with nonnegative weights associated with each of its vertices a fleet of vehicles of capacity q is located at the depot represented by vertex v1 the capacitated vehicle routing problem on trees tcvrp consists of determining vehicle collection routes starting and ending at the depot such that the weight associated with any given vertex is collected by exactly one vehicle the sum of all weights collected by a vehicle does not exceed q a linear combination of the number of vehicles and of the total distance traveled by these vehicles is minimized the tcvrp is shown to be np hard this paper presents lower bounds for the tcvrp based on the solutions of associated bin packing problems we develop a linear time heuristic upper bound procedure and present a bound on its worst case performance these lower and upper bounds are then embedded in an enumerative algorithm numerical results follow

high order adi finite difference schemes for parabolic equations in the combination technique with application in finance in this article we combine high order ho finite difference discretisations with alternating direction implicit adi schemes for parabolic partial differential equations with mixed derivatives in a sparse grid setting in each implicit leg of the adi schemes we propose a high order compact hoc discretisation such that only tridiagonal systems have to be solved with the help of ho spatial discretisations and adi schemes solutions with second order accuracy in time and fourth order accuracy in space can be computed in order to reduce the number of involved grid points we use the combination technique to construct the so called sparse grid solution the theoretical findings are illustrated by numerical examples with european basket options