on linearization of third order ordinary differential equations a new algorithm for linearization of a third order ordinary differential equation is presented the algorithm consists of composition of two operations reducing order of an ordinary differential equation and using the lie linearization test for the obtained second order ordinary differential equation the application of the algorithm to several ordinary differential equations is given

symmetry algebras of third order ordinary differential equations the main result of this paper is the complete classification of the third order ordinary differential equations according to their symmetries the same classification was done for second order by tresse determination des invariants ponctuels de l equation differentielle ordinaire du second ordre y x y y gekronte preisschrift hirzel leipzig 1896 and recently for arbitrary order linear ordinary differential equations the sections preceding the classification consist of a brief description of the concepts and methods along the lines of krause and michel lecture notes phys 382 251 1991 these sections also contain some definitions and a table listing the prolongations of a few vector fields finally two appendices give additional information relevant to equations in real variables and describe how some of the results can be easily generalized to higher orders

brief paper an analytical study on structure stability and design of general nonlinear takagi sugeno fuzzy control systems in this paper we first study analytical structure of general nonlinear takagi sugeno ts for short fuzzy controllers then establish a condition for analytically determining asymptotic stability of the fuzzy control systems at the equilibrium point and finally use the stability condition in design of the control systems that are at least locally stable the general ts fuzzy controllers use arbitrary input fuzzy sets any types of fuzzy logic and ts fuzzy rules with linear consequent and the generalized defuzzifier which contains the popular centroid defuzzifier as a special case we have mathematically proved that the general ts fuzzy controllers are nonlinear controllers with variable gains continuously changing with controllers input variables using lyapunov s linearization method we have established a necessary and sufficient condition for analytically determining local asymptotic stability of ts fuzzy control systems each of which is made up of a fuzzy controller of the general class and a nonlinear plant we show that the condition can be used in practice even when the plant model is not explicitly known we have utilized the stability condition to design with or without plant model general ts fuzzy control systems that are at least locally stable three numerical examples are given to illustrate in detail how to use our new results our results offer four important practical advantages 1 our stability condition being a necessary and sufficient one is the tightest possible stability condition 2 the condition is simple and easy to use partially because it only needs the fuzzy controller structure around the equilibrium point 3 the condition can be used for determining system local stability and designing fuzzy control systems that are stable at least around the equilibrium point even when the explicit plant models are unavailable and 4 the condition covers a very broad range of nonlinear ts fuzzy control systems for which a meaningful global stability condition seems impossible to establish

issues concerning the analysis and implementation of a class of fuzzy controllers this paper presents some important aspects concerning the analysis and implementation of a piecewise linear pwl fuzzy model with universal approximation capability the main advantages of pwl models are the availability of well established analysis methods and their suitability for simple hardware and software implementations as an example the development of a two input single output fuzzy controller is presented special attention is paid to the analysis of the system stability using piecewise quadratic lyapunov functions in addition two implementations are addressed one based on a digital signal processor dsp and the other based on a programmable logic device pld

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

three dimensional finite element methods for the stokes problem the tetrahedral finite element approximation of the stokes problem is analyzed by means of polynomials piecewise of degree k 1 for the velocity and continuous piecewise of degree k for the pressure a stability result is given for every k ge 1

algebraic multigrid for higher order finite elements two related approaches for solving linear systems that arise from a higher order finite element discretization of elliptic partial differential equations are described the first approach explores direct application of an algebraic based multigrid method amg to iteratively solve the linear systems that result from higher order discretizations while the choice of basis used on the discretization has a significant impact on the performance of the solver results indicate that amg is capable of solving operators from both poisson s equation and a first order system least squares fosls formulation of stoke s equation in a scalable manner nearly independent of basis order p for 3

new formulations positivity preserving discretizations and stability analysis for non newtonian flow models we propose a class of new discretization schemes for solving the rate type non newtonian constitutive equations the so called conformation tensor has been known to be symmetric and positive definite in a large class of constitutive equations preserving such a positivity property on the discrete level is believed to be crucially important but difficult high weissenberg number problems on numerical instabilities have been often associated with this issue in this paper we present various discretization schemes that preserve the positive definiteness of the conformation tensor regardless of the time and spatial resolutions moreover the robustness of the algorithm has been also demonstrated by the stability analysis using the discrete analogue of energy estimates new schemes presented in this paper are constructed based upon the newly discovered relationship between the rate type constitutive equations and the symmetric matrix riccati differential equations

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

multi grid methods for stokes and navier stokes equations in the present paper we introduce transforming iterations an approach to construct smoothers for indefinite systems this turns out to be a convenient tool to classify several well known smoothing iterations for stokes and navier stokes equations and to predict their convergence behaviour epecially in the case of high reynolds numbers using this approach we are able to construct a new smoother for the navier stokes equations based on incomplete lu decompositions yielding a highly effective and robust multi grid method besides some qualitative theoretical convergence results we give large numerical comparisons and tests for the stokes as well as for the navier stokes equations for a general convergence theory we refer to 29

symmetries and integrability of a fourth order euler bernoulli beam equation the complete symmetry group classification of the fourth order euler bernoulli ordinary differential equation where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement is obtained we perform the lie and noether symmetry analysis of this problem in the lie analysis the principal lie algebra which is one dimensional extends in four cases viz the linear exponential general power law and a negative fractional power law it is further shown that two cases arise in the noether classification with respect to the standard lagrangian that is the linear case for which the noether algebra dimension is one less than the lie algebra dimension as well as the negative fractional power law in the latter case the noether algebra is three dimensional and is isomorphic to the lie algebra which is sl 2 r this exceptional case although admitting the nonsolvable algebra sl 2 r remarkably allows for a two parameter family of exact solut

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

linearization of third order ordinary differential equations by point and contact transformations we present here the solution of the problem on linearization of third order ordinary differential equations by means of point and contact transformations we provide in explicit forms the necessary and sufficient conditions for linearization the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations

algebraic multigrid theory the symmetric case a rigorous two level theory is developed for general symmetric matrices and nonsymmetric ones using kaczmarz relaxation without assuming any regularity not even any grid structure of the unknowns the theory applies to algebraic multigrid amg processes as well as to the usual geometric multigrid it yields very realistic estimates and precise answers to basic algorithmic questions such as in what algebraic sense does gauss seidel or jacobi kaczmarz line gauss seidel etc relaxation smooth the error when is it appropriate to use block relaxation what algebraic relations must be satisfied by the coarse to fine interpolations what is the algorithmic role of the geometric origin of the problem the theory helps to rigorize local mode analyses and locally analyze cases where the latter is inapplicable

conditions for linearization of a projectable system of two second order ordinary differential equations a new method for linearizing a system of ordinary differential equations is introduced the method is applied to a system of two second order ordinary differential equations it is shown that for a particular class of equations the method gives more general linearization criteria than linearization via point transformations examples of systems of equations which are not linearizable via point transformations but linearizable by the new method are given

the characterization of third order ordinary differential equations admitting a transitive fiber preserving point symmetry group abstract third order ordinary differential equations admitting a transitive symmetry group of fiber preserving point transformations are completely characterized we give the general expressions of these equations as well as the canonical equations representing the equivalence classes the symmetry groups admitted by these equations are also characterized in some cases we give the transformation taking the canonical equation into the general one the linearization problem for third order equations is solved

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

the third order ordinary differential equations admitting the painleve i and ii as first integrals third order ordinary differential equations admitting the painleve equations pi and pii as first integrals are completely characterized this is done by determining necessary and sufficient conditions for these equations to belong to the equivalence class of certain canonical equations this characterization also includes equations admitting first integrals which are equivalent to pi and pii the general form of these equations is given in some cases we determine the transformation taking the equation into a canonical one

a stable and scalable hybrid solver for rate type non newtonian fluid models we present and analyze hybrid discretization schemes for rate type non newtonian fluids models the method employs higher order conforming approximations for velocity and pressure of the stokes equation and lower order approximations such as piecewise linear or piecewise constant for conformation tensor to reduce the accuracy gap the constitutive equation is discretized on a refined mesh which is obtained by subdividing the mesh for the velocity and pressure the temporal discretization is made by the standard semi lagrangian scheme the fully discrete nonlinear system is shown to be solved iteratively by applying the three steps 1 locating the characteristic feet of fluid particles 2 solving the constitutive equation and 3 solving the momentum and continuity equations stokes type equation to achieve a scalability of the solution process we employ an auxiliary space preconditioning method for the solution to the conforming finite element methods for the stokes equation this method is basically two grid method in which lower order finite element spaces are employed as auxiliary spaces it is shown to not only lead to the mesh independent convergence but also improve robustness and scalability stability analysis shows that if t o h d where d is the dimension of domain then the scheme admits a globally unique solution a number of full 3d test cases are provided to demonstrate the advantages of the proposed numerical techniques in relation to efficiency robustness and weak scalability

linearization of fourth order ordinary differential equations by point transformations the solution of the problem on linearization of fourth order equations by means of point transformations is presented here we show that all fourth order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third order derivative we provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation for ordinary differential equations of order greater than 4 we obtain necessary conditions which separate all linearizable equations into two classes

stability analysis and systematic design of fuzzy control systems abstract a method of the stability analysis and systematic design of fuzzy control systems is presented in this paper the fuzzy control system is formulated by a set of local takagi sugeno fuzzy models t s fuzzy model with aggregation the controller is designed by state feedback for each local model separately it is shown that under some conditions the controller designed can guarantee the global stability of the closed loop fuzzy system the stability of the whole closed loop fuzzy system is also proved via the piecewise lyapunov function the simulation results for the fuzzy control of a nonlinear system the ball and beam system bb system show that this method is effective

block implicit multigrid solution of navier stokes equations in primitive variables abstract a calculation procedure for rapid computation of steady multidimensional viscous flows is presented the method solves the navier stokes equations in primitive variables using a coupled block implicit multigrid procedure the procedure is applicable to finite difference formulations using staggered locations of the flow variables a smoothing technique called symmetrical coupled gauss seidel scgs is proposed and is empirically observed to provide good smoothing rates the viscous flow in a square cavity with a moving top wall is calculated for a range of reynolds numbers calculations with finite difference grids as large as 321 321 nodes have been made to test the accuracy and efficiency of the calculation scheme the cpu times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation the calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation