multiple correction and continued fraction approximation abstract the main aim of this paper is to further develop a multiple correction method formulated in a previous work 6 as applications we find a kind of hybrid type finite continued fraction approximations in two cases of landau constants and lebesgue constants in addition we refine the previous results of lu 29 and xu and you 48 for the euler mascheroni constant

unified treatment of several asymptotic expansions concerning some mathematical constants recently various approximation formulas for some mathematical constants have been investigated and presented by many authors in this paper we first find that the relationship between the coefficients pj and qj is such that xj 0qjxj ln xj 0pjxj x where is the logarithmic derivative of the gamma function often referred to as psi function and p0 q0 1 next by using this result we give a unified treatment of several asymptotic expansions concerning the eulermascheroni constant landau and lebesgue constants glaisherkinkelin constant and choisrivastava constants

extension and improvement to the krylov bogoliubov methods using elliptic functions abstract it is shown that the krylov bogoliubov methods that give the approximate oscillatory solution of the equation in terms of jacobi elliptic functions are applicable not only when c 1 0 and c 3 0 but also when c 1 0 and c 3 0 in particular the most precise of these methods the christopher brocklehurst method is discussed in detail its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters

the riemann hilbert problem and the generalized neumann kernel on multiply connected regions this paper presents and studies fredholm integral equations associated with the linear riemann hilbert problems on multiply connected regions with smooth boundary curves the kernel of these integral equations is the generalized neumann kernel the approach is similar to that for simply connected regions see r wegmann a h m murid m m s nasser the riemann hilbert problem and the generalized neumann kernel j comput appl math 182 2005 388 415 there are however several characteristic differences which are mainly due to the fact that the complement of a multiply connected region has a quite different topological structure this implies that there is no longer perfect duality between the interior and exterior problems we investigate the existence and uniqueness of solutions of the integral equations in particular we determine the exact number of linearly independent solutions of the integral equations and their adjoints the latter determine the conditions for solvability an analytic example on a circular annulus and several numerically calculated examples illustrate the results

numerical computation of the capacity of generalized condensers abstract we present a boundary integral method for numerical computation of the capacity of generalized condensers the presented method applies to a wide variety of generalized condenser geometry including the cases when the plates of the generalized condenser are bordered by piecewise smooth jordan curves or are rectilinear slits the presented method is used also to compute the harmonic measure in multiply connected domains

laplace s equation and the dirichlet neumann map a new mode for mikhlin s method mikhlin s method for solving laplace s equation in domains exterior to a number of closed contours is discussed with particular emphasis on the dirichlet neumann map in the literature there already exist two computational modes for mikhlin s method here a new mode is presented the new mode is at least as stable as the previous modes furthermore its computational complexity in the number of closed contours is better as a result highly accurate solutions in domains exterior to tens of thousands of closed contours can be obtained on a simple workstation

spiral slits map and its inverse of bounded multiply connected regions this paper presents a boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto a spiral slits region the method is an extension of the author s method for computing the circular slits map of bounded multiply connected regions see a w k sangawi et al 2012 16 several numerical examples are given to prove the effectiveness of the proposed methods

radial and circular slit maps of unbounded multiply connected circle domains infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain or to domains with both radial and circular

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

a charge simulation method for the numerical conformal mapping of interior exterior and doubly connected domains abstract a simple numerical method is described for computing the following three conformal maps a from a domain interior to a closed jordan curve onto the interior of the unit disk b from a domain exterior to a closed jordan curve onto the exterior of the unit disk and c from a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials without integration typical examples show that numerical results of high accuracy can be obtained by the simple method if the domains have no concave corners

a charge simulation method for numerical conformal mapping onto circular and radial slit domains a simple numerical method is described for computing the following two conformal maps a from a domain exterior to closed jordan curves onto a circular slit domain and b from a domain exterior to closed jordan curves onto a radial slit domain they constitute a dual problem and can be computed in a dual way the numerical method is based on the charge simulation method or the method of fundamental solutions applied to the dirichlet problem of laplace s equation in which a pair of conjugate harmonic functions are approximated by a linear combination of complex logarithmic potentials the unknown coefficients are determined by the collocation condition imposed on the real part or the imaginary part the modulus or the argument of the approximate mapping function effectiveness of the method is demonstrated by some typical examples

numerical conformal mappings of bounded multiply connected domains by the charge simulation method we present here a simple method of numerical conformal mappings from bounded multiply connected domains onto the bounded canonical slit domains of nehari i e a disk with concentric circular slits and an annulus with concentric circular slits in this method we express the mapping functions in terms of a pair of conjugate harmonic functions and approximate them using the charge simulation method by a linear combination of complex logarithmic functions some numerical examples show the effectiveness of the method

applications of numerical conformal mapping abstract the application of conformal mapping methods to the solution of free surface flow problems is considered methods of numerical conformal mapping based on fourier series are extended to handle efficiently problems with time dependent boundaries they are shown to be practicable only for moderately distorted geometries extensions of the menikoff zemach method to breaking geometries are presented these latter methods are robust at quite large distortions but degrade prematurely in time dependent problems at amplitudes smaller than achieved by our recent vortex methods

multiple correction and continued fraction approximation ii the main aim of this paper is to further develop the multiple correction method formulated in our previous works cao et al 2015 cao 2015 as applications we establish a kind of hybrid type finite continued fraction approximations related to bbp type series of the constant and other classical constants such as catalan constant or 2

numerical conformal mapping of multiply connected regions onto the fifth category of koebe s canonical slit regions abstract this paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of koebe s classical canonical slit regions the method is based on a uniquely solvable boundary integral equation with generalized neumann kernel the results of some test calculations illustrate the performance of the presented method

an elliptic perturbation method for certain strongly non linear oscillators abstract an elliptic perturbation method is presented for calculating periodic solutions of strongly non linear oscillators of the formx c1x c3x3 f x x in which the jacobian elliptic functions are employed instead of usual circular functions in the conventional perturbation procedure three types of duffing van der pol equation withf x x 1 x2 xare studied in detail the present method can give a second approximate solution and therefore its accuracy is much higher than that of other methods in which the solutions are only first approximations

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

a nystrom method for boundary integral equations in domains with corners we give a convergence and error analysis for a nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic dirichlet problem in plane domains with corners

sharp bounds of the landau constants the aim of this paper is to establish new bounds of the landau constants 1 introduction and motivation the landau constants de ned for all positive integers n by gn 1 1 2 2 1 3 2 4 2 1 3 5 2 4 6 2 1 3 2n 1 2 4 2n 2 play an important role in some extremal problems in complex analysis and in the theory of fourier series more precisely in 1913 landau 6 proved that gn is the maximum of the expression j pn k 1 akj with respect to all functions of the form f z p1 k 0 akz k which is analytic in the unit disk and satis es jf z j 1 for every jzj 1 in consequence the problem of approximation of the landau constants have attracted the attention of many authors in particular landau himself studied the asymptotic behaviour of gn and showed that gn 1 lnn then watson 7 established the following asymptotic formula gn c0 1 ln n 1 1 4 n 1 o 1 n2 n 1 here and in what follows c0 1 4 ln 2 1 06627 and 0 577215 is the euler mascheroni constant this problem of approximation of the landau constants was continued in the works of brutman 3 and falaleev 5 who proved that for every non negative integer n 1 1 ln n 1 gn 1 0663 1 ln n 1 respective 1 0662 1 ln n 3 4 gn 1 0916 1 ln n 3 4 we improve the upper bound in the following way that also shows that the constant 3 4 is the best possible date november 18 2009 1991 mathematics subject classi cation primary 41a60 secondary 26d15

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

equation of motion for point vortices in multiply connected circular domains the paper gives the equation of motion for n point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles called the circular domain the velocity field induced by the point vortices is described in terms of the schottky klein prime function associated with the circular domain the explicit representation of the equation enables us not only to solve the euler equations through the point vortex approximation numerically but also to investigate the interactions between localized vortex structures in the circular domain as an application of the equation we consider the motion of two point vortices with unit strength and of opposite signs when the multiply connected domain is symmetric with respect to the real axis the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle which we investigate in detail

numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions this paper presents a boundary integral equation method with the adjoint generalized neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions for each canonical region two integral equations are solved before one can approximate the boundary values of the mapping function cauchy s type integrals are used for computing the mapping function and its inverse for interior points this method also works for regions with piecewise smooth boundaries three examples are given to illustrate the effectiveness of the proposed method

a hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators abstract a hyperbolic perturbation method is presented for determining the homoclinic solution of certain strongly nonlinear autonomous oscillators of the form x c 1 x c 2 x 2 e f x x in which hyperbolic functions can be employed instead of the usual periodic functions in the perturbation procedure the generalized van der pol oscillator in which f x x 1 x 2 x 2 x is studied to illustrate the accuracy of the present method its predictions are compared with those of runge kutta method

existence and uniqueness of rectilinear slit maps we consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently koebe proved that for domains of finite connectivity there is up to a normalization a unique rectilinear slit map achieving any given angle assignmnent koebe s theorem is partially extended to domains of infinite connectivity a uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments and ani existence result is proved for arbitrary domains under the assumption that the angle assigrnment is continuous and has finite range in order to prove the existence result a new extremal length tool called the crossing module is introduced the crossing module allows greater freedom in the family of admissible arcs than the classical module several results known for the module are extended to the crossing module a generalization of jenkins 0 module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing module and it is shown that rectilinear slit maps satisfying this crossing module condition exist

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

fast conformal mapping of multiply connected regions abstract an iterative method is presented which constructs for an unbounded region g with m holes and sufficiently smooth boundary a circular region h and a conformal mapping from h to g with the usual normalization both h and are uniquely determined by g with a few modifications the method can also be applied to a bounded region g with m holes the canonical region h is then the unit disc with m circular holes the proposed method also determines the centers and radii of the boundary circles of h and requires at each iterative step the solution of a riemann hilbert rh problem which has a unique solution numerically the rh problem can be treated efficiently by the method of successive conjugation using the fast fourier transform fft the iteration for the solution of the rh problem converges linearly the conformal mapping method converges quadratically the results of some test calculations exemplify the performance of the method

inequalities for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 by gn k 0n1 16k 2 and ln 12 sin n t sin t dt respectively we establish sharp inequalities for gn and ln 2 in terms of the logarithmic derivative of the gamma function further we prove that the sequence gn is completely monotonic we provide best possible upper and lower bounds for the ratios gn 1 gn 1 gn and l n 1 2 l n 1 2 ln 2 and we present sharp bounds for ln 2 gn and ln 2 gn

a simplified fornberg like method for the conformal mapping of multiply connected regions comparisons and crowding we present a new fornberg like method for the numerical conformal mapping of multiply connected regions exterior to circles to multiply connected regions exterior to smooth curves the method is based on new symmetric conditions for analytic extension of functions given on circular boundaries we also briefly discuss a similar method due to wegmann and compare some computations with both methods some examples of regions which exhibit crowding of the circles are also presented

asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by gn k 0n116k 2kk 2andln 12 sin n 12 t sin 12t dt which play important roles in the theories of complex analysis and fourier series respectively certain inequalities and asymptotic expansions for the constants gn and ln have been investigated by many authors here we aim at establishing new asymptotic expansions for the constants gn and ln of landau and lebesgue respectively by mainly using bell polynomials and the partition function

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

full asymptotic expansions of the landau constants via a difference equation approach abstract we derive full asymptotic expansions for the landau constants g n as n some of the expansions are not new yet all the coefficients of the expansions are given iteratively in an explicit manner and are more efficiently evaluated as compared with the known results we obtain the asymptotic formulas old and new by applying the theory of wong and li for second order linear difference equations in deriving the expansions we have also confirmed a conjecture made by nemes and nemes

numerical conformal mapping via the bergman kernel using the generalized minimum residual method abstract the bergman kernel function is known to satisfy a certain boundary integral equation of the second kind for boundaries that possess symmetrical qualities the integral equation can be transformed into another integral equation that uses only a small part of the original boundary this paper applies an iterative procedure known as the generalized minimum residual method for the computation of the riemann mapping function via the bergman kernel the complexity of this procedure is o n 2 where n is the number of collocation points on the boundary of the region numerical implementation on some test regions is also presented

optimal rate of convergence for sequences of a prescribed form abstract we provide a highly convergent version of the generalized euler sequence and we determine the element with the optimal rate of convergence in several classes of sequences of a prescribed form the new approach extends and unifies many previous efforts in this direction in addition we answer an open problem on the asymptotic expansion of the harmonic number sequence

rapid methods for the conformal mapping of multiply connected regions abstract we present fast methods for the conformal mapping of simply doubly and multiply connected regions onto certain canonical regions in the plane our mapping procedure consists of two parts first we solve an integral equation on the boundary of the region we wish to map the solution of this integral equation is needed to determine the boundary correspondence we have chosen to use the integral equation formulation of mikhlin although it is not widely used this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels the solutions are also periodic allowing for effective use of the trapezoid rule once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region this method makes use of fast poisson solvers and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region and avoids the expense of computing many integrals we also provide results of numerical experiments

boundary integral equations with the generalized neumann kernel for laplace s equation in multiply connected regions this paper presents a new boundary integral method for the solution of laplace s equation on both bounded and unbounded multiply connected regions with either the dirichlet boundary condition or the neumann boundary condition the method is based on two uniquely solvable fredholm integral equations of the second kind with the generalized neumann kernel numerical results are presented to illustrate the efficiency of the proposed method

numerical computation of mityuk s function and radius for circular radial slit domains abstract we consider mityuk s function and radius which have been proposed in 15 as generalizations of the reduced modulus and conformal radius to the cases of multiply connected domains we present a numerical method to compute mityuk s function and radius for canonical domains that consist of the unit disk with circular radial slits our method is based on the boundary integral equation with the generalized neumann kernel special attention is given to the validation of the theoretical results on the existence of critical points and the boundary behavior of mityuk s radius

the riemann hilbert problem and the generalized neumann kernel this paper presents two new fredholm integral equations associated to the interior and the exterior riemann hilbert problems in simply connected regions with smooth boundaries the kernel of these integral equations is the generalized neumann kernel the solvability of the integral equations depends on whether l 1 are eigenvalues of the kernel which in turn depends on the index of the riemann hilbert problem the complete discussion of the solvability of the integral equations with the generalized neumann kernel is presented the integral equations can be used effectively to solve numerically the riemann hilbert problems the case of non uniquely solvable riemann hilbert problems is treated by imposing additional constraints to get a uniquely solvable problem fredholm integral equations with generalized neumann kernels are also derived for the problem of the interior and the exterior harmonic conjugate functions as applications we study the problem of conformal mapping to a nearby region and extend wegmann s iterative method to general regions numerical examples reveal that the present method offers an effective solution technique for the riemann hilbert problems when the boundaries are sufficiently smooth

proofs of two conjectures on the landau constants abstract in this paper we derive a family of asymptotic expansions for the nth landau constant g n the proof is based on a new general integral representation of the constants in terms of a hypergeometric function as a consequence we prove two conjectures related to earlier asymptotic developments of the g n s

inequalities and asymptotic expansions for the constants of landau and lebesgue the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively diverse inequalities and approximations for these constants have been investigated and developed by many authors here in this paper we establish new asymptotic expansions for the constants g n and l n 2 of landau and lebesgue respectively in terms of the digamma and polygamma functions based on our expansion for the landau constants g n we present new bounds for the landau constants g n in terms of the digamma and polygamma functions we also establish inequalities for the lebesgue constants l n 2 which are applied to derive an asymptotic expansion for l n 2 in terms of 1 n 1 furthermore by giving numerical calculations to be compared among several developed asymptotic expansions for the constants g n and l n 2 it is shown that our expansions presented here would be best ones

parallel slits map of bounded multiply connected regions abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a parallel slit region the method is based on some uniquely solvable boundary integral equations with adjoint classical adjoint generalized and modified neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method

construction of approximate analytical solutions to a new class of non linear oscillator equation the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire by assuming that harmonic balance will hold solutions are devised for a steady state limit cycle and or limit point motion a method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective fourier expansions the latter technique is actually a generalization of the method proposed by kryloff and bogoliuboff 1943

fast solver for plane potential problems with mixed boundary conditions abstract the present paper proposes a new fast algorithm for plane potential problems on simply connected domains with mixed boundary conditions dirichlet and von neumann conditions are given on complementary parts of the boundary the method is based on the solution of a corresponding riemann hilbert problem in terms of cauchy integrals they can be computed efficiently and accurately by means of fast fourier transforms once the problem has been transformed conformally onto a standard domain and singularities have been extracted analytically

numerical conformal mapping of multiply connected regions onto the second third and fourth categories of koebeʼs canonical slit domains abstract this paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region g onto the second third and fourth categories of koebe s canonical slit domains the method can be also used for calculating the conformal mappings of simply and doubly connected regions the method is an extension of the author s method for the first category of koebe s canonical slit domains see m m s nasser numerical conformal mapping via a boundary integral equation with the generalized neumann kernel siam j sci comput 31 2009 1695 1715 three numerical examples are presented to illustrate the performance of the proposed method

numerical conformal mapping via the szego kernel abstract a new method to compute the riemann mapping function is numerically implemented and tested on examples the method expresses the szego kernel as the solution of a second kind integral equation the equation has its origin in an earlier non numerical work of stein and one of the authors 8 the experimental results show the algorithm to be effective and stable

secondary schottky klein prime functions associated with multiply connected planar domains in recent years a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the schottky klein s k prime function of an underlying compact riemann surface known as the schottky double of the domain in this paper we describe additional function theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary s k prime functions the basic idea develops and extends an observation of burnside dating back to 1892 applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given other possible applications are also surveyed

generalized fourier series for the study of limit cycles the approximate solution to first order of non linear differential equations is studied using the method of harmonic balance with generalized fourier series and jacobian elliptic functions as an interesting use of the series very good analytic approximations to the limit cycles of lienard s ordinary differential equation ode x g x f x x are presented specifically it is shown that contrary to an opinion given in a well known textbook on non linear oscillations g x not only modifies the period but influences the topology in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 for 0 1 the presence of zero one or three limit cycles is found to depend on the value of ab

new bounds and asymptotic expansions for the constants of landau and lebesgue abstract the constants of landau and lebesgue are defined for all integers n 0 in order by g n k 0 n 1 16 k 2 k k 2 and l n 1 2 sin n 1 2 t sin 1 2 t d t which play important roles in the theories of complex analysis and fourier series respectively in this paper we establish new bounds and asymptotic expansions for the constants of landau and lebesgue

a modified elliptic lindstedt poincare method for certain strongly non linear oscillators abstract a modified elliptic lindstedt poincare method is presented for the steady state analysis of strongly non linear oscillators of the form x c 1 x c 3 x 3 ef x x in which a new parameter e is employed such that the value of is always small regardless of the magnitude of the original parameter e therefore the above strongly non linear oscillators with large parameter e is transformed into a small parameter system with respect to

computation of capacity this article introduces a method for computing upper and lower bounds for the logarithmic capacity of a compact plane set if the set has the holder continuity property then these bounds converge to the value of the capacity a number of examples are discussed in detail including the cantor middle third set for which we estimate c e 0 220949102189507

numerical conformal mapping via the bergman kernel a new method to compute the riemann mapping function via the bergman kernel is presented the method expresses the bergman kernel as the solution of a second kind integral equation involving the neumann kernel for symmetric regions the integral equation can be transformed into a new one that uses only a small part of the original boundary numerical implementations on some test regions are also presented

numerical conformal mapping of multiply connected regions by fornberg like methods we develop a new algorithm for computing conformal maps from regions exterior to non overlapping disks to unbounded multiply connected regions exterior to non overlapping smoothly bounded jordan regions the method is an extension of fornberg s original newton like method for mapping of the disk to simply connected regions a fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case numerical examples are reported

on moduli of rings and quadrilaterals algorithms and experiments moduli of rings and quadrilaterals are frequently applied in geometric function theory see e g the handbook by kuhnau handbook of complex analysis geometric function theory vols 1 and 2 north holland amsterdam 2005 yet their exact values are known only in a few special cases previously the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation we present here a new hp fem algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the schwarz christoffel toolbox of driscoll and trefethen we also demonstrate that the hp fem algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds

a fast method for solving certain integral equations of the first kind with application to conforma mapping abstract we describe a fast iterative method for solving fredholm integral equations of the first kind whose kernels have a logarithmic principal part the method is a fourier galerkin method and due to the singularity of the kernel the linear system of simultaneous equations is block diagonally dominant and can be solved rapidly by an iterative method we present examples of the computation of conformal mappings of simply and multiply connected domains

linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits abstract in this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a disk with circular slits the method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized neumann kernels these boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region some numerical examples are presented to illustrate the efficiency of the presented method

some sharp estimates of the constants of landau and lebesgue we establish several sharp two sided inequalities involving the constants of landau and lebesgue which occur and play important roles in two related extremal problems in complex analysis and in fourier analysis respectively the main results improves essentially the watson asymptotic expansion formulas for these constants

a note on the landau constants abstract in this paper we establish new asymptotic expansions for the n th landau constant g n an open problem is indicated as well

numerical conformal mapping via a boundary integral equation with the generalized neumann kernel we present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region g onto the five classical canonical slit domains the method is based on a uniquely solvable boundary integral equation with the generalized neumann kernel using the proposed method the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix the method can be also used for calculating the conformal mappings of simply and doubly connected regions the performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and

fractal local fields in random composites the local fields in composites and porous media can have complicated structure because of the fine geometrical inhomogeneity numeric examples show that randomly generated local fields can have complicated fractal structure contrary to the local fields in periodic composites it is demonstrated that the local oscillations of the stresses in random composites are higher than in regular ones this result implies that the damage risk is higher for irregular elastic composites than for regular ones ceteris paribus for viscous fluid this means that irregular locations of obstacles increase local oscillations of the velocity hence lead to turbulence we consider fields governed by the 2d poisson equation 2 u 1 in a perforated domain d corresponding to the host material the holes of d correspond to the torsion problem in elasticity and the hard disks to longitudinal flow of viscous fluid the corresponding dirichlet problem in randomly generated multiply connected domains is solved a method of functional equations is applied following mityushev s scheme for the riemann hilbert type problems in multiply connected domains it is justified that the rate of convergence for functional equations is of order 2 n r 2 where n is the connectivity of the domain whose linear size is normalized to unity and r the radii of holes this observation shows that for large n and sufficiently small r few iterations for the functional equations can give an acceptable numerical result