univalent harmonic mappings and linearly connected domains abstract we investigate the relationship between the univalence of f and of h in the decomposition f h g of a sense preserving harmonic mapping defined in the unit disk d c among other results we determine the holomorphic univalent maps h for which there exists c 0 such that every harmonic mapping of the form f h g with g c h is univalent the notion of a linearly connected domain appears in our study in a relevant way

a block equation solver for large unsymmetric linear equation systems with dense coefficient matrices the presented solver uses the crout l u decomposition method disk memory space requirements are reduced to an absolute minimum and disk input output is spectacularly limited very extensive pivoting improves numerical accuracy and stability experimental test runs prove the overall efficiency of the solver

on lengths areas and lipschitz continuity of polyharmonic mappings abstract in this paper we continue our investigation of polyharmonic mappings in the complex plane first we establish two landau type theorems we also show a three circles type theorem and an area version of the schwarz lemma finally we study lipschitz continuity of polyharmonic mappings with respect to the distance ratio metric

harmonic univalent mappings into a half plane with nonreal vertical slits a class of harmonic univalent mappings is constructed by applying the method of clunie and sheil small these mappings assuming their values in a half plane with a vertical boundary omit two vertical half lines symmetric w r t the real axis several basic properties are proved

on the quasi isometries of harmonic quasiconformal mappings abstract we prove versions of the ahlfors schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the poincare metric

a schwarz pick inequality for harmonic quasiconformal mappings and its applications abstract the main result of this paper is the sharp generalized schwarz pick inequality for euclidean harmonic quasiconformal mappings with convex ranges which generalizes a result given by mateljevic as its applications we obtain the property of quasi isometry with respect to the poincare distance and an analogue of the koebe theorem for this class of mappings

estimates on bloch constants for planar harmonic mappings the schwarz lemma and bloch constants for planar bounded harmonic mappings are considered sharper form and better estimates are obtained our results improve the one made by dorff and nowak as well as by chen gauthier and hengartner

temporal derivatives in the finite element method on continuously deforming grids when solving certain time dependent partial differential equations using a finite element technique on a deforming grid it is shown that there is a need to differentiate the trial solution with respect to the position of each of the moveable node points a result is presented which enables these nonstandard derivatives to be expressed in terms of the standard spatial derivatives of the trial functions provided that certain conditions are satisfied these conditions are investigated and shown to be satisfied for a large class of trial spaces even when the necessary conditions are not satisfied by all of the trial functions being used the result is shown to be still of great use in evaluating these awkward nodal derivatives 1 u x t y u x t a u

on characterizations of bloch type hardy type and lipschitz type spaces in this paper we establish a bloch type growth theorem for generalized bloch type spaces and discuss relationships between dirichlet type spaces and hardy type spaces on certain classes of complex valued functions then we present some applications to non homogeneous yukawa pdes we also consider some properties of the lipschitz type spaces on certain classes of complex valued functions finally we will study a class of composition operators on these spaces

on certain geometric properties of polyharmonic mappings abstract the main purpose of this paper is twofold first the three quarter theorem is established for a large class of polyharmonic mappings second a class of polyharmonic mappings with positive jacobian and being univalent in the unit disk d are constructed

the landau bloch type theorems for planar harmonic mappings with bounded dilation abstract in this paper we first establish the sharp version of landau bloch type theorem for holomorphic functions with bounded derivative by applying schwarz pick lemma next we derive the sharp version of landau bloch type theorem for planar harmonic mappings with bounded dilation by applying geometric method furthermore all extremal functions are also provided under suitable restriction

on the univalence of the log biharmonic mappings in this paper we establish the univalence and starlikeness connection between log biharmonic mappings and logharmonic mappings

on some properties of solutions of the biharmonic equation in the present paper the properties of the linear complex operator l f zfz z fz which is defined on the class of complex valued c1 functions in the plane are investigated it is shown that harmonicity and biharmonicity are invariant under the linear operatorl results concerning starlikeness and convexity of biharmonic functions versus the corresponding harmonic functions are considered the operator l can be manipulated to express the conditions in the definitions of starlikeness and convexity in a convenient way

bloch constant and landau s theorem for planar p harmonic mappings abstract in this paper our main aim is to introduce the concept of planar p harmonic mappings and investigate the properties of these mappings first we discuss the p harmonic bloch mappings two estimates on the bloch constant are obtained which are generalizations of the main results in colonna 1989 9 as a consequence of these investigations we establish a bloch and landau s theorem for p harmonic mappings

moving finite element model for one dimensional infiltration in unsaturated soil the numerical integration of richards equation for infiltration into homogeneous unsaturated soil is carried out by a moving finite element mfe method accurate solution of this one dimensional nonlinear partial differential equation by standard finite difference fd or finite element fe methods can be obtained by using a fine discretization on space and time this can lead to large computer times with the mfe method grid points are moved during computation along the wetting front so that accuracy can be maintained by using a small number of nodes for simulating cases several meters in depth for these cases the mfe method is faster than fd and fe methods the numerical mfe model is verified through comparison with the quasi analytical solution by philip 1957 moreover we compare for some other test problems the computational efficiency of this method with that of two efficient pressure head based fd and fe fixed grid formulations

norm of the pre schwarzian derivative bloch s constant and coefficient bounds in some classes of harmonic mappings abstract the aim of the paper is a study of a family of sense preserving complex valued harmonic functions f that are normalized in the open unit disk and such that its analytic part is convex in one direction we prove the univalence and establish estimates on pre schwarzian derivative and the bloch constant for co analytic part of harmonic mapping the bounds of the coefficients of co analytic part are also given

landau s theorem for log p harmonic mappings abstract the main aim of this paper is to prove the existence of landau bloch constant for log p harmonic mappings

lengths areas and lipschitz type spaces of planar harmonic mappings abstract in this paper we give bounds for length and area distortion for harmonic k quasiconformal mappings and investigate certain lipschitz type spaces on harmonic mappings

a note on harmonic quasiconformal mappings abstract in this note we show that a harmonic quasiconformal mapping f u i v with respect to the poincare metric of the upper half plane r 2 onto itself such that v x y v y or u x y u x is a conformal mapping

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

landau s theorem for planar harmonic mappings chen gauthier and hengartner obtained two versions of landau s theorem for bounded planar harmonic mappings later dorff and nowak improved their estimates furthermore grigoryan and huang independently improved their results in this note we improve these last results by obtaining better coefficient estimates for bounded and normalized planar harmonic mappings in particular our theorems are sharp when m 1 which are consistent with landau s theorem when m 1

simple time and space adaptation in one dimensional evolutionary partial differential equations a simple technique for time and space adaptation in one dimensional evolutionary partial differential equations suggested by sanz serna and christie 1 is tested it is found that the equidistribution of hu xi greatly improves on the equidistribution of arc length used by those authors the time step control is found to perform poorly in the integration of rough solutions and the reasons for this behaviour are analysed

landau s theorem for certain biharmonic mappings abstract in this paper we show the existence of landau and bloch constants for biharmonic mappings of the form l f here l represents the linear complex operator l z z z z defined on the class of complex valued c 1 functions in the plane and f belongs to the class of biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

precise coefficient estimates for close to convex harmonic univalent mappings the class sh consists of harmonic univalent and sense preserving functions f in the open unit disk u z z 1 such that f h g where h z z n 2 anzn and g z n 1 a nzn let s0h ch and c0h denote the subclass of sh with a 1 0 the subclass of sh with f being a close to convex mapping and the intersection of s0h and ch respectively in this paper for f c0h and f ch we prove that the harmonic analogue of the bieberbach conjecture and the generalization of the bieberbach conjecture are true

a variable node finite element method abstract a new finite element method with variable nodes for integrating partial differential equations in one space dimension is presented the method utilizes the approximation power of piecewise polynomial functions better than a fixed node finite element method because it distributes grid points in a nonuniform way to suit the peculiarities of the solution at each instant of time the method uses the gradient and curvature of the approximating piecewise polynomial itself to determine the position of the nodes rather than physical properties which vary from problem to problem the method has been implemented in a program vfe which has the exceptional feature of a variable space step as well as a variable time step since the method concentrates nodes in regions where small inter nodal spacing is needed it is very efficient in integrating equations with shock like solutions finally the method can be extended to systems of partial differential equations in one dimension and to equations in two dimensions

generalized bloch spaces integral means of hyperbolic harmonic mappings in the unit ball abstract in this paper we investigate the properties of hyperbolic harmonic mappings in the unit ball b n in r n n 2 firstly we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the bloch space b b n and the generalized bloch space l b a 0 b n respectively secondly we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients the obtained results are the generalizations of hardy and littlewood s related ones in the setting of hyperbolic harmonic mappings finally we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric

bohr s inequalities for the analytic functions with lacunary series and harmonic functions abstract we determine the bohr radius for the class of all functions f of the form f z z m k 0 a k p z k p analytic in the unit disk z 1 and satisfy the condition f z 1 for all z 1 in particular our result also contains a solution to a recent conjecture of ali et al 9 for the bohr radius for odd analytic functions solved by the authors in 17 we consider a more flexible approach by introducing the p bohr radius for harmonic functions which in turn contains the classical bohr radius as special case also we prove several other new results and discuss p bohr radius for the class of odd harmonic bounded functions

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

on the univalence of polyharmonic mappings abstract a 2p times continuously differentiable complex valued function f u i v in a simply connected domain is polyharmonic or p harmonic if it satisfies the polyharmonic equation p f 0 every polyharmonic mapping f can be written as f z k 1 p z 2 p 1 g p k 1 z where each g p k 1 is harmonic in this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f z and that of g p z the notion of stable univalence and logpolyharminc mappings are also considered

landau s theorem for biharmonic mappings abstract in this paper we show the existence of landau constant for biharmonic mappings of the form f z z 2 g z k z z 1 where g and k are harmonic

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

landau s theorem for polyharmonic mappings abstract in this paper we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk d then we obtain two versions of landau s theorem for polyharmonic mappings f and for the mappings of the type l f where l is the differential operator of abdulhadi abu muhanna and khuri examples and numerical estimates are given

some properties of a class of elliptic partial differential operators abstract we prove schwarz pick type estimates and coefficient estimates for a class of functions induced by the elliptic partial differential operators then we apply these results to obtain a landau type theorem

distortion theorems for bloch functions we establish various distortion theorems for both normalized locally schlicht bloch functions and normalized bloch function with branch points these distortion theorems give lower bounds on either f z or re f z most of our distortion theorems are sharp and all extremal functions identified the main tools used in establishing these distortion theorems are the classical form of julia s lemma and a new version of julia s lemma that applies to certain multiple valued analytic functions

on the linear combinations of harmonic univalent mappings abstract in this paper we derive several sufficient conditions of the linear combinations of harmonic univalent mappings to be univalent and convex in the direction of the real axis furthermore some illustrative examples and imagine domains of the linear combinations satisfying the desired conditions are enumerated

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

solving ax b a method with reduced storage requirements it is shown that a recently developed method for solving ax b can be arranged so that it requires approximately one quarter of the in core storage capacity of other comparable solution procedures for the solution of large systems

a simple adaptive technique for nonlinear wave problems abstract a method for the numerical integration of the nonlinear schrodinger equation is derived which uses variable time steps and a moving spatial grid the benefits of adaptation are clearly demonstrated in the numerical experiments reported the simple technique employed to move the nodes can be applied with little coding effort to general one dimensional systems of pdes

biorthogonal series solution of stokes flow problems in sectorial regions in this paper we develop an eigenfunction expansion method for solving stokes flow problems in sectorial cavities that arise in fluid dynamics the method leads to the development of a set of eigenfunctions adjoint eigenfunctions biorthogonality conditions and an algorithm for the computation of the coefficients of the eigenfunction expansion the resulting infinite system of linear equations is then solved by truncation these biorthogonality conditions are properties satisfied by the eigenfunctions and adjoint eigenfunctions which are used to compute the coefficients of the eigenfunction expansion solution the biorthogonality conditions are derived for a class of fourth order boundary value problems with variable coefficients that arise from separating variables of the governing stokes equation the method is applied to the slow steady flow in a two dimensional sectorial cavity the fluid is set into motion by the uniform translation of a covering plate or belt