solving boundary value problems for delay differential equations by a fixed point method a general linear boundary value problem for a nonlinear system of delay differential equations dde in short is reduced to a fixed point problem v a v with a properly chosen generally nonlinear operator a the unknown fixed point v is approximated by piecewise linear function v h defined by its values v i v h t i at grid points t i i 0 1 n where n is a given positive integer and h max 1 i n t i t i 1 under suitable assumptions the existence of a fixed point of a is equivalent to existence of so called e h approximate fixed points of v h a v h which can be found by minimization of l 2 n norm of residuum v h a v h with respect to the variables v i these e h approximate fixed points are used for obtaining approximate solutions of the original boundary value problem for a system of dde numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems one for the periodic distributed delay lotka volterra competition system and the second one modeling two coupled identical neurons with time delayed connections show an efficiency of this kind of approach

an interpolating polynomial method for numerical conformal mapping a simple algorithm is presented for numerical approximation of conformal mappings for simply connected planar regions the desired mapping is represented by a normalized polynomial of degree n determined by the boundary values which it is presumed to take at the nth roots of unity then its values at the n intermediate 2nth roots of unity are used to correct the initial guess each iteration which consists essentially of matrix multiplication and boundary projection costs o n log n arithmetic operations numerical results are provided to confirm linear convergence of the algorithm

existence of sign changing solutions for nonlocal kirchhoff schrodinger type equations in r3 abstract in this paper we investigate the existence of sign changing solution for a class nonlocal kirchhoff schrodinger type equation a b r 3 u 2 d x u v x u f x u x r 3 where a and b are positive constants with the help of the constraint variational method and via a direct approach we show the existence of sign changing solution and prove that the sign changing solution has precisely two nodal domains this work can be regarded as the complement for some results of the literature

the existence of positive periodic solutions of a class of lotka volterra type impulsive systems with infinitely distributed delay in this paper the existence of positive periodic solutions of a class of periodic lotka volterra type impulsive systems with distributed delays is studied by using the continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are obtained which improve and generalize some existing results

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

existence of ground states for a kirchhoff type problem without 4 superlinear condition abstract we consider the existence of ground state solutions for the kirchhoff type problem a b r n u 2 d x u v x u u p 2 u x r n u h 1 r n where a b 0 n 1 2 3 and 2 p 2 here we are interested in the case that 2 p 4 since the existence of ground state for 4 p 2 is easily obtained by a standard variational argument our method is based on a poho z aev type identity

accessory parameters in conformal mapping exploiting the isomonodromic tau function for painleve vi we present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral the schwarz christoffel accessory parameter problem relevant when all sides have zero curvature is also captured within our approach the method exploits the isomonodromic tau function associated with the painleve vi equation recently these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the young diagrams after showing how to extract the monodromy data associated with the target domain we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem the viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

an iterative method for conformal mapping abstract in this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region g onto a region h with smooth boundary the method is based on a certain riemann hilbert problem it turns out that this problem is the linearized version of a singular integral equation of the second kind hence the method is a newton method whenever the boundaries of g and h are sufficiently smooth its convergence is locally quadratic if g is the unit circle the solution of the linearized problem can be represented explicitly in terms of integral transforms from this one derives a quadratically convergent newton like numerical method that avoids the numerical solution of systems of linear equations and therefore in comparison with other methods based on integral equations is quite economical in terms of computer time and storage requirements

a fredholm integral equation of the second kind for conformal mapping abstract we present a fredholm integral equation of the second kind for the derivative of the boundary correspondence function of the conformal mapping of a jordan region with piecewise twice differentiable boundary onto the unit disk one of our derivations of the equation leads to a more general regularisation procedure for some singular integral equations of the first kind the integral equation can be solved numerically by applying the fourier method the convergence to in l 2 of the solution of the discrete equation so obtained follows by simple theorems examples show that the equation often yields better numerical results than both a related method proposed by henrici and warschawski s equation

the conjugate function method and conformal mappings in multiply connected domains the conjugate function method is an algorithm for numerical computation of conformal mappings for simply and doubly connected domains in this paper the conjugate function method is generalized for

the accuracy of numerical conformal mapping methods a survey of examples and results this paper shows how the geometry of the region affects the conditioning and the accuracy of numerical conformal mapping methods for simply connected regions especially fourier series methods both explicit examples of popular test cases and more general estimates are discussed the severe ill conditioning that is known as the crowding phenomenon is discussed and its effect on a conformally transplanted boundary value problem is illustrated remarks on various numerical methods are included

a nonoverlapping domain decomposition method for symm s equation for conformal mapping symm s equation is a first kind integral equation for computing conformal maps of simply connected regions the package confpack solves symm s equation by an indirect boundary element method using an accurate corner representation this solution technique is extended here to include nonoverlapping domain decomposition degrees of freedom are introduced on one or more interfaces and different unknowns are used leading to a system of second kind equations the corresponding linear system can be expressed in schur complement form the accurate treatment of corners is preserved in the new formulation the results of serial and parallel matlab implementations of the new algorithm show significant speedups as the number of unknowns grows

existence of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

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

eigenfrequencies of fractal drums a method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented the approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method the numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in l banjai l n trefethen a multipole method for schwarz christoffel mapping of polygons with thousands of sides siam j sci comput 25 3 2003 1042 1065 the linear system arising from the spectral discretization is large and dense to circumvent this problem we devise a fast method for the inversion of such a system consequently the eigenvalue problem is solved iteratively we obtain eight digits for the first eigenvalue of the koch snowflake and at least five digits for eigenvalues up to the 20th numerical results for two more fractals are shown

stage structured control on a class of predator prey system in almost periodic environment in this paper we consider a kind of stage structured predator prey systems with two predators and general functional responses by applying the comparison theorem lyapunov functional and mawhin s

numerical conformal mapping of circular arc polygons abstract howell l h numerical conformal mapping of circular arc polygons journal of computational and applied mathematics 46 1993 7 28 solutions to the schwarzian differential equation are conformal maps from the upper half plane to circular arc polygons plane regions bounded by straight line segments and arbitrary arcs of circles we develop methods for numerically integrating this equation both directly and through the use of a related linear differential equation particular attention is given to the behavior near corner singularities we also derive alternate versions of the transformation which map from the unit disk and from an infinite strip while the former may be of primarily theoretical interest the latter can be used to map highly elongated regions such as channels for internal flow problems such regions are difficult or impossible to map from the disk or the half plane due to the so called crowding phenomenon

representation of conformal maps by rational functions the traditional view in numerical conformal mapping is that once the boundary correspondence function has been found the map and its inverse can be evaluated by contour integrals we propose that it is much simpler and 10 1000 times faster to represent the maps by rational functions computed by the aaa algorithm to justify this claim first we prove a theorem establishing root exponential convergence of rational approximations near corners in a conformal map generalizing a result of d j newman in 1964 this leads to the new algorithm for approximating conformal maps of polygons then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part there must be a singularity or loss of univalence exponentially close to the boundary and polynomial approximations cannot be accurate unless of exponentially high degree this motivates the application of the new algorithm to smooth domains where it is again found to be highly effective

a multiplicity result for some nonlocal eigenvalue problem with exponential growth condition in this paper a new variational tool is used to establish a multiplicity result for some equation containing a nonlocal term and involving the n laplacian operator and whose nonlinear term can have a critical exponential growth the equation has a nonvariational nature

periodic solutions for delay differential equations model of plankton allelopathy abstract in this paper we propose a modified delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other by using the continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are obtained for the existence of positive periodic solutions for this model

infinitely many small energy solutions for a modified kirchhoff type equation in rn abstract in the present paper we study the following schrodinger kirchhoff quasilinear problem q a b r n u 2 d x u u u 2 v x u f x u x r n where a 0 b 0 n 3 and f c r n r r which includes the modified nonlinear schrodinger equation we prove the existence of infinitely many small energy solutions for equation q by applying clark s theorem to a perturbation functional

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

on a schrodinger kirchhoff type equation involving the p x laplacian abstract in this paper we deal with a schrodinger type equation involving a nonlocal kirchhoff type coefficient and depending on two real parameters working within the framework of variable exponent spaces and using the variational approach we obtain several results of existence and multiplicity of solutions depending on the range of the parameters

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

numerical conformal mapping using cross ratios and delaunay triangulation we propose a new algorithm for computing the riemann mapping of the unit disk to a polygon also known as the schwarz christoffel transformation the new algorithm crdt for cross ratios of the delaunay triangulation is based on cross ratios of the prevertices and also on cross ratios of quadrilaterals in a delaunay triangulation of the polygon the crdt algorithm produces an accurate representation of the riemann mapping even in the presence of arbitrary long thin regions in the polygon unlike any previous conformal mapping algorithm we believe that crdt solves all difficulties with crowding and global convergence although these facts depend on conjectures that we have so far not been able to prove we demonstrate convergence with computational experiments the riemann mapping has applications in two dimensional potential theory and mesh generation we demonstrate crdt on problems in long thin regions in which no other known algorithm can perform comparably

the nehari manifold for a kirchhoff type problem involving sign changing weight functions abstract this paper examines a class of kirchhoff type equations that involve sign changing weight functions using nehari manifold and fibering map the existence of multiple positive solutions is established

conjugate function method for numerical conformal mappings we present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so called canonical domains which in our case are rectangles or annuli the method is based on conjugate harmonic functions and properties of quadrilaterals several numerical examples are given

kirchhoff systems with dynamic boundary conditions abstract we are interested in the study of the global non existence of solutions of hyperbolic nonlinear problems governed by the p kirchhoff operator under dynamic boundary conditions when p p n with p n 2 the systems involve nonlinear external forces and may be affected by a perturbation of the type u p 2 u several models already treated in the literature are covered in special subcases and concrete examples are provided for the source term f and the external nonlinear boundary damping q

multiple solutions for p kirchhoff equations in rn abstract in this paper we prove the existence of multiple ground state solutions for the nonhomogeneous p kirchhoff elliptic equation 0 1 m r n u p v x u p d x p u v x u p 2 u f x u g x in r n where v x c r n and v x as x the nonlinear function f x u is continuous and satisfies some conditions the solutions will be obtained by the mountain pass theorem ekeland s variational principle and krasnoselskii s genus theory in struwe 2000 1

on a periodic multi species ecological model a periodic predator prey model with m predators and n preys is proposed in this paper which can be seen as the modification of the traditional lotka volterra model by using comparison theorem the ultimately bounded region of the system is obtained by using comparison theorem and brouwer fixed point theorem sufficient conditions with guarantee the existence of a positive periodic solution of the system is obtained finally by constructing a suitable lyapunov function some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of system the results obtained here generalized the main results of j d zhao w c chen global asymptotic stability of a periodic ecological model applied mathematics and computation 147 3 2004 881 892

revisiting the crowding phenomenon in schwarz christoffel mapping we address the problem of conformally mapping the unit disk to polygons with elongations the elongations cause the derivative of the conformal map to be exponentially large in some regions this crowding phenomenon creates difficulties in standard numerical methods for the computation of the conformal map we make use of the schwarz christoffel representation of the mapping and show that a simple change to the existing algorithms introduced by trefethen siam j sci statist comput 1 1980 pp 82 102 makes it feasible to accurately compute conformal maps to polygons even in the presence of extreme crowding for an efficient algorithm it is essential that a good initial guess for the parameters of the schwarz christoffel map be available a uniformly close initial guess can be obtained from the cross ratios of certain quadrilaterals as introduced in the crdt algorithm of driscoll and vavasis siam j sci comput 19 1998 pp 1783 1803 we present numerical experiments and compare our algorithms with the crdt which has been particularly designed to combat crowding

existence and multiplicity of solutions for an indefinite kirchhoff type equation in bounded domains we study the indefinite kirchhoff type problem where is a smooth bounded domain in and we require that f is sublinear at the origin and superlinear at infinity using the mountain pass theorem and ekeland variational principle we obtain the multiplicity of non trivial non negative solutions we improve and extend some recent results in the literature

permanence in nonautonomous multi species predator prey system with feedback controls a nonautonomous multi species predator prey system with feedback controls is proposed in this paper where the competition among the predator species and among the prey species is considered also some important factors such as the effect of toxins and the age structure are also considered average conditions are obtained for permanence and global attractivity in the system the results obtained here generalized the main results of zhao and jiang j d zhao j f jiang permanence in nonautonomous lotka volterra system with predator prey appl math computat 152 2004 99 109

spatio temporal complexity of a delayed diffusive model for plant invasion abstract in this paper we develop a diffusive plant invasion model with delay under the homogeneous neumann boundary condition first the existence and uniqueness of a non negative solution persistence property and local asymptotic stability of the constant steady states are established then by analyzing the associated characteristic equation the stability of steady states and the existence of hopf bifurcation are demonstrated under special circumstance the discontinuous hopf bifurcation is also investigated furthermore the existence and non existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity finally in order to verify our theoretical results some numerical simulations are also included it shows that the numerically observed behaviors are in good agreement with the theoretically proposed results on the basis of numerical simulations we provide some useful comparisons for the readers between our results and the existing ones and discuss the effects of the delay and the diffusion term on dynamic behaviors numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct

polynomials and lemniscates of indefiniteness for a large indefinite linear system there exists the option to directly precondition for the normal equations matrix nearness problems are formulated to assess the attractiveness of this alternative polynomial preconditioning leads to polynomial approximation problems involving lemniscate like sets both in the plane and in mathbb c n times n cn n a natural matrix analytic extension for lemniscates is introduced operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse for the speed of convergence and lemniscate asymptotics the notion of quasilemniscate arises in the l 2 l2 norm algorithms for solving the problem are devised

on fornberg s numerical method for conformal mapping it is shown that fornberg s method 1 is a discretized version of an iterative process which is essentially the same we introduced and studied in 7 8 the main difference consists in the determination of the shift u k in each iterative step while we give an explicit representation of u k fornberg determines u k from a linear equation which involves a certain integral operator r we prove some properties and estimates for this operator which explain several experimental results reported in 1 especially the fact that the conjugate gradient method is so efficient the results of 8 concerning the convergence can be applied also for fornberg s method finally we give estimates for the numerical error

the treatment of corner and pole type singularities in numerical conformal mapping techniques abstract this paper is a report of recent developments concerning the nature and the treatment of singularities that affect certain numerical conformal mapping techniques the paper also includes some new results on the nature of singularities that the mapping function may have in the complement of the closure of the domain under consideration

on a p kirchhoff equation via fountain theorem and dual fountain theorem abstract in this paper we show the existence of infinite solutions to the kirchhoff type quasilinear elliptic equation m u p x u p d x p 1 p u x u p 2 u f x u in a smooth bounded domain r n with nonlinear boundary condition u p 2 u u p 2 on the method we used here is based on fountain theorem and dual fountain theorem

adaptive reference elements via harmonic extensions and associated inner modes abstract a non intrusive extension to the standard p version of the finite element method is proposed meshes with hanging nodes are handled by adapting the reference elements so that the resulting discretisation is always conforming the shape functions on these adaptive reference elements are not polynomials but either harmonic extensions of the boundary restrictions of the standard shape functions or solutions to a local poisson problem the numerical experiments are taken from computational function theory and the efficiency of the proposed extension resulting in exponential convergence in the quantities of interest is demonstrated

numerical conformal mapping methods based on function conjugation abstract a unifying treatment of methods for computing conformal maps from the unit disk onto a jordan region is presented integral and integro differential equations involving the conjugation operator for the boundary correspondence function are first derived using an arbitrary auxiliary function having certain properties then various iteration methods for solving these equations are described in this generality so that the basic ideas become manifest specific methods are then treated as examples of the general theory among them are in particular the successive conjugation methods of theodorsen melentiev and kulisch timman and friberg the projection method of bergstrom and the newton methods of vertgeim wegmann and hubner which make use of the easy construction of the solutions of riemann hilbert problems many of these methods are treated in greater generality than in the literature the connections with the methods of fornberg menikoff zemach chakravarthy anderson and challis burley are also outlined

solving theodorsen s integral equation for conformal maps with the fast fourier transform and various nonlinear iterative methods we investigate several iterative methods for the numerical solution of theodorsen s integral equation the discretization of which is either based on trigonometric polynomials or function families with known attenuation factors all our methods require simultaneous evaluations of a conjugate periodic function at each step and allow us to apply the fast fourier transform for this in particular we discuss the nonlinear jor iteration the nonlinear sor iteration a nonlinear second order euler iteration the nonlinear chebyshev semi iterative method and its cyclic variant under special symmetry conditions for the region to be mapped onto we establish local convergence in the case of discretization by trigonometric interpolation and give simple formulas for the optimal parameters e g the underrelaxation factor and the asymptotic convergence factor weaker related results for the general non symmetric case are presented too practically our methods extend the range of application of theodorsen s method and improve its effectiveness strikingly

existence of a positive solution for a kirchhoff problem type with critical growth via truncation argument abstract in this paper we shall show an existence result of a positive solution for a kirchhoff problem type in a bounded domain of r n that is for the problem m u 2 d x u f x u u 2 2 u in u 0 on we shall study the asymptotic behavior of this solution when converges to infinity our approach is based on the variational method an appropriated truncated argument and a priori estimates to obtain the solution

global nonexistence for nonlinear kirchhoff systems in this paper we consider the problem of non continuation of solutions of dissipative nonlinear kirchhoff systems involving the p x laplacian operator and governed by nonlinear driving forces f f t x u as well as nonlinear external damping terms q q t x u u t both of which could significantly dependent on the time t the theorems are obtained through the study of the natural energy eu associated to the solutions u of the systems thanks to a new approach of the classical potential well and concavity methods we show the nonexistence of global solutions when the initial energy is controlled above by a critical value that is when the initial data belong to a specific region in the phase plane several consequences interesting in applications are given in particular subcases the results are original also for the scalar standard wave equation when p 2 and even for problems linearly damped

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

a modified schwarz christoffel mapping for regions with piecewise smooth boundaries a method where polygon corners in schwarz christoffel mappings are rounded is used to construct mappings from the upper half plane to regions bounded by arbitrary piecewise smooth curves from a given curve a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points the schwarz christoffel mapping for that polygon is then constructed and modified to round the corners since such a modification causes effects on the polygon outside the rounded corners the parameters in the mapping have to be redetermined this is done by comparing side lengths in tangent polygons to the given curve and the curve produced by the modified schwarz christoffel mapping the set of equations that this comparison gives can normally be solved using a quasi newton method the resulting function maps the upper half plane on a region bounded by a curve that apart from possible vertices is smooth i e one time continuously differentiable that passes through the tangent points on the given curve has the same direction as the given curve in these points and changes direction monotonically between them furthermore where the original curve has a vertex the constructed curve has a vertex with the same inner angle the method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines such as channels with parallel walls at the ends these properties are kept in the region produced by the constructed mapping

convergence proofs and error estimates for an iterative method for conformal mapping the iterative method as introduced in 8 and 9 for the determination of the conformal mapping of the unit disc onto a domaing is here described explicitly in terms of the operatork which assigns to a periodic functionu its periodic conjugate functionk u it is shown that whenever the boundary curve ofg is parametrized by a function with lipschitz continuous derivative dot eta ne 0 then the method converges locally in the sobolev spacew of 2 periodic absolutely continuous functions with square integrable derivative if is in a holder classc 2 the order of convergence is at least 1 if is inc l 1 withl 1 0 1 then the iteration converges inc l for analytic boundary curves the convergence takes place in a space of analytic functions for the numerical implementation of the method the operatork can be approximated by wittich s method which can be applied very effectively using fast fourier transform the sobolev norm of the numerical error can be estimated in terms of the numbern of grid points it iso n 1 l if is inc l 1 ando exp n 2 if is an analytic curve the number in the latter formula is bounded by logr wherer is the radius of the largest circle into which can be extended analytically such that z 0 for z r the results of some test calculations are reported

conformal mappings to a doubly connected polycircular arc domain the explicit construction of the conformal mapping of a concentric annulus to a doubly connected polygonal domain was first reported by akhiezer in 1928 the construction of an analogous formula for the case of a polycircular arc domain i e for a doubly connected domain whose boundaries are a union of circular arc segments has remained an important open problem in this paper we present this explicit formula we first introduce a new method for deriving the classical formula of akhiezer and then show how to generalize the method to the case of a doubly connected polycircular arc domain as an analytical check of the formula a special exact solution for a doubly connected polycircular arc mapping is derived and compared with that obtained from the more general construction as an illustrative example a doubly connected polycircular arc domain arising in a classic potential flow problem considered in the last century by lord rayleigh is considered in detail

stabilization for the kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control abstract the first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string in order to describe the geometrical nonlinearity due to finite transverse deformation the exact expression of the strain is used the mathematical modeling is derived first by using hamilton s principle and variational lemma and the derived nonlinear pde system is the kirchhoff type equation with boundary feedback control next we show the existence and uniqueness of strong solutions of the pde system via techniques of functional analysis mainly a theorem of compactness for the analysis of the approximation of the faedo galerkin method and estimate a decay rate for the energy the theoretical results are assured by numerical results of the solution s shape and asymptotic behavior for the system

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

a domain decomposition method for approximating the conformal modules of long quadrilaterals this paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals the method has been studied already by us and also by d gaier and w k hayman but only in connection with a special class of quadrilaterals viz quadrilaterals where a the defining domain is bounded by two parallel straight lines and two jordan arcs and b the four specified boundary points are the four corners where the arcs meet the straight lines our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above

almost periodic dynamics of a delayed three level food chain model abstractthis paper is concerned with the dynamics of a three level food chain model with time delays by means of mawhin s continuation theorem of coincidence degree theory we investigate the existence of positive almost periodic solutions in a three level food chain model with time delays a set of sufficient conditions for the existence of at least one positive almost periodic solution of the model is obtained in addition the global asymptotical stability of the almost periodic solution of the above model is also studied finally two examples and simulations are also given to illustrate the main results in this paper

discrete riemann hilbert problems interpolation of simply closed curves and numerical conformal mapping abstract let g be a simply connected region with 0 g and with a twice lipschitz continuously differentiable boundary curve and let z 1 n be an even number of n 2 n equidistant grid points on the unit circle sfnc z sfnc 1 with z 1 1 then there exists for all sufficiently large n a polynomial p n of degree n 1 normalized by the condition that the coefficient p 0 0 and the coefficients p 1 and p n 1 are real such that p n satisfies the interpolation condition p n z for all 1 n in a neighbourhood of the normalized conformal mapping function there is exactly one such interpolating polynomial the sequence of these p n converges to the conformal mapping function as n if is three times lipschitz continuously differentiable then the p n are also conformal mappings of the unit circle onto regions which approximate g an important tool for the theoretical investigation is a discrete analogon of the riemann hilbert problem we present two fast procedures for the numerical solution of the discrete riemann hilbert problem a conjugate gradient method with computational cost o n log n and a toeplitz matrix method with cost o n log 2 n using this one can calculate the interpolating polynomials by a newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function

positive solution for a superlinear kirchhoff type problem with a parameter abstract this paper is devoted to a nonlinear kirchhoff type problem depending on a real function and a nonnegative parameter on a smooth bounded domain of r n n 2 we show that if the nonlinearity is subcritical and superlinear at zero and infinity then there exists a positive value such that the kirchhoff type problem has at least one positive solution for any value of the parameter less than the positive value the main tools are variational methods critical point theory and iterative techniques

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

dynamic behaviors of a delay differential equation model of plankton allelopathy in this paper we consider a modified delay differential equation model of the growth of n species of plankton having competitive and allelopathic effects on each other we first obtain the sufficient conditions which guarantee the permanence of the system as a corollary for periodic case we obtain a set of delay dependent condition which ensures the existence of at least one positive periodic solution of the system after that by means of a suitable lyapunov functional sufficient conditions are derived for the global attractivity of the system for the two dimensional case under some suitable assumptions we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation examples show the feasibility of the main results

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

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

modified schwarz christoffel mappings using approximate curve factors the schwarz christoffel mapping from the upper half plane to a polygonal region in the complex plane is an integral of a product with several factors where each factor corresponds to a certain vertex in the polygon different modifications of the schwarz christoffel mapping in which factors are replaced with the so called curve factors to achieve polygons with rounded corners are known since long times among other requisites the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis in this paper the term approximate curve factor is defined such that many of the already known curve factors are included as special cases additionally by alleviating the requisite on the argument from exact to asymptotic equality new types of curve factors are introduced while traditional curve factors have a c 1 regularity c regular approximate curve factors can be constructed resulting in smooth boundary curves when used in conformal mappings applications include modelling of wave scattering in waveguides when using approximate curve factors in modified schwarz christoffel mappings numerical conformal mappings can be constructed that preserve two important properties in the waveguides first the direction of the boundary curve can be well controlled especially towards infinity where the application requires two straight parallel walls second a smooth c boundary curve can be achieved

periodic solutions for delay lotka volterra competition systems abstract by using the continuation theorem of coincidence degree theory sufficient and realistic conditions are obtained for the existence of positive periodic solutions for the periodic distributed delay lotka volterra competition system du i t dt u i t r i t a ii t u i t j 1 j i n a ij t 0 t ij k ij s u j t s ds i 1 2 n and the periodic state dependent delay lotka volterra competition system du i t dt u i t r i t a ii t u i t j 1 j i n a ij t u j t j t u 1 t u n t i 1 2 n where ri aii 0 aij 0 j i i j 1 2 n are continuous periodic functions tij 0 j i i j 1 2 n kij c tij 0 0 0 tijkij s ds 1 j i i j 1 2 n i c rn 1 r and i i 1 2 n are periodic with respect to their first arguments respectively

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

an integral equation method for the numerical conformal mapping of interior exterior and doubly connected domains a numerical method based on the integral equation formulation of symm is described for computing approximations to the mapping functions which accomplish the following conformal maps a the mapping of a domain interior to a closed jordan curve onto the interior of the unit disc b the mapping of a domain exterior to a closed jordan curve onto the exterior of the unit disc c the mapping of a doubly connected domain bounded by two closed jordan curves onto a circular annulus the numerical method is based on approximating the unknown source density by cubic splines and singular functions and is particularly suited for the mapping of difficult domains having sharp corners

pole type singularities and the numerical conformal mapping of doubly connected domains abstract let f be the function which maps conformally a given doubly connected domain onto a circular annulus and let h z f z f z 1 z in this paper we consider the problem of determining the main singularities of the function h in compl our purpose is to provide information regarding the location and nature of such singularities and to explain how this information can be used to improve the efficiency of certain expansion methods for numerical conformal mapping