geometric quantization of dirac manifolds we extend a prequantization procedure to dirac manifolds by using singular distributions obtained from 2 cocycles associated with dirac structures given a dirac manifold m d we describe a prequantization formula in terms of a lie algebroid connection and show that it is a representation of a poisson algebra consisting of admissible functions on m d on the space of global sections of a hermitian line bundle over m if and only if the curvature derived from the lie algebroid connection is represented by a skew symmetric operation which is naturally defined for m d moreover we describe a necessary and sufficient condition for the prequantization formula to be the representation in terms of a lie algebroid cohomology we introduce the notion of a polarization for m d and construct a representation of a subalgebra of admissible functions lastly we discuss procedures for quantization in two cases where m is compact and where m is not compact

extended framework of hamilton s principle applied to duffing oscillation abstract the paper begins with a novel variational formulation of duffing equation using the extended framework of hamilton s principle ehp unlike the original hamilton s principle this formulation properly accounts for initial conditions and it recovers all the governing differential equations as its euler lagrange equation thus it serves base for the development of novel computational methods involving finite element representation over time for its feasibility we provide the simplest temporal finite element method by adopting linear temporal shape functions numerical examples are included to verify and investigate performance of non iterative algorithm in the developed method

a nurbs based hybrid collocation galerkin method for the analysis of boundary represented solids abstract the paper is concerned with a new numerical method nurbs based hybrid collocation galerkin method nurbs hcgm to solve the in plane motion problem of elastic solids it combines the merits of the so called scaled boundary finite element method sb fem and the isogeometric collocation method for the analysis the boundary scaling technique of sb fem is adopted it leads to a formulation where only the boundary of a structure is discretized here the nurbs basis functions are employed for the description of the geometry of the boundary as well as for the approximation of the displacements at the boundary this is in accordance with the boundary representation modeling technique which is commonly employed in computer aided design software the inner domain is described by a radial scaling parameter applying the weak form only in circumferential direction the governing partial differential equations of elasticity are transformed to an ordinary differential equation ode of euler type where the unknown displacements are a function of the radial scaling parameter in the present work a nurbs based collocation scheme is introduced to solve this equation nurbs basis functions are suggested for the approximation of the displacements in scaling direction the higher continuity provided by nurbs allows to use collocation to solve the ode directly instead of using the weak form in scaling direction the proposed approach is validated by comparison with the eigenvalue solution of the ode it is remarked that the eigenvalue solution is restricted to linear problems whereas the proposed method could be extended to nonlinear problems in general the presented formulation will allow to model patches bounded by an arbitrary number of contour boundaries the accuracy of the proposed approach is analyzed and estimated with respect to analytical solutions the computational cost is investigated with the help of numerical examples and is compared to isogeometric galerkin approach

radial spline collocation method for static analysis of beams conventional spline collocation method scm uses equally spaced knots in the computation of approximation however when load distribution geometry material property of structure are discontinuous it will greatly reduce accuracy the goal of this paper is mainly to develop unequally spaced knots to improve the disadvantage of using equally spaced knots applying the concept of radial basis functions rbf the spline function is transformed into radial spline function rsf consequently rsf can conveniently calculate the values for unequally spaced knots so that the complicated beam problems with discontinuous characteristics can be solved accurately and efficiently by the proposed radial spline collocation method rscm in this paper the proposed rscm also can be extended to solve other structural problems such as frame plate shell etc

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

a method of centers based on barrier functions for solving optimal control problems with continuum state and control constraints this paper describes a method of centers based on barrier functions for solving optimal control problems with continuum inequality constraints on the state and control the method decomposes the original problem into a sequence of easily solved optimal control problems with control constraints only the method requires only approximate solution of these problems

explicit and implicit meshless methods for linear advection diffusion type partial differential equations simple mesh grid free explicit and implicit numerical schemes for the solution of linear advection diffusion problems is developed and validated herein unlike the mesh or grid based methods these schemes use well distributed quasi random points and approximate the solution using global radial basis functions the schemes can be seen as generalized finite differences with random points instead of a regular grid system this allows the computation of problems with complex shaped boundaries in higher dimensions with no need for complex mesh grid structure and with no extra implementation difficulties

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

solving high order ordinary differential equations with radial basis function networks this paper is concerned with the application of radial basis function networks rbfns for numerical solution of high order ordinary differential equations odes two unsymmetric rbf collocation schemes namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process are developed to solve high order odes directly and the latter is found to be considerably superior to the former good accuracy and high rate of convergence are obtained with the proposed indirect method

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

a radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory abstract the free vibration analysis of functionally graded plates is performed by the multiquadric radial basis function method and a higher order shear deformation theory the multiquadric method is a truly meshless method allowing a fast and simple domain and boundary discretization we use an homogenization technique for material properties based on the mori tanaka scheme numerical tests show that the method is reliable and produces good results

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

numerical solution of the nonlinear fredholm integral equations by positive definite functions a numerical method for solving the nonlinear fredholom integral equations is presented the method is based on interpolation by radial basis functions rbf to approximate the solution of the fredholm nonlinear integral equations several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

numerical solution of the controlled duffing oscillator by hybrid functions a numerical method for solving the controlled duffing oscillator is presented the method is based upon hybrid functions approximation the properties of hybrid functions which consists of block pulse functions plus legendre polynomials are given the method is based upon expanding various time functions in the system as their truncated hybrid functions the operational matrix of product is introduced this matrix together with the operational matrix of integration are utilized to reduce the solution of controlled duffing oscillator to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

a meshfree method for the numerical solution of the rlw equation in this paper we present a meshfree technique for the numerical solution of the regularized long wave rlw equation this approach is based on a global collocation method using the radial basis functions rbfs different kinds of rbfs are used for this purpose accuracy of the new method is tested in terms of l 2 and l error norms in case of non availability of the exact solution performance of the new method is compared with existing methods stability analysis of the method is established propagation of single and double solitary waves wave undulation and conservation properties of mass energy and momentum of the rlw equation are discussed

some extensions of radial basis functions and their applications in artificial intelligence abstract radial basis functions have recently found interesting applications in artificial intelligence and in particular in the problem of learning to perform a particular task from a set of examples however in many practical cases the radial basis functions method cannot be applied in a straight forward manner because it does not take into account some features that are typical of the problem of learning from examples in this paper we show some extensions of the standard theory introduced in order to deal with a large number of examples and with problems in which different variables play very different roles we present some examples and also point out some open problems

existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis abstract isogeometric analysis is a numerical simulation method which uses the nurbs based representation of cad models nurbs stands for non uniform rational b splines and is a generalization of the concept of b splines the isogeometric method uses the tensor product structure of 2 or 3 dimensional nurbs functions to parameterize domains which are structurally equivalent to a rectangle or a hexahedron the special case of singularly parameterized nurbs surfaces and nurbs volumes is used to represent non quadrangular or non hexahedral domains without splitting which leads to a very compact and convenient representation if the parameterization of the physical domain is available the test functions for the isogeometric analysis are obtained by composing the inverse of the domain parameterization with the nurbs basis functions in the case of singular parameterizations however some of the resulting test functions are not well defined at the singular points and they do not necessarily satisfy the required integrability assumptions consequently the stiffness matrix integrals which occur in the numerical discretizations may not exist after summarizing the basics of the isogeometric method we discuss the existence of the stiffness matrix integrals for 1 2 and 3 dimensional second order elliptic partial differential equations we consider several types of singularities of nurbs parameterizations and derive conditions which guarantee the existence of the required integrals in addition we present cases with diverging integrals and show how to modify the test functions in these situations

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

a numerical solution of the nonlinear controlled duffing oscillator by radial basis functions in this research a new numerical method is applied to investigate the nonlinear controlled duffing oscillator this method is based on the radial basis functions rbfs to approximate the solution of the optimal control problem by using the collocation method we apply legendre gauss lobatto points for rbfs center nodes in order to use the numerical integration method more easily then the method of lagrange multipliers is used to obtain the optimum of the problems for this purpose different applications of rbfs are used the differential and integral expressions which arise in the dynamic systems the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients illustrative examples are included to demonstrate the validity and applicability of the technique

coisotropic regularization of singular lagrangians we present an alternative approach to the usual treatments of singular lagrangians it is based on a hamiltonian regularization scheme inspired on the coisotropic embedding of presymplectic systems a lagrangian regularization of a singular lagrangian is a regular lagrangian defined on an extended velocity phase space that reproduces the original theory when restricted to the initial configuration space a lagrangian regularization does not always exists but a family of singular lagrangians is studied for which such a regularization can be described explicitly these regularizations turn out to be essentially unique and provide an alternative setting to quantize the corresponding physical systems these ideas can be applied both in classical mechanics and field theories several examples are discussed in detail

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

poisson reduction and quantization for the n 1 photon for a dynamical system in which the constraints are given by the vanishing of a singular momentum map j reduction in the usual group theoretic sense may not be possible nonetheless one may still reduce j 1 0 at least on the level of poisson algebras an example of such a singular constrained system is the n 1 photon that is a massless spinless particle in n 1 dimensional minkowski space time we apply the generalized reduction procedure to the n 1 photon explicitly constructing the poisson algebra of gauge invariant observables this technique also enables us to completely analyze the effects of the singularities in j 1 0 on the system we then quantize obtaining results which are in agreement with a quantization of the extended phase space and the subsequent imposition of the constraint

hybrid legendre polynomials and block pulse functions approach for nonlinear volterra fredholm integro differential equations this paper introduces an approach for obtaining the numerical solution of the nonlinear volterra fredholm integro differential nvfid equations using hybrid legendre polynomials and block pulse functions these hybrid functions and their operational matrices are used for representing matrix form of these equations the main characteristic of this approach is that it reduces nvfid equations to a system of algebraic equations which greatly simplifying the problem numerical examples illustrate the validity and applicability of the proposed method

the quantum theory of second class constraints kinematics the problem of second class quantum constraints is here set up in the context ofc algebras utilizing the connection with state conditions as given by the heuristic quantization rules that is a constraint set is said to be first class if all its members can satisfy the same state condition and second class otherwise several heuristic models are examined and they all agree with this definition given then a second class constraint set we separate out its first class part as all those constraints which are compatible with the others and we propose an algebraic construction for imposition of the constraints this construction reduces to the normal one when the constraints are first class moreover the physical automorphisms assumed as conserving the constraints will also respect this construction the final physical algebra obtained is free of constraints gauge invariant unital and with the right choice simple thisc algebra also contains a factor algebra of the usual observables i e the commutator algebra of the constraints the general theory is applied to two examples the elimination of a canonical pair from a boson field theory as in the two dimensional anomalous chiral schwinger model of rajaraman 14 and the imposition of quadratic second class constraints on a linear boson field theory

hybrid collocation galerkin approach for the analysis of surface represented 3d solids employing sb fem abstract this paper presents a numerical method to solve the three dimensional elasticity problem of surface represented solids a surface oriented formulation is derived in which the three dimensional solid is described by its boundary surfaces and a radial scaling center scaling the boundary surfaces with respect to the scaling center yields a parameterization of the complete solid the definition of the boundary surfaces is sufficient for the description of the solid thus the formulation conforms ideally to the boundary representation modeling technique used in cad in the present approach no tensor product structure of three dimensional objects is exploited to parameterize the physical domain the weak form of the equation of motion and the weak form of the neumann boundary conditions are enforced only at the boundary surfaces it leads to a transformation of the governing partial differential equations of elasticity to an ordinary differential equation ode of euler type solving the ode leads to the displacement in the radial scaling direction in the present approach the isogeometric galerkin finite element method is employed to describe the geometry of the boundary surfaces and also to approximate the displacement response of the boundary surfaces it exploits two dimensional nurbs objects to parameterize the boundary surfaces the final euler type ode is solved by the nurbs based collocation method the displacement response in the radial scaling direction is approximated by the nurbs basis functions hence the method can be extended to nonlinear problems the accuracy of the method is validated against analytical solutions in general the proposed formulation is able to model solids bounded by an arbitrary number of surfaces

parameterization of computational domain in isogeometric analysis methods and comparison parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis in this paper we investigate this problem in the 2d case i e how to parametrize the computational domains by planar b spline surface from the given cad objects four boundary planar b spline curves firstly two kinds of sufficient conditions for injective b spline parameterization are derived with respect to the control points then we show how to find good parameterization of computational domain by solving a constraint optimization problem in which the constraint condition is the injectivity sufficient conditions of planar b spline parametrization and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar b spline parameterization by using this method the resulted parameterization has no self intersections and the isoparametric net has good uniformity and orthogonality after introducing a posteriori error estimation for isogeometric analysis we propose r refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution comparison examples with known exact solutions are also presented

pseudospectral legendre based optimal computation of nonlinear constrained variational problems it is well known that spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques in this paper the pseudospectral legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied the technique is based on spectral collocation methods in which the trajectory x t is approximated by the nth degree interpolating polynomial using legendre gauss lobatto points as the collocation points and lagrange polynomials as trial functions the integral involved in the formulation of the problem is approximated based on legendre gauss lobatto integration rule thereby reducing the problem to a nonlinear programming one to which existing well developed algorithms may be applied the method is easy to implement and yields very accurate results illustrative examples are included to confirm the convergence of the pseudospectral legendre method moreover a numerical experiment on a nonsmooth problem indicates that by applying a smoothing filter procedure to the pseudospectral legendre approximation one can recover the nonsmooth solution within spectral accuracy

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

isogeometric analysis enhanced by the scaled boundary finite element method abstract by leveraging the information of a typical cad model in the analysis the intensive process of discretization can be circumvented this unification has led to the isogeometric analysis iga hughes et al 2005 however as the cad model provides information only of the boundary a 2d 3d stress analysis is still one major step away in this work the concepts of isogeometric analysis and the scaled boundary finite element method sbfem are combined the sbfem requires only the boundary information and hence provides a seamless integration with the cad modeling within the proposed framework the nurbs basis functions are used to discretize the unknown fields in the circumferential direction whilst analytical solution is sought in the radial direction we further extend the framework to problems with singularities and to dynamic analysis the accuracy and the convergence properties of the proposed method are demonstrated with benchmark problems in the context of linear elasticity and linear elastic fracture mechanics

solving a system of nonlinear integral equations by an rbf network in this paper a novel learning strategy for radial basis function networks rbfn is proposed by adjusting the parameters of the hidden layer including the rbf centers and widths the weights of the output layer are adapted by local optimization methods a new local optimization algorithm based on a combination of the gradient and newton methods is introduced the efficiency of some local optimization methods to update the weights of rbfn is studied in solving systems of nonlinear integral equations

a scaled boundary isogeometric formulation for the elasto plastic analysis of solids in boundary representation abstract this contribution deals with the nonlinear analysis of boundary represented solids with elasto plastic material behavior based on the so called scaled boundary isogeometric formulation sb iga the proposed approach combines the features of the scaled boundary finite element method and isogeometric analysis based on the original boundary representation of the cad model a formulation is provided where the geometrical description of the boundary is sufficient to define the entire surface the domain is parameterized by a radial scaling parameter emanating from a scaling center and a parameter in circumferential direction along the boundary non star shaped domains are tackled by standard sub structuring here conforming discretizations are considered for the two dimensional case according to the isogeometric paradigm nurbs basis functions are employed for the approximation of the solution the displacement response is derived based on a multiplicative decomposition of the approximation in circumferential and radial scaling direction the boundary value problem is solved with the galerkin method the newton raphson iterative scheme is employed to obtain the nonlinear response several benchmark tests demonstrate the accuracy and computational efficiency of the formulation

on the quantization of presymplectic dynamical systems via coisotropic imbeddings we study certain aspects of the problem of quantizing a presymplectic dynamical system such a system is quantized by imbedding the presymplectic manifoldm under consideration into a symplectic manifoldx and then geometrically quantizing the latter it is known that this procedure will yield consistent results provided the imbeddingm x is coisotropic we show that such imbeddings always exist and are locally unique furthermore we investigate the extent to which the resulting quantum dynamics is independent of the choice of coisotropic imbedding that is we examine to what extent the presymplectic phase spacem determines the quantum representation space and the quantization of observables the quantization is carried out within the geometric quantization framework of kostant and souriau

quadrature rules for numerical integration based on haar wavelets and hybrid functions in this paper haar wavelets and hybrid functions have been applied for numerical solution of double and triple integrals with variable limits of integration this approach is the generalization and improvement of the methods siraj ul islam et al 2010 9 where the numerical methods are only applicable to the integrals with constant limits apart from generalization of the methods 9 the new approach has two major advantages over the classical methods based on quadrature rule i no need of finding optimum weights as the wavelet and hybrid coefficients serve the purpose of optimal weights automatically ii mesh points of the wavelets algorithm are used as nodal values instead of considering the n nodes as unknown roots of polynomial of degree n the new methods are more efficient the novel methods are compared with existing methods and applied to a number of benchmark problems accuracy of the methods are measured in terms of absolute errors

numerical simulation of reaction diffusion neural dynamics models and their synchronization desynchronization application to epileptic seizures abstract cognitive disorders especially epilepsy are closely linked with synchronization desynchronization of neurons in the brain in this paper the dynamical modeling and behavior analysis of a fitzhugh nagumo neuron and also synchronization control of a network of fitzhugh nagumo neurons which promise the understanding of cognitive processing are studied to numerically simulate these models and in order to overcome their difficulties such as computational complexity multi dimensionality non linearity having large spatial and temporal domains and also having a discrete initial data we propose the use of a strongly meshless technique based on wendland compactly supported radial basis function in conjunction with an operator splitting algorithm the main advantage of the proposed algorithm is its computational complexity for example o n x 1 3 n x 2 3 of the inversion of all matrices which is much lower than the complexity of pure radial basis functions method o n x 1 3 n x 2 3 of the inversion of matrices numerical experiments are presented showing that the proposed approaches are extremely accurate and fast

multi degree smooth polar splines a framework for geometric modeling and isogeometric analysis we develop a multi degree polar spline framework with applications to both geometric modeling and isogeometric analysis first multi degree splines are introduced as piecewise non uniform rational b splines nurbs of non uniform or variable polynomial degree and a simple algorithm for their construction is presented then an extension to two dimensional polar configurations is provided by means of a tensor product construction with a collapsed edge suitable combinations of these basis functions encoded in a so called isogeometric analysis suitable extraction operator yield ckck smooth polar splines for any k 0k 0 we show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality explicit constructions for k 0 1 2 k 0 1 2 are presented optimal approximation behavior is observed numerically and examples of applications to free form design smooth hole filling and high order partial differential equations demonstrate the applicability of the developed framework

solution and control of pde via global optimization methods abstract based on the concept of equivalent solutions not to be confused with approximations to the exact solution a new consideration is given to ill posed 1 2 and overdetermined pde problems and to problems with nonexistent solutions 3 then a new method based on full global optimization techniques is developed for solution and control of processes described by partial differential equations the ideas are illustrated by examples and a case study is presented in comparison with the quasi reversibility method 4

hybrid function method for solving fredholm and volterra integral equations of the second kind numerical solutions of fredholm and volterra integral equations of the second kind via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions integration of the cross product a special product matrix and a related coefficient matrix with optimal order are applied to solve these integral equations the main characteristic of this technique is to convert an integral equation into an algebraic hence the solution procedures are either reduced or simplified accordingly the advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function for the solutions of integral equations we propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions the high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples the hybrid function method is superior to other piecewise constant orthogonal functions w f blyth r l may p widyaningsih volterra integral equations solved in fredholm form using walsh functions anziam j 45 e 2004 c269 c282 m h reihani z abadi rationalized haar functions method for solving fredholm and volterra integral equations j comp appl math 200 2007 12 20 for these problems

numerical solutions of linear time varying descriptor systems via hybrid functions state analysis of linear time varying descriptor systems via hybrid functions are proposed in this paper based upon some useful properties of hybrid functions a special product matrix and a related coefficient matrix are applied to solve the descriptor systems the unknown coefficient matrix will be in generalized lyapunov equation form which is solved via the kronecker product method the advantages of hybrid functions are not only the values of n and m are adjustable but also it can yield more accurate numerical solutions than orthogonal functions and orthogonal polynomials to the problems of solving the descriptor systems we propose that the available optimal values of n and m can minimize the relative errors in the numerical solutions the high accuracy and the wide applicability of hybrid function approach will be demonstrated with numerical examples

covariant geometric quantization of nonrelativistic time dependent mechanics we provide geometric quantization of the vertical cotangent bundle v q q r equipped with the canonical poisson structure and treated as a momentum phase space of nonrelativistic time dependent mechanics we show that this quantization is equivalent to fiberwise quantization of symplectic fibers of v q r and that the quantum algebra of time dependent mechanics is an instantwise algebra quantization of the classical evolution equation defines a connection on this instantwise algebra and describes quantum evolution in time dependent mechanics as a parallel transport

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems ii the reduction via dirac and canonical quantization procedure the generalized theory of the r structure on affine operator lie algebras is used to construct a complete theory of lax integrable nonlinear dynamical systems in multidimensions the operator bi hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator korteweg de vries and benney kaup dynamical systems as an important by product of the developed algebraic theory the dirac canonical quantization problem is solved almost completely for the neumann bogoliubov type oscillatory dynamical system on spheres associated via moser with the spectral moment map on an affine associative metrized lie coalgebra with a one parameter gauge two cocycle some remarks are given on the problem of extending the developed algebraic theory to the case of lax integrable dynamical systems on discrete manifolds

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

isogeometric analysis for second order partial differential equations on surfaces we consider the numerical solution of second order partial differential equations pdes on lower dimensional manifolds specifically on surfaces in three dimensional spaces for the spatial approximation we consider isogeometric analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by b splines or nurbs our analysis addresses linear nonlinear time dependent and eigenvalues problems involving the laplace beltrami operator on surfaces moreover we propose a priori error estimates under h refinement in the general case of second order pdes on the lower dimensional manifolds we highlight the accuracy and efficiency of isogeometric analysis with respect to the exactness of the geometrical representations of the surfaces

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

a matrix function solution for the scaled boundary finite element equation in statics the scaled boundary finite element method is a fundamental solution less boundary element method based on finite elements it leads to semi analytical solutions for displacement and stress fields which permits problems with singularities or in infinite domains to be handled conveniently and accurately however the present eigenvalue method for solving the scaled boundary finite element equation requires additional treatments for multiple eigenvalues with parallel eigenvectors which results from logarithmic terms in the solutions a matrix function solution for the scaled boundary finite element equation in statics is presented in this paper it is numerically stable for multiple or near multiple eigenvalues as it is based on real schur decomposition power functions logarithmic functions and their transitions as occurring in fracture mechanics composites and two dimensional unbounded domains are represented semi analytically no a priori knowledge on the types and orders of singularity are required when simulating stress singularities

improved accuracy of multiquadric interpolation using variable shape parameters given n scattered data points we examined the problem of finding n variable multiquadric mq shape parameters or r2 values because the problem of finding the optimal r2 values is a nonlinear one we optimized these parameters numerically by minimizing the root mean square rms errors the resulting r2 values varied over many orders of magnitude we have tested this approach on a number of univariate and bivariate franke s problems and found that the rms error reduction was substantial

scaled boundary parametrizations in isogeometric analysis abstract this paper deals with a special class of parametrizations for isogeometric analysis iga the so called scaled boundary parametrizations are easy to construct and particularly attractive if only a boundary description of the computational domain is available the idea goes back to the scaled boundary finite element method sb fem which has recently been extended to iga we take here a different viewpoint and study these parametrizations as bivariate or trivariate b spline functions that are directly suitable for standard galerkin based iga our main results are first a general framework for this class of parametrizations including aspects such as smoothness and regularity as well as a generalization to domains that are not star shaped second using the poisson equation as an example we explain the relation between standard galerkin based iga and the scaled boundary iga by means of the laplace beltrami operator and derive an equivalence theorem further results concern the separation of integrals in both approaches and an analysis of the singularity in the scaling center among the computational examples we present a rotor geometry that stems from a screw compressor machine and compare different parametrization strategies

constraints reduction and quantization theorems are proved that establish the unitary equivalence of the extended and reduced phase space quantizations of a constrained classical system with symmetry several examples are presented

reduced phase space quotienting procedure for gauge theories we present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration velocity space local expressions for a basis of this kernel are obtained using phase space procedures the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented we show that this reduction procedure is equivalent to the standard dirac method as long as the dirac conjecture holds that the dirac hamiltonian containing the primary first class constraints with their lagrange multipliers can be enlarged to an extended dirac hamiltonian which includes all first class constraints without any change of the dynamics the quotienting procedure is always equivalent to the extended dirac theory even when it differs from the standard dirac theory the differences occur when there are ineffective constraints and in these situations we conclude that the standard dirac method is preferable at least for classical theories an example is given to illustrate these features as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the pseudospectral legendre method for discretizing optimal control problems this paper presents a computational technique for optimal control problems including state and control inequality constraints the technique is based on spectral collocation methods used in the solution of differential equations the system dynamics are collocated at legendre gauss lobatto points the derivative x spl dot t of the state x t is approximated by the analytic derivative of the corresponding interpolating polynomial state and control inequality constraints are collocated at legendre gauss lobatto nodes the integral involved in the definition of the performance index is discretized based on the gauss lobatto quadrature rule the optimal control problem is thereby converted into a mathematical programming program thus existing well developed optimization algorithms may be used to solve the transformed problem the method is easy to implement capable of handling various types of constraints and yields very accurate results illustrative examples are included to demonstrate the capability of the proposed method and a comparison is made with existing methods in the literature

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

the scaled boundary finite element method alias consistent infinitesimal finite element cell method for elastodynamics the scaled boundary finite element method alias the consistent infinitesimal finite element cell method is developed starting from the governing equations of linear elastodynamics only the boundary of the medium is discretized with surface finite elements yielding a reduction of the spatial dimension by one no fundamental solution is necessary and thus no singular integrals must be evaluated general anisotropic material is analysed without any increase in computational effort boundary conditions on free and fixed surfaces and on interfaces between different materials are enforced exactly without any discretization this method is exact in the radial direction and converges to the exact solution in the finite element sense in the circumferential directions for a bounded medium symmetric static stiffness and mass matrices with respect to the degrees of freedom on the boundary result without any additional assumption a stress singularity is represented very accurately as the condition on the boundary in the vicinity of the point of singularity is satisfied without spatial discretization

numerical solution of the controlled duffing oscillator by the pseudospectral method abstract this paper presents a numerical method for solving the controlled duffing oscillator the method can be extended to nonlinear calculus of variations and optimal control problems the method is based upon constructing the m th degree interpolation polynomials to approximate the state and the control vector using legendre gauss lobatto collocation points the differential and integral expressions which arise in the system dynamics the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients

a nurbs based galerkin approach for the analysis of solids in boundary representation abstract the paper is concerned with a new numerical method to solve the elasticity problem of solids in boundary representation a formulation is derived where the geometrical description of the boundary is sufficient for defining the equations of elasticity of the complete solid while the interior of the domain is described by a radial scaling parameter the scaling of the boundary with respect to the specified scaling center leads to the complete solid this idea fits perfectly to the boundary representation modeling technique commonly employed in cad in the present approach the tensor product structure of the solid will be reduced by one dimension to parametrize the physical domain i e the three dimensional solid exploits only two dimensional nurbs objects which parametrize the boundary surfaces for the analysis the weak form of the equilibrium equations is enforced for the entire solid in particular the weak form is employed in the scaling direction and the circumferential direction applying the isogeometric paradigm the nurbs functions that describe the boundary of the geometry form also the basis for the approximation of the displacement at the boundary the displacement response in the radial scaling direction on the other hand is approximated by a one dimensional nurbs overall the galerkin projection of the weak form yields a linear system of equilibrium equations whose solution gives rise to the displacement response the accuracy of the method is validated by means of analytical reference data in conclusion the proposed method allows the analysis of solids which are bounded by an arbitrary number of contour boundaries

edged based smoothed point interpolation method for acoustic radiation with perfectly matched layer abstract it is known that for exterior acoustics the domain based finite element method always encounters the accuracy loss for high wavenumbers and the difficulty in dealing with the sommerfeld radiation condition in this work by using the polynomial and radial bases with specific node selection schemes the point interpolation method with the generalized gradient smoothing technique is employed to improve the accuracy for acoustic computations the appropriately softened acoustical stiffness of the discrete model is constructed based on the edge based smoothing domains via the gradient smoothing technique in addition the perfectly matched layer technique enables the smoothed point interpolation method to investigate exterior acoustics the complex coordinate transformation is utilized to determine the proper parameters for the perfectly matched layer through theoretical and numerical analyses several typical acoustic problems are analyzed revealing that better accuracy is obtained by the present method compared with the finite element method especially for high wavenumbers

a meshless method on non fickian flows with mixing length growth in porous media based on radial basis functions a comparative study the present study aims to introduce a solution for parabolic integro differential equations arising in heat conduction in materials with memory which naturally occur in many applications two radial basis functions rbfs collocation schemes are employed for solving this equation the first method tested is an unsymmetric method and the second one which appears to be more efficient is a symmetric one the convergence of these two schemes is accelerated as we use the cartesian nodes as the center nodes

a kind of improved univariate multiquadric quasi interpolation operators in this paper a kind of improved univariate multiquadric quasi interpolation operators is proposed by using hermite interpolating polynomials error analysis shows that the convergence rate of the operators depends heavily on the shape parameter c which indicates that our operators could provide the desired smoothness and precision by choosing a suitable value of c numerical examples show that the operators provide a high degree of accuracy moreover operators are applied to the fitting of discrete solutions of initial value problems

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme