an implicit fourth order compact finite difference scheme for one dimensional burgers equation abstract a fourth order compact finite difference method is proposed in this paper to solve one dimensional burgers equation the newly proposed method is based on the hopf cole transformation which transforms the original nonlinear burgers equation into a linear heat equation and transforms the dirichlet boundary condition into the robin boundary condition the linear heat equation is then solved by an implicit fourth order compact finite difference scheme a compact fourth order formula is also developed to approximate the robin boundary conditions while the initial condition for the heat equation is approximated using simpson s rule to maintain overall fourth order accuracy numerical experiments have been conducted to demonstrate the efficiency and high order accuracy of this method the numerical results also show that the method is unconditionally stable as there is no constraint on time step size

a fast adaptive diffusion wavelet method for burger s equation abstract a fast adaptive diffusion wavelet method is developed for solving the burger s equation the diffusion wavelet is developed in 2006 coifman and maggioni 2006 and its most important feature is that it can be constructed on any kind of manifold classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the burger s equation the diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of burger s equation in this paper we have considered one dimensional and two dimensional burger s equation with dirichlet and periodic boundary conditions for each test problem the cpu time taken by fast adaptive diffusion wavelet method is compared with the cpu time taken by finite difference method and observed that the proposed method takes lesser cpu time we have also verified the convergence of the given method

a numerical solution of burgers equation by time discretization of adomian s decomposition method in this paper burgers equation one dimensional quasi linear parabolic pde which is used in turbulence and shock wave theory was solved by modified adomian s decomposition method with discretization in time the numerical results obtained by this method for various values of viscosity have been compared with the exact solution to show the efficiency of the method

weak formulation based haar wavelet method for solving differential equations the haar wavelet based discretization method for solving differential equations is developed nonlinear burgers equation is considered as a test problem both strong and weak formulations based approaches are discussed the discretization scheme proposed is based on the weak formulation an attempt is made to combine the advantages of the fem and haar wavelets the obtained numerical results have been validated against a closed form analytical solution as well as fem results good agreement with the exact solution has been observed

numerical solutions of the modified burgers equation by petrov galerkin method abstract the modified burgers equation mbe is solved numerically by the petrov galerkin method using a linear hat function as the trial function and a cubic b spline function as the test function product approximation has been used in this method a linear stability analysis of the scheme shows it to be unconditionally stable the accuracy of the presented method is demonstrated by two test problems the numerical results are found in good agreement with the exact solutions

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

a numerical solution of burgers equation by pseudospectral method and darvishi s preconditioning in this paper we solve the burgers equation by pseudospectral method this equation is one dimensional quasilinear partial differential equation to reduce round off error of pseudospectral method we use darvishi s preconditioning the numerical results obtained by this method for different values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other because norm infinity of error is very small

a parameter uniform implicit difference scheme for solving time dependent burgers equations abstract a numerical study is made for solving one dimensional time dependent burgers equation with small coefficient of viscosity burgers equation is one of the fundamental model equations in the fluid dynamics to describe the shock waves and traffic flows for high coefficient of viscosity a number of solution methodology exist in the literature 6 7 8 9 and 14 but for the sufficiently low coefficient of viscosity the exist solution methodology fail and a discrepancy occurs in the literature in this paper we present a numerical method based on finite difference which works nicely for both the cases i e low as well as high viscosity coefficient the method comprises a standard implicit finite difference scheme to discretize in temporal direction on uniform mesh and a standard upwind finite difference scheme to discretize in spacial direction on piecewise uniform mesh the quasilinearzation process is used to tackle the non linearity an extensive amount of analysis has been carried out to obtain the parameter uniform error estimates which show that the resulting method is uniformly convergent with respect to the parameter to illustrate the method numerical examples are solved using the presented method and compare with exact solution for high value of coefficient of viscosity

a numerical solution of burgers equation by finite element method constructed on the method of discretization in time abstract in this paper the galerkin finite element method constructed on the method of discretization in time was applied to solve the one dimensional nonlinear burgers equation the system of nonlinear equations obtained for each time step was solved by using the newton method in order to show the efficiency of the presented method the numerical solutions obtained for various values of viscosity were compared with the exact solutions it was seen that they were in excellent agreement

bounds on the average number of products in the minimum sum of products expressions for multiple value input two valued output functions the authors derive an upper and a lower bound on the average number of products in the minimum sum of products expressions sopes the upper bound is obtained by using the minimization results of functions with fewer variables the lower bound is based on an assumption so it is incorrect until this assumption is proven a lower bound l sub p n u and an upper bound u sub p n u on s sub p n u are derived where s sub p n u is the average number of products in minimum sope for p valued input two valued output functions n is the number of the inputs and u is the number of minterms the values of s sub p n u are obtained by minimizing randomly generated functions and they are compared to the calculated values of u sub p n u and l sub p n u these bounds are useful for estimating the size of programming logic arrays

numerical solution of burgers equation with restrictive taylor approximation in this paper we have applied restrictive taylor approximation classical explicit finite difference method to the burgers equation with a set of initial and boundary conditions to obtain its numerical solution the stability region and truncation error of the new method are discussed the accuracy of the proposed method is demonstrated by the two test problems the numerical results are found in good agreement with the exact solutions

efficient numerical techniques for burgers equation this paper presents new efficient numerical techniques for solving one dimensional quasi linear burgers equation burgers equation is used as a model problem in the study of turbulence boundary layer behavior shock waves convection dominated diffusion phenomena gas dynamics acoustic attenuation in fog and continuum traffic simulation using a non linear cole hopf transformation the burgers equation is reduced to one dimensional diffusion equation the linearized diffusion equation is semi discretized by using method of lines mol which leads to a system of ordinary differential equations in time resulting system of ordinary differential equations is solved by backward differentiation formulas bdf of order one two and three and the analysis of numerical errors are presented numerical results for modest values of kinematic viscosity are compared with the exact solution to demonstrate the efficiency of proposed numerical methods

a comparison of differential quadrature methods for the solution of partial differential equations abstract discrete ordinate methods for the solution of partial differential equations are examined and compared a novel approach for the discrete representation of differential operators based on split range polynomial expansions is introduced the utility of the method is demonstrated for the case of differentiation of functions involving steep gradients the solution of burgers equation is presented to illustrate the effectiveness of the technique for the solution of nonlinear partial differential equations exhibiting nearly discontinuous solutions comparisons are made with other differential quadrature methods

space time spectral element methods for one dimensional nonlinear advection diffusion problems abstract the following space time galerkin spectral element methods are developed and applied to solve the burgers equation with small viscosity a coupled methods consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations b two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly one uses spectral elements in the explicit part and the other uses the adams bashforth multistep method a subcycling technique in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods a stability analysis of the four methods is performed and subsequent results are debated a convergence study and a comparison of computer execution time for the four methods is made and the results are discussed comparative study leads to the conclusion that the space time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time the number of subcycles should be kept low 2 3 in order to avoid significant loss of accuracy the coupled explicit method is also applied to the solution of the one dimensional coupled continuity momentum and energy equations for non isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius

a finite element approach for solution of burgers equation in this study for the burgers equation which is used a model problem in turbulence and shock wave theory was given a finite element solution considering the numerical results obtained for different values of viscosity compared with cole s analytical solution it was seen that they were in good agreement

quadratic b spline finite element method for numerical solution of the burgers equation abstract in this paper the one dimensional nonlinear burgers equation was converted to a set of nonlinear ordinary differential equations by the method of discretization in time and then each of them was solved by applying the quadratic b spline finite element method in order to show the efficiency of the presented method two problems were considered the numerical solutions obtained for various values of viscosity were compared with the exact solutions it is seen that the results obtained by this way are quite satisfactory

a simple similarity transformation iterative scheme applied to korteweg de vries equation in this paper we apply a simple iterative scheme and give an algorithm to solve the reduced korteweg de vries equation by a scaled similarity transformation the obtained numerical solutions are valid for the whole solution domain results obtained by the method have been compared with the solution of marchant and smyth t r marchant n f smyth initial boundary value problems for the kdv equations ima j appl math 47 1991 247 264 and are found to be in good agreement with each other

the semi approximate approach for solving burgers equation with high reynolds number this paper presents the solution of burgers equation involving very high reynolds number re 5 we use semi implicit time difference scheme to reduce burgers equation to two point non linear ordinary differential equation and solved it by matched asymptotic expansion method

a numerical solution of burgers equation based on least squares approximation abstract burgers equation which is one dimensional non linear partial differential equation was converted to p non linear ordinary differential equations by using the method of discretization in time each of them was solved by the least squares method for various values of viscosity at different time steps the numerical solutions obtained were compared with the exact solutions it was seen that both of them were in excellent agreement while the exact solution was not available for viscosity smaller than 0 01 it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

new space time spectral and structured spectral element methods for high order problems abstract we propose new space time spectral and structured spectral element methods for high order problems by matrix decomposition we introduce a family of new basis functions which leads to a diagonal system for the fourth order problems with constant coefficients based on new basis functions in space and a dual petrov galerkin formulation in time we propose a new space time spectral method for a time dependent problem and proved their spectral accuracy both in space and time numerical results demonstrate their high efficiency and coincide well with theoretical analysis further to ultimately promote the numerical performance and efficiency we exploit the idea of locally simultaneous diagonalization to provide new structured spectral element methods for high order problems besides to increase the flexibility a c 1 conforming rectangular spectral method in analogy to argyris or bell triangular elements are proposed for fourth order equations which serves as a preparative work towards conforming quadrilateral spectral elements for high order equations

an algorithm to derive the complement of a binary function with multiple valued inputs a recursive algorithm to obtain a complement of a sum of products expression for a binary function with p valued inputs is presented it produces at most pn 2 products for n variable functions whereas a conventional elementary algorithm produces o tn n 1 t 2 products where t 2p 1 it is 10 20 times faster than the elementary one when p 2 and n 8 for large practical problems it produces many fewer products than the disjoint sharp algorithm used by mini appplications of the algorithm to pla minimization are also presented

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

an efficient robust numerical method for singularly perturbed burgers equation abstract in this article we propose a parameter uniformly convergent numerical method for viscous burgers equation in order to find a numerical approximation to burgers equation we linearize the equation to obtain sequence of linear pdes the linear pdes are solved by a finite difference scheme which comprises of the backward difference scheme for the time derivative and upwind finite difference scheme for the spatial derivatives layer adapted nonuniform meshes are invoked at each time level to exhibit layer nature of the solution the nonuniform meshes are obtained by equidistribution of a positive integrable monitor function which involves the derivative of the solution it is shown that the methods converges uniformly with respect to the perturbation parameter numerical experiments are carried out to validate the e uniform error estimate of o n 1 t

a weak galerkin finite element method for burgers equation abstract we propose a weak galerkin wg finite element method for solving the one dimensional burgers equation based on a new weak variational form both semi discrete and fully discrete wg finite element schemes are established and analyzed we prove the existence of the discrete solution and derive the optimal order error estimates in a discrete h 1 norm and the standard l 2 norm respectively numerical experiments are presented to illustrate our theoretical analysis

on the numerical solution of nonlinear burgers type equations using meshless method of lines abstract in this paper a meshless method of lines mmol is proposed for the numerical solution of nonlinear burgers type equations this technique does not require a mesh in the problem domain and only a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions rbfs the scheme is tested for various examples the results obtained by this method are compared with the exact solutions and some earlier work

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

a numerical solution of burgers equation in this paper the burgers equation which is one dimensional quasi linear parabolic partial differential equation was solved by a variational method constructed on the method of discretization in time the numerical results obtained by these ways for various values of viscosity have been compared with the exact solution it was seen that they were in good agreement with each other

on the optimal design of multiple valued plas a description is given of the design and analysis of three types of multivalued plas programmable logic arrays type 1 plas realize functions directly in the form of the max of min of literal functions and constants in type 2 plas the body of the pla is binary and the output is encoded as a multiple valued logic value type 3 plas are the same as type 2 plas except for the use of 2 bit decoders and a permutation network on the input using the number of columns required to realize a given function as a measure to compare plas it is shown that type 3 plas are superior to type 2 which in turn are superior to type 1

a direct variational methods applied to burgers equation in this study a compact approximate method in limiting form for calculating the solution of burgers equation with appropriate boundary conditions is presented the results obtained by present method are found to be in good agreement with those due to earlier authors and offers appreciable advantages for burgers like nonlinear problems

a novel numerical scheme for solving burgers equation abstract in this paper we propose a novel numerical scheme for solving burgers equation the scheme is based on a cubic spline quasi interpolant and multi node higher order expansion which make the algorithm simple and easy to implement the numerical experiments show that the proposed method produces high accurate results

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

on the size of plas required to realize binary and multiple valued functions upper and lower bounds are shown for the average number of product terms required in the minimal realization as a function of the number of nonzero output values the variance in addition to the bounds allows conclusions to be drawn about how pla size determines the set of realizable functions although the bounds are most accurate when there are few nonzero values they are adequate for analyzing commercially available plas the analysis shows that for all but one commercially available pla the number of nonzero values is a statistically meaningful criterion for determining whether or not a given function is likely to be realized

a finite difference approach for solution of burgers equation abstract in this paper we have applied restrictive pade approximation classical implicit finite difference method to the burgers equation with a set of initial and boundary conditions to obtain its numerical solution the stability region and truncation error of the new method are discussed the accuracy of the proposed method is demonstrated by the two test problems the numerical results obtained by this method for various values of viscosity have been compared with the exact solution to show the efficiency of the method the numerical results are found in good agreement with the exact solutions

decomposition of a scalar operator into a product of unitary operators with two points in spectrum abstract we consider products of unitary operators with at most two points in their spectra 1 and e i we prove that the scalar operator e i i is a product of k such operators if 1 1 k 3 k 1 1 k 3 for k 5 also we prove that for e i 1 only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class as a corollary we show that every unitary operator on an infinite dimensional space is a product of finitely many such operators

numerical solution of one dimesional burgers equation explicit and exact explicit finite difference methods this paper presents finite difference solution and analytical solution of the finite difference approximations based on the standard explicit method to the one dimensional burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics results obtained by these ways for some modest values of viscosity have been compared with the exact fourier one it is shown that they are in good agreement with each other

the generalized boundary element approach to burgers equation the generalized boundary element method is presented for the numerical solution of burgers equation the new method is based on the set of boundary integral equations derived for each subdomain by using the fundamental solution for the linearized differential operator of the equation the resulting system of quasi non linear equations is solved implicitly with use of a simple iterative procedure the adaptability and the accuracy of the proposed method are demonstrated by three examples and a comparison of the numerical results with the exact solution or other existing solutions is shown for the first example

block marching in time with dq discretization an efficient method for time dependent problems abstract in this paper an efficient method is presented to solve time dependent problems the proposed method marches in the time direction block by block in each block there are several time levels and the numerical results at these time levels are obtained simultaneously the global method of differential quadrature is applied in each block to discretize both the spatial and temporal derivatives the proposed method is validated by a sample problem which has an exact solution for easy comparison it was found that as compared to the conventional 4 stage runge kutta method the present method could give much higher order of accuracy for numerical results and require much less computational effort

moving least squares particle hydrodynamics method for burgers equation abstract in this paper a meshless method named moving least squares particle hydrodynamics mlsph method is applied to obtain numerical solutions of the burgers equation with a set of initial and boundary conditions the influence of kernel function and smoothing length on the accuracy of mlsph method is investigated several various forms of one and two dimensional burgers equations are successfully calculated it is concluded that the proposed method can obtain second order accuracy the way of non uniform distribution is applied to present the desired performance near the shocks the numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods the satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation

a difference scheme for burgers equation in an unbounded domain the numerical solution of the one dimensional burgers equation in an unbounded domain is considered two artificial boundaries are introduced to make the computational domain finite on both artificial boundaries two exact boundary conditions are proposed respectively to reduce the original problem to an initial boundary value problem in a finite computational domain a difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain at each time level only a strictly diagonal dominated tridiagonal system of linear algebraic equations needs to be solved it is proved that the difference scheme is uniquely solvable and unconditional convergent with the convergence order 3 2 in time and order 2 in space in an energy norm a numerical example demonstrates the theoretical results

the new numerical method for solving the system of two dimensional burgers equations in this paper the system of two dimensional burgers equations are solved by local discontinuous galerkin ldg finite element method the new method is based on the two dimensional hopf cole transformations which transform the system of two dimensional burgers equations into a linear heat equation then the linear heat equation is solved by the ldg finite element method the numerical solution of the heat equation is used to derive the numerical solutions of burgers equations directly such a ldg method can also be used to find the numerical solution of the two dimensional burgers equation by rewriting burgers equation as a system of the two dimensional burgers equations three numerical examples are used to demonstrate the efficiency and accuracy of the method

numerical solution of burgers equation by quadratic b spline finite elements in this paper a galerkin quadratic b spline finite element method is used to solve numerically the one dimensional burgers equation reduced by the hopf cole transformation the performance of the method is tested on two model problems involving moderately reynolds numbers with known exact solutions the obtained numerical results show that the method is efficient robust and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails computed results are compared with other numerical results in the literature a stability analysis of the method is also investigated

worst and best irredundant sum of products expressions in an irredundant sum of products expression isop each product is a prime implicant pl and no product can be deleted without changing the function among the isops for some function f a worst isop wsop is an isop with the largest number of pls and a minimum isop msop is one with the smallest number we show a class of functions for which the minato morreale isop algorithm produces wsops since the ratio of the size of the wsop to the size of the msop is arbitrarily large when it the number of variables is unbounded the minato morreale algorithm can produce results that are very far from minimum we present a class of multiple output functions whose wsop size is also much larger than its msop size for a set of benchmark functions we show the distribution of isops to the number of pls among this set are functions where the msops have almost as many pls as do the wsops these functions are known to be easy to minimize also there are benchmark functions where the fraction of isops that are msops is small and msops have many fewer pls than the wsops such functions are known to be hard to minimize for one class of functions we show that the fraction of isops that are msops approaches 0 as n approaches infinity suggesting that such functions are hard to minimize

multiple valued decomposition of generalized boolean functions and the complexity of programmable logic arrays generalized boolean functions are shown to be useful for the design of programmable logic arrays pla s and the complexity of three types of pla s is obtained by the theory of multiple valued decomposition a two level pla consists of an and array and an or array and they are cascaded to perform a two level and or circuit a pla with decoders consists of decoders an and array and an or array a three level pla consists of a d array an and array and an or array and they are cascaded to perform a three level or and or circuit it is shown that a generalized boolean function f x1 x2 xr x bni b where b 0 1 is represented by a generalized boolean expression of 2ni valued variables x i and f can be directly realized by a pla with decoders or a three level pla to realize a function of n variables n 2r the following sizes are shown to be sufficient for a two level pla n 2n for a pla with two bit decoders 4 n 4 2n for a three level pla 2n 3n l 2n 2n2especially in the case of pla with two bit decoders the following sizes are shown to be necessary and sufficient for an arbitrary symmetric function 3 2 n 3n and for a parity function n 2n

numerical solution of the burgers equation by local discontinuous galerkin method abstract in this paper the burgers equation is transformed into the linear diffusion equation by using the hopf cole transformation the obtained linear diffusion equation is discretized in space by the local discontinuous galerkin method the temporal discretization is accomplished by the total variation diminishing runge kutta method numerical solutions are compared with the exact solution and the numerical solutions obtained by adomian s decomposition method finite difference method b spline finite element method and boundary element method the results show that the local discontinuous galerkin method is one of the most efficient methods for solving the burgers equation even with small viscosity coefficient it can get the satisfied solution

a numerical solution of burgers equation by modified adomian method in this paper the modified adomian s decomposition method for calculating a numerical solution of the one dimensional quasi linear the burgers equation is presented time discretization has been used in decomposition without using any transformation in the burgers equation such as hopf cole transformation the numerical results obtained by this way for various values of viscosity have been compared with the exact solution to show the efficiency of the method

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

an exponentially fitted method for solving burgers equation in this paper an exponentially fitted method is used to numerically solve the one dimensional burgers equation the performance of the method is tested on the model involving moderately large reynolds numbers the obtained numerical results show that the method is efficient stable and reliable for solving burgers equation accurately even involving high reynolds numbers for which the exact solution fails copyright 2009 john wiley sons ltd

on the complexity of mod 2l sum pla s consideration is given to the realization of logic functions by using plas with an exclusive or exor array where a function is represented by mod 2 exor sum of products esops and both true and complemented variables are used the authors propose a new pla structure using an exor array they derive upper bounds on the number of products of this type of pla that are useful for estimating the size of a pla as well as for assessing the minimality of the solutions obtained by heuristic esop minimization algorithms computer simulation using randomly generated functions shows that plas with the exor array require on the average fewer products than conventional plas for symmetric functions the authors conjecture that the plas with an exor array require at most as many products as the conventional plas the proposed plas can be made easily testable by adding a small amount of hardware

an implementation of a moving finite element method abstract the moving finite element method an adaptive gridding procedure for systems of partial differential equations whose solutions contain steep gradients has been implemented in a very straightforward way accurate results can be obtained using fewer nodes than that required by fixed mesh solution methods performance of the method using some minor changes to a standard ordinary differential equation algorithm is illustrated with solutions to burger s equation

generating essential primes for a boolean function with multiple valued inputs detecting essential primes is important in multiple valued logic minimization in this correspondence we present a fast algorithm that can generate all essential primes without generating a prime cover of the boolean function a new consensus operation called asymmetric consensus acons is defined in terms of acons we prove a necessary and sufficient condition for detecting essential primes for a boolean function with multiple valued inputs the detection of essential primes can be performed by using a tautology checking algorithm we exploit the unateness of a boolean function to speed up tautology checking the notion of unateness considered is more general than that has appeared in the literature

a space time fully decoupled wavelet galerkin method for solving two dimensional burgers equations abstract a space time fully decoupled formulation for solving two dimensional burgers equations is proposed based on the coiflet type wavelet sampling approximation for a function defined on a bounded interval by applying a wavelet galerkin approach for spatial discretization nonlinear partial differential equations are first transformed into a system of ordinary differential equations in which all matrices are completely independent of time and never need to be updated in the time integration finally the mixed explicit implicit scheme is employed to solve the resulting semi discretization system by numerically studying three widely considered test problems results demonstrate that the proposed method has a much better accuracy and a faster convergence rate than many existing numerical methods most importantly the study also indicates that the present wavelet method is capable of solving the two dimensional burgers equation at high reynolds numbers

an algorithm based on exponential modified cubic b spline differential quadrature method for nonlinear burgers equation in this paper the authors developed a new differential quadrature method exponential modified cubic b spline differential quadrature method expo mcb dqm by using exponential modified cubic b spline functions as test functions in the traditional differential quadrature method 32 the new method is tested on one and two dimensional nonlinear burgers equations to check the efficiency and accuracy of the proposed method five numerical problems have been considered the numerical results of the method are compared with some existing methods and found that the proposed numerical method produces more accurate results than existing methods stability analysis of the algorithm is also done by using matrix stability analysis method

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

numerical solution of burgers equation by cubic hermite collocation method abstract in this paper numerical solution of the non linear burgers equation are obtained by using cubic hermite collocation method chcm the advantage of the method is continuity of the dependent variable and its derivative throughout the solution range a linear stability analysis shows that the numerical scheme based on crank nicolson approximation in time is unconditionally stable this method is applied on some test problems with different choice of collocation points to validate the accuracy of the method the obtained numerical results show that the method is efficient robust and reliable even for high reynolds numbers for which the exact solution fails moreover the method can be applied to a wide class of nonlinear partial differential equations

input variable assignment and output phase optimization of pla s a pla minimization system having the following features is presented 1 minimization of both two level pla s and pla s with two bit decoders 2 optimal input variable assignment to the decoders 3 optimal output phase assignment and 4 essential prime implicants detection without generating all the prime implicants