about the use of standard integration schemes for x fem in solid mechanics plasticity abstract the introduction of plasticity in an extended finite element x fem framework is very recent and arises several technical and scientific difficulties due to the moving position of integration points as the crack proceeds if a fixed grid is not used currently the existing analyses in plasticity with x fem are limited to monotonic plasticity approximated by non linear elasticity using the hrr field decomposition however such an analysis cannot be applied in fatigue where cyclic plasticity plays a major role in crack propagation the aim of this study is to tackle one of the difficulties induced by cyclic plastic computation the projection of the internal variables during crack propagation the discontinuous heaviside function in the elements cut by the crack is replaced by a new continuous enrichment function this new function is polynomial and its degrees and coefficients depend on the type of element and on the position of the interface in the element the automatic determination of the new enrichment and its introduction in a linear and non linear framework have been implemented for linear elements in two dimensions and three dimensions and tested with convergence tests in elasticity and plasticity

conditional lie backlund symmetry second order differential constraint and direct reduction of diffusion systems abstract the equivalence relation between conditional lie backlund symmetry higher order differential constraint and direct reduction is presented in the case of evolution systems the second order nonlinear conditional lie backlund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional lie backlund symmetry of evolution systems consequently the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems

a fixed mesh method for incompressible flow structure systems with finite solid deformations a fixed mesh algorithm is proposed for simulating flow structure interactions such as those occurring in biological systems in which both the fluid and solid are incompressible and the solid deformations are large several of the well known difficulties in simulating such flow structure interactions are avoided by formulating a single set of equations of motion on a fixed eulerian mesh the solid s deformation is tracked to compute elastic stresses by an overlapping lagrangian mesh in this way the flow structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system these forces which depend on the solid elastic stress distribution are computed on the lagrangian mesh by a standard finite element method and then transferred to the fixed eulerian mesh where the joint momentum and continuity equations are solved by a finite difference method the constitutive model for the solid can be quite general for the force transfer standard immersed boundary and immersed interface methods can be used and are demonstrated we have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum the interface is still effectively sharp for this approach the spatial convergence of the method is observed to be between first and second order as in most immersed boundary methods for membrane flows the algorithm is demonstrated by the simulations of an advected elastic disk a flexible leaflet in an oscillating flow and a model of a swimming jellyfish

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

new exact solutions for a free boundary system a method for constructing explicit exact solutions to nonlinear evolution equations is further developed the method is based on consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation the method is then applied to a free boundary problem based on the process of precipitant assisted protein crystal growth

zeros of orthogonal laurent polynomials and solutions of strong stieltjes moment problems the strong stieltjes moment problem for a bisequence c n n consists of finding positive measures m with support in 0 such that 0 t nd m t c nfor n 0 1 2 orthogonal laurent polynomials associated with the problem play a central role in the study of solutions when the problem is indeterminate the odd and even sequences of orthogonal laurent polynomials suitably normalized converge in c 0 to distinct holomorphic functions the zeros of each of these functions constitute together with the origin the support of two solutions m and m 0 we discuss how odd and even subsequences of zeros of the orthogonal laurent polynomials converge to the support points of m and m 0

iterations of the non classical symmetries method and conditional lie backlund symmetries iterations of the non classical symmetries method give rise to new nonlinear equations which inherit the lie point symmetry algebra of the original equation invariant solutions of these heir equations supply new solutions of the original equation we show that particular cases of such invariant solutions correspond to zhdanov s conditional lie backlund symmetries

integration of singular enrichment functions in the generalized extended finite element method for three dimensional problems a mapping method is developed to integrate weak singularities which result from enrichment functions in the generalized extended finite element method the integration scheme is applicable to 2d and 3d problems including arbitrarily shaped triangles and tetrahedra implementation of the proposed scheme in existing codes is straightforward numerical examples for 2d and 3d problems demonstrate the accuracy and convergence properties of the technique copyright 2008 john wiley sons ltd

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

non classical reductions of initial value problems for a class of nonlinear evolution equations we classify initial value problems for a class of one dimensional evolution equations which can be reduced to cauchy problems for some systems of first order ordinary differential equations the technique applied relies heavily on higher conditional symmetries of the equations under study which means in particular that the obtained reductions cannot be derived within the framework of the standard lie approach

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

symmetries of the fisher kolmogorov petrovskii piskunov equation with a nonlocal nonlinearity in a semiclassical approximation abstract the classical group analysis approach used to study the symmetries of integro differential equations in a semiclassical approximation is considered for a class of nearly linear integro differential equations in a semiclassical approximation an original integro differential equation leads to a finite consistent system of differential equations whose symmetries can be calculated by performing standard group analysis the approach is illustrated by the calculation of the lie symmetries in explicit form for a special case of the one dimensional nonlocal fisher kolmogorov petrovskii piskunov population equation

the galilean relativistic principle and nonlinear partial differential equations the second order partial differential equations invariant under transformations of galilei rotation scale and projection are described

exact solutions to nonlinear equations with quadratic nonlinearity based on the concepts of the generalized conditional symmetry and of the invariant subspace a constructive way to derive nonlinear equations with quadratic nonlinearity that admit exact solutions as well as many new examples is presented this type of solution found by galaktionov describes the blowup phenomena examples of exact solutions that do not correspond to the invariant subspace are also presented and explained by the nonlinear generalized conditional symmetries

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

on the reduction of the nonlinear multi dimensional wave equations and compatibility of the d alembert hamilton system abstract the necessary conditions of the compatibility of the d alembert hamilton system in minkowsky space r 1 n are established the problem of reduction of p 1 n invariant wave equations to ordinary differential equations is discussed

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

higher order extended finite elements with harmonic enrichment functions for complex crack problems summary in this paper we analyze complex crack problems in elastic media using harmonic enrichment functions in a higher order extended finite element implementation the numerically computed enrichment function of a crack is the solution of the laplace equation with discontinuous dirichlet boundary condition along the crack and its interaction with branches or other cracks is realized by imposing vanishing neumann boundary conditions along those cracks the classical finite element displacement approximation is enriched by adding the enrichment function of a crack through the framework of partition of unity a nested subgrid mesh is used in the laplace solve with a rasterized approximation of a crack which simplifies the numerical integration no partitioning of finite elements is required harmonic enrichment functions readily permit the extension to handle multiple interacting and branched cracks without any special treatment around the junction points several numerical examples are presented that arm the accuracy and eectiveness of the method when applied to complex crack

on new ways of group methods for reduction of evolution type equations new exact solutions of the evolution type equations are constructed by means of a non point contact symmetries also we analyzed the discrete symmetries of maxwell equations in vacuum and decoupled ones to the four independent equations that can be solved independently

nonlinear reaction diffusion systems with a non constant diffusivity q conditional symmetries nonclassical symmetries for a general class of two component reaction diffusion systems with non constant diffusivities are studied the work is a natural continuation of our paper conditional symmetries and exact solutions of nonlinear reaction diffusion systems with non constant diffusivities cherniha and davydovych 2012 1 in order to extend the results on so called no go case using the notion of q conditional symmetries of the first type an exhaustive list of reaction diffusion systems admitting such symmetry is derived the results obtained are compared with those derived earlier the symmetries for reducing reaction diffusion systems to two dimensional dynamical systems ode systems and finding exact solutions are applied as result multiparameter families of exact solutions in the explicit form for nonlinear reaction diffusion systems with an arbitrary power law diffusivity are constructed and their properties for possible applicability are established

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

equivalent legendre polynomials numerical integration of discontinuous functions in the finite element methods abstract the efficiency of numerical integration of discontinuous functions is a challenge for many different computational methods to overcome this problem we introduce an efficient and simple numerical method for the integration of discontinuous functions on an arbitrary domain in the proposed method the discontinuous function over the domain is replaced by a continuous equivalent function using legendre polynomials the method is applicable to all ansatz spaces with continuous basis functions and it allows to use standard numerical integration methods in the entire domain this imposed continuous function is called equivalent legendre polynomial elp several numerical examples serve to demonstrate the efficiency and accuracy of the proposed method based on 2 d and 3 d problems of linear elastostatics the introduced method is further evaluated in the concept of the fictitious domain finite element methods

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

orthogonal laurent polynomials and continued fractions associated with log normal distributions abstract this paper describes properties and computational procedures related to orthogonal laurent polynomials continued fractions two point pade approximants strong moment problems and l gaussian quadrature associated with log normal distribution functions t defined by t q1 2 2 e lnt 2 2 0

equivalent polynomials for quadrature in heaviside function enriched elements summary one of the advantages of partition of unity fems like the extended fem is the ability of modeling discontinuities independent of the mesh structure the enrichment of the element functional space with discontinuous or non differentiable functions requires when the element stiffness is computed partitioning into subdomains for quadrature however the arbitrary intersection between the base mesh and the discontinuity plane generates quadrature subdomains of complex shape this is particularly true in three dimensional problems where quite sophisticate methodologies have been presented in the literature for the element stiffness evaluation the present work addresses the problem of heaviside function enrichments and is based on the replacement of the discontinuous enrichment function with the limit of an equivalent polynomial defined on the entire element domain this allows for the use of standard gaussian quadrature in the elements crossed by the discontinuity the work redefines conceptually the first version of the equivalent polynomial methodology introduced in 2006 allowing a much broader applicability as a consequence equivalent polynomials can be computed for all continuum element families in one two and three dimensions copyright 2014 john wiley sons ltd

explicit time marching schemes for solving the magnetic field volume integral equation a method for constructing explicit marching on in time mot schemes to solve the time domain magnetic field volume integral equation td mfvie on inhomogeneous dielectric scatterers is proposed the td mfvie is cast in the form of an ordinary differential equation ode and the unknown magnetic field is expanded using curl conforming spatial basis functions inserting this expansion into the td mfvie and spatially testing the resulting equation yield an ode system with a gram matrix this system is integrated in time for the unknown time dependent expansion coefficients using a linear multistep method the gram matrix is sparse and well conditioned for galerkin testing and consists of only four diagonal blocks for point testing the resulting explicit mot schemes which call for the solution of this matrix system at every time step are more efficient than their implicit counterparts which call for inversion of a fuller matrix system at lower frequencies numerical results compare the efficiency accuracy and stability of the explicit mot schemes and their implicit counterparts for low frequency excitations the results show that the explicit mot scheme with point testing is significantly faster than the other three solvers without sacrificing from accuracy

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

similarity reductions of the 1 3 dimensional burgers equation we consider the 1 3 dimensional burgers equation u t u x x u y y u z z uu x which has considerable interest in mathematical physics we complete the list of similarity reductions that are obtained from the lie symmetries admitted by this equation to achieve this goal we employ two and three dimensional subalgebras of the lie symmetry algebra in addition to the one dimensional subalgebras that appears in the literature we solve analytically where it is possible the reduced ordinary differential equations and hence we obtain closed form solutions for the original equation finally we derive certain non lie solutions

generalized gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method new gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method for discontinuous functions we construct gauss like quadrature rules over arbitrarily shaped elements in two dimensions without the need for partitioning the finite element a point elimination algorithm is used in the construction of the quadratures which ensures that the final quadratures have minimal number of gauss points for weakly singular integrands we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques

generalized gaussian quadrature rules on arbitrary polygons in this paper we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons these quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required we have used this algorithm for the construction of symmetric and non symmetric quadrature rules over convex and concave polygons while in the case of symmetric quadratures our results are comparable to available rules the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions to demonstrate the efficiency of the new quadrature rules we have tested them for the integration of rational polygonal shape functions over a regular hexagon for a relative error of 10 8 in the computation of stiffness matrix entries one needs at least 198 evaluation points when the region is partitioned whereas 85 points suffice with our quadrature rule copyright 2009 john wiley sons ltd

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

on a symmetry in strong distributions a strong stieltjes distribution d t is called symmetric if it satises the property t d 2 t 2 t d t for t 2 a b 0 1 2 2 z and 0 in this article some consequences of symmetry on the moments the orthogonal l polynomials and the quadrature formulae associated with the distribution are given c 1999 elsevier science b v all rights reserved msc 30e05 30b70 41a55

harmonic enrichment functions a unified treatment of multiple intersecting and branched cracks in the extended finite element method a unifying procedure to numerically compute enrichment functions for elastic fracture problems with the extended finite element method is presented within each element that is intersected by a crack the enrichment function for the crack is obtained via the solution of the laplace equation with dirichlet and vanishing neumann boundary conditions a single algorithm emanates for the enrichment field for multiple cracks as well as intersecting and branched cracks without recourse to special cases which provides flexibility over the existing approaches in which each case is treated separately numerical integration is rendered to be simple there is no need for partitioning of the finite elements into conforming subdivisions for the integration of discontinuous or weakly singular kernels stress intensity factor computations for different crack configurations are presented to demonstrate the accuracy and versatility of the proposed technique copyright 2010 john wiley sons ltd

on the elimination of quadrature subcells for discontinuous functions in the extended finite element method the introduction of discontinuous non differentiable functions in the extended finite element method allows to model discontinuities independent of the mesh structure however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity line is commonly adopted in the paper it is shown how standard gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation the technique is illustrated and developed in one two and three dimensions for crack and material discontinuity problems

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

quadrature schemes for arbitrary convex concave volumes and integration of weak form in enriched partition of unity methods abstract quadrature schemes are constructed based on moment fitting equations to integrate polynomials over arbitrary convex concave volumes that arise among others in enriched partition of unity finite element methods epum the building block of the scheme involves the divergence theorem of multivariable calculus which is used to integrate the base functions an efficient and robust point distribution method is proposed and the quadrature weights at the corresponding points are obtained by solving a least squares problem the method is applied initially to integrate given polynomial functions over complex volumes and further to simulate simple three dimensional fluid dynamic problems which involve very complex volumes when solved with epum accuracy of the present quadrature construction scheme is demonstrated by comparing the results with the available exact numerical solutions and efficiency of the method is proved by comparing the computational time with that of the widely used tessellation method

modeling crack discontinuities without element partitioning in the extended finite element method in this paper we model crack discontinuities in two dimensional linear elastic continua using the extended nite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals for crack modeling in the x fem the standard finite element approximation is enriched with a discontinuous function and the near tip crack functions each element that is fully cut by the crack is decomposed into two simple convex or nonconvex polygons whereas the element that contains the crack tip is treated as a nonconvex polygon on using euler s homogeneous function theorem and stokes s theorem to numerically integrate homogeneous functions on convex and nonconvex polygons the exact contributions to the sti ness matrix from discontinuous enriched basis functions are computed for contributions to the sti ness matrix from weakly singular integrals due to enrichment with asymptotic crack tip functions we only require a one dimensional quadrature rule along the edges of a polygon hence neither element partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed structured fi nite element meshes consisting of rectangular elements as well as unstructured triangular meshes are used we demonstrate the exibility of the approach and its excellent accuracy in stress intensity factor computations for two dimensional crack problems

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

exact solutions of a class of nonlinear boundary value problems with moving boundaries presents a method for the construction of the exact solutions of nonlinear bvps in melting and evaporation processes the case of smooth functions of thermal conductivity is studied and illustrations of some specific nonlinearities given

new exact solutions of a nonlinear cross diffusion system a system of two reaction diffusion equations with cross diffusion and quadratic nonlinearities is considered the system is a particular case the well known model proposed by shigesada et al 1979 j theor biol 79 83 new non lie ansatze special type substitutions reducing the system to systems of first order ordinary differential equations are obtained and applied to find its exact solutions several families of exact solutions are constructed their asymptotic behavior is investigated and some biological interpretation of the solutions is provided lie symmetry of this system is also discussed

symmetry groups and separation of variables of a class of nonlinear diffusion convection equations in this paper a class of nonlinear diffusion convection equations ut d u uxn x p u ux which has quite a large number of physical applications is analysed by using symmetry group methods which include the classical method the potential symmetry method and the generalized conditional symmetry method a complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits lie s point symmetry groups and potential symmetry groups the separation of variables for the equation is investigated using the generalized conditional symmetry approach for some interesting cases exact solutions using the method of separation of variables are discussed in detail

pc fractions and orthogonal laurent polynomials for log normal distributions abstract log normal distributions have received considerable recent attention partly through their applications to several branches of science economics and business and partly in their relation to orthogonal laurent polynomials l polynomials pade approximants and continued fractions this paper investigates pc fractions and orthogonal l polynomials associated with log normal distributions our main results include a explicit expressions for the continued fractions and their approximants and the l polynomials b proofs of the existence of the limits of subsequences of approximants and of l polynomials c closed form expressions for the limits of these subsequences and d identities between related approximants and related l polynomials several of the limit functions are seen to be essentially theta funcions

the characterization of reaction convection diffusion processes by travelling waves it has long been known that the heat equation displays infinite speed of propagation this is to say that if the initial data are nonnegative and have nonempty compact support the solution of an initial value problem is positive everywhere after any infinitesimal time however since the nineteen fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon for these equations the generalized solution of an initial value problem with compactly supported initial data will have bounded support with respect to the spatial variable at all times in this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reaction convection diffusion equation formula is determined the assumptions on the coefficientsa bandcare such that the classification unifies and generalizes previously known results the technique employed is comparision of an arbitrary solution of the equation with suitably constructed travelling wave solutions and subsolutions basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling wave solution with bounded support concurrently the search for a travelling wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint

non linear reaction diffusion systems with variable diffusivities lie symmetries ansatze and exact solutions abstract this work first considers the classical lie symmetry analysis of a class of systems of two quasilinear reaction diffusion equations having variable diffusivities subsequently non lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems in particular families of exact solutions of a diffusive lotka volterra type system are constructed

galilei invariant nonlinear systems of evolution equations all systems of n 1 dimensional quasilinear second order evolution equations invariant under chain of algebras ag 1 n contained in implied by ag1 1 n contained in implied by ag2 1 n are described the results obtained are illustrated by the examples of the nonlinear schrodinger equations hamilton jacobi type systems and reaction diffusion equations

high order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles a high order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle by converting the implicitly defined geometry into the graph of an implicitly defined height function the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one dimensional root finding and one dimensional gaussian quadrature the computed quadrature scheme yields strictly positive quadrature weights and inherits the high order accuracy of gaussian quadrature a range of different convergence tests demonstrate orders of accuracy up to 20th order also presented is an application of the quadrature algorithm to a high order embedded boundary discontinuous galerkin method for solving partial differential equations on curved domains

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

quadrature rules for triangular and tetrahedral elements with generalized functions quadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains these quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries specifically the integrands considered here consist of a quadratic polynomial multiplied by a heaviside or dirac delta function operating on a linear polynomial this form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values compared with methods involving explicit integration over subdomains the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical jacobian information describing the dependence of the integrals on the nodal level set values the accuracy of the method is demonstrated for a simple conduction problem with the neumann and robin type boundary conditions copyright 2007 john wiley sons ltd

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

a conformal decomposition finite element method for arbitrary discontinuities on moving interfaces summary interface capturing methods using enriched finite element formulations are well suited for solving multimaterial transport problems that contain weak or strong discontinuities the conformal decomposition fem decomposes multimaterial elements of a non conforming background mesh into sub elements that conform to material interfaces captured using a level set method as the interface evolves interfacial nodes move and background nodes may change material the present work describes approaches for handling moving interfaces in the context of the conformal decomposition fem for both weakly and strongly discontinuous fields dynamic discretization methods using extrapolation and moving mesh approaches are considered and developed with first order and second order time integration methods the moving mesh approach is demonstrated to be a stable method that preserves both weak and strong discontinuities on a variety of one dimensional and two dimensional test problems while achieving the expected second order error convergence rate in space and time copyright 2014 john wiley sons ltd

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

interface handling for three dimensional higher order xfem computations in fluid structure interaction three dimensional higher order extended finite element method xfem computations still pose challenging computational geometry problems especially for moving interfaces this paper provides a method for the localization of a higher order interface finite element fe mesh in an underlying three dimensional higher order fe mesh additionally it demonstrates how a subtetrahedralization of an intersected element can be obtained which preserves the possibly curved interface and allows therefore exact numerical integration the proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their extended axis aligned bounding boxes the intersection method is applied to a highly reduced number of intersection candidates the resulting linearized interface is used as input for an elementwise constrained delaunay tetrahedralization which computes an appropriate subdivision for each intersected element the curved interface is recovered from the linearized interface in the last step the output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element application of the interface algorithm currently concentrates on fluid structure interaction problems on low order and higher order fe meshes which may be composed of any arbitrary element types such as hexahedra tetrahedra wedges etc nevertheless other xfem problems with explicitly given interfaces or discontinuities may be tackled in addition multiple structures and interfaces per intersected element can be handled without any additional difficulties several parallelization strategies exist depending on the desired domain decomposition approach numerical test cases including various geometrical exceptions demonstrate the accuracy robustness and efficiency of the interface handling

on a new method for constructing exact solutions of the nonlinear differential equations of mathematical physics a simple new method for constructing solutions of multidimensional nonlinear wave equations is proposed

the symmetric strong moment problem sequences of polynomials that occur as denominators in the two point pade table for two series expansions are considered in the special case when the series coefficients are solutions of a strong symmetric stieltjes moment problem the continued fractions whose convergents generate these polynomials as denominators are presented together with determinant representations for the polynomials and the continued fraction coefficients the log normal distribution is used as an example

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant