numerical estimation of coercivity constants for boundary integral operators in acoustic scattering coercivity is an important concept for proving existence and uniqueness of solutions to variational problems in hilbert spaces but while coercivity estimates are well known for many variational problems arising from partial differential equations they are still an open problem in the context of boundary integral operators arising from acoustic scattering problems where rigorous coercivity results have so far only been established for combined integral operators on the unit circle and sphere the fact that coercivity holds even in these special cases is perhaps surprising as formulations of helmholtz problems are generally thought to be indefinite the main motivation for investigating coercivity in this context is that it has the potential to give error estimates for the galerkin method which are both explicit in the wavenumber k and valid regardless of the approximation space used thus they apply to hybrid asymptotic numerical methods recently developed for the high frequency case one way to interpret coercivity is by considering the numerical range of the operator the numerical range is a well established tool in spectral theory and algorithms exist to approximate the numerical range of finite dimensional matrices we can therefore use galerkin projections of the boundary integral operators to approximate the numerical range of the original operator we prove convergence estimates for the numerical range of galerkin projections of a general bounded linear operator on a hilbert space to justify this approach by computing the numerical range of the combined integral operator in acoustic scattering for several interesting convex nonconvex smooth and polygonal domains we numerically study coercivity estimates for varying wavenumbers we find that coercivity holds uniformly in the wavenumber k for a wide variety of domains finally we consider a trapping domain for which there exist resonances also called scattering poles very close to the real line to demonstrate that coercivity for a certain wavenumber k seems to be strongly dependent on the distance to the nearest resonance

non similar series solution for boundary layer flow of a third order fluid over a stretching sheet this study deals with the boundary layer flow of a third order fluid induced by a stretching sheet by means of similarity transformation the governing non linear partial differential equation is reduced to a non linear ordinary differential equation the analytical solution of the problem is obtained using homotopy analysis method ham finally the solutions are also compared with the existing results in the literature

nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

twenty five years of multifractals in fully developed turbulence a tribute to giovanni paladin the paper on the multifractal nature of fully developed turbulence and chaotic systems by benzi et al 1984 j phys a math gen 17 3521 has been a starting point of many investigations on the different faces of self similarity and intermittency in turbulent phenomena since then the multifractal model has become a useful tool for the study of small scale turbulence in particular for detailed predictions of different eulerian and lagrangian statistical properties on the occasion of the 50th birthday of our unforgettable friend and colleague giovanni paladin 1958 1996 we review here the basic concepts and some applications of the multifractal model for turbulence

on a perturbation treatment of a model for mhd viscous flow we discuss the solution of a nonlinear ordinary differential equation that appears in a model for mhd viscous flow caused by a shrinking sheet we propose an accurate numerical solution and derive simple analytical expressions our results suggest that a recent perturbation treatment of the same problem exhibits a pathological behaviour and conjecture its probable cause

cascade structures and scaling exponents in a dynamical model of turbulence measurements and comparison a detailed examination of the cascade statistics and scaling exponents is carried out for a dynamical system model of fully developed turbulence called the goy shell model the convergence in time of the probability density functions and moments of the velocity fluctuations and their scaling exponents is studied with particular care with a large sample size 5 ifmmode times else texttimes fi 10 9 we demonstrate that there exists a finite cutoff for the velocity fluctuations at each inertial range wave number shell and the properties of the cutoff determine the scaling exponents of all moments this cutoff represents the most intermittent structures in the cascade dynamics and exhibits a power law dependence on wave number the accurately determined scaling exponents permit a detailed comparison with various phenomenological models describing the statistics of the energy cascade the consideration of the first and second derivatives of the scaling exponents with respect to the order of the moments p provides the evidence that the hierarchical structure model she and leveque phys rev lett 72 336 1994 predicts the best functional dependence on p of the scaling exponents in the goy shell model

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

card shuffling and the polynomial numerical hull of degree emk the polynomial numerical hull of degree k is a set in the complex plane associated with a matrix a that is designed to give more information than the spectrum can provide about the behavior of a under the action of polynomials we give an example of the application of the polynomial numerical hull of degree k to explain the cutoff phenomenon that is often observed in markov processes in particular we consider the much publicized result that it takes seven riffle shuffles to randomize a deck of cards d bayer and p diaconis ann appl probab 2 1992 pp 294 313 while this result cannot be explained by eigenvalues it is shown to be very well predicted by the 1 norm polynomial numerical hulls of various degrees associated with a matrix derived from the probability transition matrix

nelkin scaling for the burgers equation and the role of high precision calculations nelkin scaling the scaling of moments of velocity gradients in terms of the reynolds number is an alternative way of obtaining inertial range information it is shown numerically and theoretically for the burgers equation that this procedure works already for reynolds numbers of the order of 100 or even lower when combined with a suitable extended self similarity technique at moderate reynolds numbers for the accurate determination of scaling exponents it is crucial to use higher than double precision similar issues are likely to arise for three dimensional navier stokes simulations

a general class of coupled nonlinear differential equations arising in self similar solutions of convective heat transfer problems we establish existence and uniqueness results for a general class of coupled nonlinear third order differential equations arising in flow and heat transfer problems we consider solutions over the semi infinite interval as special cases we recover the existence and uniqueness results of solutions for the following physically meaningful scenarios among others i flow and heat transfer over a stretching sheet ii flow and heat transfer over a nonlinearly stretching porous sheet iii linear convective flow and heat transfer over a porous nonlinearly stretching sheet and iv nonlinear convective heat transfer over a porous nonlinearly stretching sheet in all the cases the effects of viscous dissipation and the internal heat generation absorption on the flow and heat transfer characteristics are included moreover the obtained results are applicable to several problems dealing with flow and heat transfer phenomena

analysis of nonlinear bvps motivated by fluid film flow over a surface we study a coupled nonlinear boundary value problem which has been shown to have applications to fluid flow and heat transfer in a fluid film over a stretching surface for set values of the model parameters one of which determines the size of the problem domain for arbitrary values of these parameters we are able to establish the existence and uniqueness of a class of monotone solutions perturbation solutions are then constructed and used to approximate certain invariants for the solutions we then study a related boundary value problem formed by imposing an additional boundary condition on one of the governing equations which results in an ill posed problem and we arrive at conditions allowing for solutions to this four parameter problem to agree with the solutions to the three parameter problem

a note on a certain boundary layer equation abstract this paper deals with a differential equation of mixing layer the existence of a solution is established by using weyl technique

upper and lower bounds on norms of functions of matrices abstract given an n by n matrix a we look for a set s in the complex plane and positive scalars m and m such that for all functions p bounded and analytic on s and throughout a neighborhood of each eigenvalue of a the inequalities m inf f l s f a p a p a m inf f l s f a p a hold we show that for 2 by 2 matrices if s is the field of values then one can take m 1 and m 2 we show that for a perturbed jordan block a matrix a that is an n by n jordan block with eigenvalue 0 except that its n 1 entry is with 0 1 if s is the unit disk then m m 1 we argue however that in general due to the behavior of minimal norm interpolating functions it may be very difficult or impossible to find such a set s for which the ratio m m is of moderate size

effects of slip on sheet driven flow and heat transfer of a non newtonian fluid past a stretching sheet the flow and heat transfer of an electrically conducting non newtonian fluid due to a stretching surface subject to partial slip is considered the constitutive equation of the non newtonian fluid is modeled by that for a third grade fluid the heat transfer analysis has been carried out for two heating processes namely i with prescribed surface temperature pst case and ii prescribed surface heat flux phfcase in presence of a uniform heat source or sink suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations the issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations the important finding in this communication is the combined effects of the partial slip magnetic field heat source sink parameter and the third grade fluid parameters on the velocity skin friction coefficient and the temperature field it is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers

on the determination of the steady film profile for a non newtonian thin droplet abstract a shooting method is used to determine a solution to a third order ode modeling the steady profile of a non newtonian thin droplet we compare a direct approach to an iterative approach using a secant method we obtain a nonlinear relationship between the contact angle and the position of the contact line r from this nonlinear relationship we use curve fitting to obtain an empirical law of the form tan r f k where k is the power law coefficient and f is a nonlinear function of k

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

numerical ranges of large toeplitz matrices abstract a bananch algebraic approach is proposed to study the asymptotic behaviour of the numerical ranges of certain finite approximation matrices of infinite operators the approach works for large classes of approximation methods it is examined in detail here for the finite sections of toeplitz operators and of operators which are generated by toeplitz operators the basic ingredient is a precise knowledge of the finite section method for toeplitz and related operators

on similarity solutions of a boundary layer problem with an upstream moving wall this work deals with the problem of a boundary layer on a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream it has previously been shown that the solution of this boundary value problem is crucially dependent on the parameter which is the ratio of the velocity of the plate to the velocity of the free stream in particular it was proved that a solution exists only if this parameter does not exceed a certain critical value and numerical evidence was adduced to show that this solution is nonunique using crocco formulation the present work proves this nonuniqueness also considered are the analyticity of solutions and the derivation of upper bounds on the critical value of wall velocity parameter

new conditions for non stagnation of minimal residual methods in the solution of large linear systems a condition guaranteeing that a minimal residual krylov subspace method makes some progress i e that it does not stagnate is that the symmetric part of the coefficient matrix be positive definite this condition results in a well established worst case bound for the convergence rate of the iterative method due to elman this bound has been extensively used e g when the linear system comes from discretized partial differential equations to show that the convergence of gmres is independent of the underlying mesh size in this paper we introduce more general non stagnation conditions which do not require the symmetric part of the coefficient matrix to be positive definite and that guarantee for example the non stagnation of restarted gmres for certain values of the restarting parameter

on analytic properties of periodic solutions for equation mathcal h u_x u u p 0 the paper focuses on 2l periodic solutions ul x of the nonlocal equation p 2 is integer is the hilbert transform we give numerical evidence for the existence of a continuous family of these solutions parametrized by l for p 3 4 5 the features of the solutions ul x are discussed in particular we offer strong numerical arguments that ul x can be continued analytically from the real axis to some strip in the complex plane ul x ul z the singularities which arise under analytical continuation of these solutions into the complex plane are considered it is proved theorem 1 that ul z cannot be any power of a meromorphic function the formula which describes the asymptotic behavior of ul z in a neighborhood of the singularity is given it agrees with the numerical results computations also allow us to locate the closest to the real axis singularity of ul z and estimate the width of the analyticity strip

linear preservers of polynomial numerical hulls of matrices abstract let m n be the algebra of all n n complex matrices 1 k n 1 be an integer and m n m n be a linear operator in this paper it is shown that preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix u m n such that either a u a u for all a m n or a u a t u for all a m n

radiation effects for the blasius and sakiadis flows with a convective surface boundary condition abstract this study aims to analyze the effects of thermal radiation on the laminar boundary layer about a flat plate in a uniform stream of fluid blasius flow and about a moving plate in a quiescent ambient fluid sakiadis flow both under a convective surface boundary condition the resulting similarity energy equation is solved numerically and the variations of dimensionless surface temperature and fluid solid interface characteristics for different values of prandtl number pr radiation parameter nr and parameter a which characterizes our convection processes are graphed and tabulated quite different and interesting behaviours were encountered for a blasius flow compared with a sakiadis flow a comparison with previously published results on special case of the problem shows excellent agreement

hydromagnetic flow of a second grade fluid over a stretching sheet an analysis is carried out to study the flow characteristics in an electrically conducting second grade fluid permeated by a uniform magnetic field over a stretching sheet in order to solve the fourth order non linear differential equation a fourth boundary condition is augmented and a proper sign for the normal stress modulus is used it is observed that for a physical flow problem the solution is unique the exact solutions for the velocity fields for various combinations of the parameters m the magnetic parameter r the suction reynolds number and l 1 the viscoelastic parameter are obtained furthermore numerical solutions for the velocity components are obtained and discussed

a new set of solutions to a singular second order differential equation arising in boundary layer theory abstract the paper studies a boundary value problem of a nonlinear second order differential equation that governs the laminar flow of newtonian and non newtonian fluids it presents a new sufficient condition for the existence of solutions to the boundary value problem

analytic structure of solutions of the one dimensional burgers equation with modified dissipation we use the one dimensional burgers equation to illustrate the effect of replacing the standard laplacian dissipation term by a more general function of the laplacian of which hyperviscosity is the best known example in equations of hydrodynamics we analyze the asymptotic structure of solutions in the fourier space at very high wave numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing reynolds numbers from algebraic recursion relations involving iterated integrations we give a detailed analysis of their analytic structure for two different types of dissipation a hyperviscous and an exponentially growing dissipation term our results obtained in the limit of vanishing reynolds numbers are validated by high precision numerical simulations at non zero reynolds numbers we then study the bottleneck problem an intermediate asymptotics phenomenon which in the case of the burgers equation arises when ones uses dissipation terms such as hyperviscosity growing faster at high wave numbers than the standard laplacian dissipation term a linearized solution of the well known boundary layer limit of the burgers equation involving two numerically determined parameters gives a good description of the bottleneck region

a new solution branch for the blasius equation a shrinking sheet problem in this work a similarity equation of the momentum boundary layer is studied for a moving flat plate with mass transfer in a stationary fluid the solution is applicable to the practical problem of a shrinking sheet with a constant sheet velocity theoretical estimation of the solution domain is obtained it is shown that the solution only exists with mass suction at the wall surface the equation with the associated boundary conditions is solved using numerical techniques greatly different from the continuously stretching surface problem and the blasius problem with a free stream quite complicated behavior is observed in the results it is seen that there are three different solution zones divided by two critical mass transfer parameters f 0 1 1 7028 and f 0 2 1 7324 when f 0f 0 2 there is a terminating point for the solution domain and the terminating point corresponds to a special algebraically decaying solution for the current problem the current results provide a new solution branch of the blasius equation which is greatly different from the previous study and provide more insight into the understanding of the blasius equation

fluid flow over a nonlinearly stretching sheet numerical solutions are obtained for a class of nonlinear third order differential equations arising in fluid flows over a nonlinearly stretching sheet using a similarity transformation which is different from that of the linearly stretching sheet problem furthermore using the schauder theory existence of a solution of the third order differential equation over a large but finite interval is established moreover the analytical solutions are obtained and are compared with the corresponding numerical solutions the results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of nonlinear stretching on the flow characteristics

a new algorithm for solving classical blasius equation in this paper a reliable algorithm is employed to investigate the classical blasius equation the algorithm is based mainly on applying the adomian decomposition method to the transformation of the blasius equation the results demonstrate reliability and efficiency of the proposed algorithm

polynomial numerical hulls of the direct sum of a normal matrix and a jordan block abstract let j k be the k k jordan block with eigenvalue and let n be an m m normal matrix in this paper we study the polynomial numerical hulls of order 2 and n 1 for a j k n where n m k we obtain a necessary and sufficient condition such that v 2 a has an interior point also we analytically characterize v 2 j 2 n and we show that if n is co linear then v 2 j 2 n a n v 2 j 2 a finally we study v n 1 a and we show that if n is neither co linear nor co circular then v n 1 a has at most one point more than a

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

the polynomial numerical hulls of jordan blocks and related matrices abstract the polynomial numerical hull of degree k for a square matrix a is a set designed to give useful information about the norms of polynomial functions of the matrix it is defined as z c p a p z for all p of degree k or less whilethese sets have been computed numerically for a number of matrices the computations have not been verified analytically in most cases in this paper we show analytically that the 2 norm polynomial numerical hulls of degrees 1 through n 1 for an n by n jordan block are disks about the eigenvalue with radii approaching 1 as n and we prove a theorem characterizing these radii rk n in the special case where k n 1 this theorem leads to a known result in complex approximation theory for n even rn 1 n is the positive root of 2rn r 1 0 and for n odd it satisfies a similar formula for large n this means that rn 1 n 1 log 2n n log log 2n n these results are used to obtain bounds on the polynomial numerical hulls of certain degrees for banded triangular toeplitz matrices and for block diagonal matrices with triangular toeplitz blocks

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

thermal boundary layer problems in a semi infinite flat plate by the differential transformation method this paper investigates the velocity and temperature field in the thermal boundary layer over a semi infinite flat plate the differential transformation method is used to determine some solutions of these velocity and thermal boundary layer problems numerical results for the dimensionless temperature profiles are presented graphically for different values of prandtl number it is found that the results obtained by the present method are in good agreement with those provided by other numerical methods

a finite difference method for the falkner skan equation we present a finite difference method for the solution of the falkner skan equation the method uses a coordinate transformation to map a semi infinite physical domain to the unit interval 0 1 after using a suitable change of variables the transformed third order boundary value problem for this domain is solved using a finite difference scheme the numerical solutions obtained by the present method are in agreement with those obtained by previous authors

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

a simple perturbation approach to blasius equation in this paper we couple the iteration method with the perturbation method to solve the well known blasius equation the obtained approximate analytic solutions are valid for the whole solution domain comparison with howarth s numerical solution reveals that the proposed method is of high accuracy the first iteration step leads to 6 8 accuracy and the second iteration step yields the 0 73 accuracy of initial slop

a new analytical solution branch for the blasius equation with a shrinking sheet a similarity equation of the momentum boundary layer is analytically studied for a moving flat plate with mass transfer in a stationary fluid by a newly developed technique namely homotopy analysis method ham the equation shows its significance for the practical problem of a shrinking sheet with a constant velocity and only admits the existence of the solution with mass suction at the wall surface the present work provides analytically new solution branch of the blasius equation with a shrinking sheet in different solution areas including both multiple solutions and unique solution with the aid of an introduced auxiliary function the analytical results show that quite complicated behavior with three different solution areas controlled by two critical mass transfer parameters exists which agrees well with the numerical techniques and greatly differs from the continuously stretching surface problem and the blasius problem with a free stream the new analytical solution branch of the blasius equation with a shrinking sheet enriches the solution family of the blasius equation and helps to deeply understand the blasius equation

viscous flow over a nonlinearly stretching sheet an analysis is carried out to study the flow and heat transfer characteristics in a viscous fluid over a nonlinearly stretching sheet the equations of conservation of momentum mass and energy which govern the flow and heat transfer are partially decoupled using a similarity transformation which is different from that of the linearly stretching sheet problem however the results for the linearly stretching sheet problem can be obtained as a special case the basic nonlinear differential equation for the velocity field f and the differential equation with variable coefficient for the temperature field q are solved numerically by using a fourth order runge kutta integration scheme the numerical results thus obtained for the flow and heat transfer characteristics are presented by a table and graphs furthermore it is shown that the heat flow is always from the stretching sheet to the fluid

singularly perturbed bvp to estimation of diaphragm deflection in mems capacitive microphone in this paper we implement the adomian decomposition method adm to approximate the deflection of polysilicon diaphragm with small flexural rigidity of micro electro mechanical system mems capacitive microphone we prove the existence and the uniqueness of the solution of considered problem with the help of the theory of semi group special attention is given to prove the convergence of the implemented method numerical simulation and comparison between the approximate solutions by using the adm with the variational iteration method are given

polynomial numerical hulls of the direct sum of two jordan blocks abstract let j n be the n n jordan block with a positive real eigenvalue and a j n j n in this paper we study the polynomial numerical hull of degree k h k a when k 2 and k 2 n 1 if 0 3 2 we show that r 2 n r 2 n h 2 a where r 2 n represents the radius of the circular disk h 2 j n 0 and if 2 then there exists 0 such that h 2 a x i y c x 2 y 2 also we give explicit formulas for computing the sets h 3 a and h 2 a r when n 2 although the spectrum and numerical range are continuous set valued functions with respect to the hausdorff metric by an example we show that polynomial numerical hulls may not be even lower semicontinuous

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

rational scaled generalized laguerre function collocation method for solving the blasius equation in this paper we propose a collocation method for solving the blasius equation the blasius equation is a third order nonlinear ordinary differential equation this approach is based on a rational scaled generalized laguerre function collocation method we also present the comparison of this work with some well known results and show that the present solution is accurate

complex singularities and the lorenz attractor the lorenz attractor is one of the best known examples of applied mathematics however much of what is known about it is a result of numerical calculations and not of mathemat ical analysis as a step toward mathematical analysis we allow the time variable in the three dimensional lorenz system to be complex hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane knowledge of singularities being fundamental to any investigation in the complex plane we build upon earlier work and give a complete and consistent formal development of complex singular ities of the lorenz system using the psi series the psi series contain two undetermined constants in addition the location of the singularity is undetermined as a consequence of the autonomous nature of the lorenz system we prove that the psi series converge using a technique that is simpler and more powerful than that of hille thus implying a two parameter family of singular solutions of the lorenz system we pose three ques tions answers to which may bring us closer to understanding the connection of complex singularities to lorenz dynamics

the use of generalized laguerre polynomials in spectral methods for nonlinear differential equations the expansion of products of generalized laguerre polynomials l n x in terms of a series of generalized laguerre polynomials is considered the expansion coefficients which are equal to triple product integrals of generalized laguerre polynomials are expressed in terms of a three index recurrence relation this is reduced to a one index relation which facilitates computation of the expansion coefficients the results are useful in the solution of nonlinear differential equations when it is desired to express products of generalized laguerre polynomials as a linear series of these functions as an application we use the results to compute a spectral solution of a nonlinear boundary value problem namely the blasius equation on a semi infinite interval by using a truncated series containing the first eight polynomials l1 2n x a solution is obtained within 4 accuracy

extended self similarity in turbulent flows we report on the existence of a hitherto undetected form of self similarity which we call extended self similarity ess ess holds at high as well as at low reynolds number and it is characterized by the same scaling exponents of the velocity differences of fully developed turbulence

the blasius function computations before computers the value of tricks undergraduate projects and open research problems the blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick semi infinite flat plate the blasius function is the solution to 2 f xxx f f xx 0 on x in 0 infty subject to f 0 f x 0 0 f x infty 1 we use this famous problem to illustrate several themes first although the flow solves a nonlinear partial differential equation pde toepfer successfully computed highly accurate numerical solutions in 1912 his secret was to combine a runge kutta method for integrating an ordinary differential equation ode initial value problem with some symmetry principles and similarity reductions which collapse the pde system to the ode shown above this shows that pde numerical studies were possible even in the precomputer age the truth both a hundred years ago and now is that mathematical theorems and insights are an arithmurgist s best friend and they can vastly reduce the computational burden second we show that special tricks applicable only to a given problem can be as useful as the broad general methods that are the fabric of most applied mathematics courses the importance of particularity in spite of these triumphs many properties of the blasius function f x are unknown we give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician

numerical solutions of the classical blasius flat plate problem abstract this paper presents a numerical study of the non linear differential equation af ff 0 where a prime denotes differentiation with respect to the similarity variable and a is a parameter for a 1 and a 2 this equation is a form of the blasius relation for the flat plate flow in fluid mechanics several numerical solution are obtained using a runge kutta algorithm for high order initial value problems for 1 a 2

laguerre collocation method for the flow and heat transfer due to a permeable stretching surface embedded in a porous medium with a second order slip and viscous dissipation a numerical method is given for studying the effect of viscous dissipation on the steady flow with heat transfer of newtonian fluid towards a permeable stretching surface embedded in a porous medium with a second order slip the governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations by using similarity variables the proposed method is based on replacement of the unknown function by truncated series of well known laguerre expansion of functions an approximate formula of the derivative is introduced special attention is given to study the convergence analysis and derive an upper bound of the error of the presented approximate formula the introduced method converts the proposed equation by means of collocation points to a system of algebraic equations with laguerre coefficients thus by solving this system of equations the laguerre coefficients are obtained graphically results are shown for non dimensional velocities and temperature the effects of the porous parameter the suction injection parameter eckert number first and second order velocity slip parameter and the prandtl number on the flow and temperature profiles are presented moreover the local skin friction and nusselt numbers are presented a comparison of numerical results is made with the earlier published results under limiting cases

characterizations of the polynomial numerical hull of degree k six characterizations of the polynomial numerical hull of degree k are established for bounded linear operators on a hilbert space it is shown how these characterizations provide a natural distinction between interior and boundary points one of the characterizations is used to prove that the polynomial numerical hull of any fixed degree k for a toeplitz matrix whose symbol is piecewise continuous approaches all or most of that of the infinite dimensional toeplitz operator as the matrix size goes to infinity

on a certain boundary value problem arising in shrinking sheet flows abstract this paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic mhd viscous flow influenced by a shrinking sheet with suction for the cases of two dimensional m 1 and axisymmetric m 2 shrinking the influences of the parameter m as well as the effects of suction parameter s and hartmann number m 2 on similar entrainment velocity f and flow characteristics are studied to this purpose the resulting nonlinear ordinary differential equation is solved numerically using the 4th order runge kutta method in combination with a shooting procedure the obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f 0 and the entrainment velocity f as function of the mass transfer parameter s can also be obtained

application of fixed point method to obtain semi analytical solution to blasius flow and its variation the classical blasius flow and its variation sakiadis flow are investigated by the fixed point method fpm and the semi analytical solutions are obtained different from early perturbation series solutions the present semi analytical solutions are valid in the whole region without using matching procedure moreover these solutions can correctly provide the asymptotic property of the governing equations the comparison of the present results with numerical results is carried out which reveals that fpm is a convenient and effective tool for nonlinear problems

numerical transformation methods blasius problem and its variants blasius problem is the simplest nonlinear boundary layer problem we hope that any approach developed for this epitome can be extended to more difficult hydrodynamics problems with this motivation we review the so called topfer transformation which allows us to find a non iterative numerical solution of the blasius problem by solving a related initial value problem and applying a scaling transformation the applicability of a non iterative transformation method to the blasius problem is a consequence of its partial invariance with respect to a scaling group several problems in boundary layer theory lack this kind of invariance and cannot be solved by non iterative transformation methods to overcome this drawback we can modify the problem under study by introducing a numerical parameter and require the invariance of the modified problem with respect to an extended scaling group involving this parameter then we apply initial value methods to the most recent developments involving variants and extensions of the blasius problem

viscous flow and heat transfer over a nonlinearly stretching sheet this paper presents a numerical analysis for flow and heat transfer in a viscous fluid over a nonlinear stretching sheet by employing a novel numerical procedure and two cases are studied namely i the sheet with constant surface temperature cst case and ii the sheet with prescribed surface temperature pst case the governing partial differential equations are converted into highly nonlinear ordinary differential equations by a similarity transformation the variations of dimensionless surface temperature dimensionless surface temperature gradient as well as flow and heat transfer characteristics with the governing parameters are graphed and tabulated

linear maps preserving the polynomial numerical radius of matrices abstract let n 2 be a fixed integer and denote by m n the algebra of all n n complex matrices fix also an integer k such that 1 k n we prove that if m n m n is a linear map which preserves the polynomial numerical radius of order k there exist then a unitary n n complex matrix u and a complex number of modulus one such that either t u t u for all t m n or t u t t u for all t m n

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

fields of values and iterative methods abstract the performance of an iterative scheme to solve x t x c t c n n c c n is often judged by spectral properties of t if t is not normal it is however well known that only conclusions about the assymptotic behavior of an iterative method can be drawn from spectral information to anticipate the progress of the iteration after a finite number of steps the knowledge of the eigenvalues alone is often useless in addition the spectrum of t may be highly sensitive to perturbations if t is not normal an iterative method which on the basis of some spectral information is predicted to converge rapidly for t may well diverge if t is slightly perturbed in practice the convergence of the iteration x m t x m 1 c is therefore frequently measured by some norm verbar t verbar rather than by the spectral radius t but apart from the fact that norms lead to error estimates which are often too pessimistic they cannot be used to analyze more general schemes such as e g the chebyshev iterative methods here we discuss another tool to analyze the behavior of an iterative method namely the field of values w t the collection of all rayleigh quotients of t w t contains the eigenvalues of t and the numerical radius t max z w t z defines a norm on c n n the field of values represents therefore an intermediate concept to judge an iterative scheme by it is related to the spectral approach but has also certain norm properties

a constructive approach to regularity of lagrangian trajectories for incompressible euler flow in a bounded domain the 3d incompressible euler equations are an important research topic in the mathematical study of fluid dynamics not only is the global regularity for smooth initial data an open issue but the behaviour may also depend on the presence or absence of boundaries for a good understanding it is crucial to carry out besides mathematical studies high accuracy and well resolved numerical exploration such studies can be very demanding in computational resources but recently it has been shown that very substantial gains can be achieved first by using cauchy s lagrangian formulation of the euler equations and second by taking advantage of analyticity results of the lagrangian trajectories for flows whose initial vorticity is holder continuous the latter has been known for about 20 years serfati in j math pures appl 74 95 104 1995 but the combination of the two which makes use of recursion relations among time taylor coefficients to obtain constructively the time taylor series of the lagrangian map has been achieved only recently frisch and zheligovsky in commun math phys 326 499 505 2014 podvigina et al in j comput phys 306 320 342 2016 and references therein here we extend this methodology to incompressible euler flow in an impermeable bounded domain whose boundary may be either analytic or have a regularity between indefinite differentiability and analyticity non constructive regularity results for these cases have already been obtained by glass et al ann sci ec norm sup 45 1 51 2012 using the invariance of the boundary under the lagrangian flow we establish novel recursion relations that include contributions from the boundary this leads to a constructive proof of time analyticity of the lagrangian trajectories with analytic boundaries which can then be used subsequently for the design of a very high order cauchy lagrangian method

the cauchy lagrangian method for numerical analysis of euler flow a novel semi lagrangian method is introduced to solve numerically the euler equation for ideal incompressible flow in arbitrary space dimension it exploits the time analyticity of fluid particle trajectories and requires in principle only limited spatial smoothness of the initial data efficient generation of high order time taylor coefficients is made possible by a recurrence relation that follows from the cauchy invariants formulation of the euler equation zheligovsky and frisch 2014 44 truncated time taylor series of very high order allow the use of time steps vastly exceeding the courant friedrichs lewy limit without compromising the accuracy of the solution tests performed on the two dimensional euler equation indicate that the cauchy lagrangian method is more and occasionally much more efficient and less prone to instability than eulerian runge kutta methods and less prone to rapid growth of rounding errors than the high order eulerian time taylor algorithm we also develop tools of analysis adapted to the cauchy lagrangian method such as the monitoring of the radius of convergence of the time taylor series certain other fluid equations can be handled similarly