a vertex formalism for local ruled surfaces we develop a vertex formalism for topological string amplitudes on ruled surfaces with an arbitrary number of reducible fibers embedded in a calabi yau threefold our construction is based on large n duality and localization with respect to a degenerate torus action we also discuss potential generalizations of our formalism to a broader class of calabi yau threefolds using the same underlying principles

two dimensional q differential transformation and its application the one dimensional q differential transformation was introduced in 8 for solving the ordinary q differential equations here we present the definition and operation of the two dimensional q differential transform a distinctive feature of the q differential transform is its ability to solve linear and nonlinear partial q differential equations

on the solution of characteristic value problems arising in linear stability analysis semi analytical approach abstract the linear stability analysis is normally governed by equations that constitute an eigenvalue characteristic value problem in this paper for the first time two semi analytical algorithms 1 differential transform method dtm and 2 adomian decomposition method adm are examined for solving a characteristic value problem occurring in linear stability analysis in this paper the characteristic value problem of couette taylor flow is selected because its simple geometry continues to be a paradigm for theoretical studies of hydrodynamic stability the results show that dtm handles the solution conveniently and accurately however this paper limits the use of adm for solving characteristic value problems the results indicate that the present algorithm based on dtm could be used as a promising method for solving characteristic value problems

numerical method for non linear wave and diffusion equations by the variational iteration method this paper applies he s variational iteration to the wave equations in an infinite one dimensional medium and some non linear diffusion equations a suitable choice of an initial solution can lead to the needed exact solution by a few iterations copyright 2007 john wiley sons ltd

application of he s variational iteration method and adomian s decomposition method to the fractional kdv burgers kuramoto equation in this work the fractional kdv burgers kuramoto equation is studied he s variational iteration method vim and adomian s decomposition method adm are applied to obtain its solution comparison with ham is made to highlight the significant features of the employed methods and their capability of handling completely integrable equations

new exact solutions of heat conduction in metals we consider the way in which a solution to a class of nonlinear partial differential equationss u ut k u ux x approaches the similarity form the problem we solve is chosen for two main reasons first the equation above is of widespread use in modeling physical situations and second it provides a tractable but significant example of a free boundary problem

on the diffusion in a lattice gas model group theoretic approach motivated by some recent results concerning the model of a noninteracting one dimensional lattice gas with an order preservation of particles where multiple occupancy of the sites is not excluded we give new symmetries and new reductions of the corresponding continuum nonlinear partial differential equation closed form analytic solutions are found

new algorithms for taylor coefficients of indefinite integrals and their applications new recurrence algorithms for calculating the taylor coefficients of indefinite integral are introduced and proved in this paper these algorithms are simple in form only involve the analytic operations addition and multiplication and can be directly implemented in any symbolic language also no specific restriction on the kernel is required several types of volterra equations such as integral integro differential and system of integro differential equations with nondegenerate kernel are then solved to demonstrate the accuracy and efficiency of the proposed scheme

a spectral analysis of the decoupling algorithm for semiconductor simulation the location in the complex plane of the eigenvalues of the derivatives of the partial nonlinear maps which constitute the fixed point mapping t defining gummel s method for decoupling the steady state semiconductor equations is examined for the general n dimensional case in particular it is proven that the size of these eigenvalues is unconditionally smaller than 1 it is shown that for up to three dimensional models all the derivatives are suitably compact this implies that the spectral radii of these mappings are equal to the size of the largest eigenvalue for the current continuity equations this implies an essential improvement over earlier contraction mapping results furthermore the precise location of the eigenvalues of the derivatives of the maps defined through the current continuity equations for the general one dimensional case is found for a special one dimensional case it is proven that the size of the eigenvalues of the total mapping t defining gummel s method is unconditionally smal

conditional lie backlund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations abstract the functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations are studied by applying the conditional lie backlund symmetry method a complete list of canonical forms for such equations are presented exact solutions to the resulting equations are constructed the asymptotic behaviors and blow up properties of some solutions are also discussed

o t 2 h 4 finite difference scheme for decoupled system of two quasilinear parabolic equations abstract let lh be the five point finite difference operator which has o h2 local truncation error at the points h and 1 h next to the ends of interval 0 1 and o h4 at the interior mesh points 2h 3h 1 2h the operator generates the pentadiagonal coherent matrix which is not of positive type and not diagonally dominant however the matrix satisfies the maximum principle the operator lh has been used in a number of publications to solve elliptic and parabolic equations in the paper lh is applied to approximate the second derivatives uxx and vxx in the two diffusion equations pu t au xx t x u v and qvt bvxx g t x u v it is proved that the obtained semi discrete scheme is o h4 globally convergent for approximation of the derivatives with respect to t the o 2 trapezoidal rule is used the fully discrete scheme obtained in this way constitutes a system of algebraic equations associated with a pentadiagonal matrix to solve this system of equations an implicit iterative method based on an algorithm for pentadiagonal matrices is proposed numerical results illustrating the method are presented

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

solving multi term linear and non linear diffusion wave equations of fractional order by adomian decomposition method abstract multi term linear and non linear diffusion wave equations of fractional order are solved using adomian decomposition method some numerical examples are presented

on the two dimensional differential transform method in this paper two dimensional differential transform method of solution of the initial value problem for partial differential equations pdes have been studied new theorems have been added and some linear and nonlinear pdes solved by using this method the method can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational work in this work additionally analytical form solutions of two diffusion problems have been obtained and the solutions are compared very well with those obtained by decomposition method

on the solution of partial differential equations with specified boundary conditions abstract the decomposition method is applied to solution of partial differential equations in two and three dimensions with specified boundary conditions

exact similarity solutions to some nonlinear diffusion equations the author gives new closed form similarity solutions to n dimensional radially symmetric nonlinear diffusion equations with power law and exponential diffusivities these solutions are generalisations of known similarity solutions

approximate analytical solutions of fractional gas dynamic equations in this article differential transform method dtm has been successfully applied to obtain the approximate analytical solutions of the nonlinear homogeneous and non homogeneous gas dynamic equations shock wave equation and shallow water equations with fractional order time derivatives the true beauty of the article is manifested in its emphatic application of caputo fractional order time derivative on the classical equations with the achievement of the highly accurate solutions by the known series solutions and even for more complicated nonlinear fractional partial differential equations pdes the method is really capable of reducing the size of the computational work besides being effective and convenient for solving fractional nonlinear equations numerical results for different particular cases of the equations are depicted through graphs

the variational iteration method a powerful scheme for handling linear and nonlinear diffusion equations in this work we introduce a framework for obtaining exact solutions to linear and nonlinear diffusion equations exact solutions are developed for some diffusion processes of power law diffusitivies he s variational iteration method vim is used for analytic treatment of these equations the powerful vim method is capable of handling both linear and nonlinear equations in a direct manner

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

exact solutions of nonlinear diffusion equations by variational iteration method in this paper he s variational iteration method is applied to obtain exact solutions of some nonlinear diffusion equations the variational iteration method is used to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the solutions obtained are compared with those obtained by the adomian decomposition method and homotopy perturbation method showing excellent agreement but the variational iteration method is more effective he s variational iteration method can be introduced to overcome the difficulties arising in calculating adomian polynomials

solutions of the system of differential equations by differential transform method in this study three dimensional differential transform method has been introduced and fundamental theorems have been defined for the first time moreover as an application of two and three dimensional differential transform exact solutions of linear and non linear systems of partial differential equations have been investigated the results of the present method are compared very well with those obtained by decomposition method differential transform method can easily be applied to linear or non linear problems and reduces the size of computational work with this method exact solutions may be obtained without any need of cumbersome work and it is an useful tool for analytical and numerical solutions

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

large time solution of an inhomogeneous nonlinear diffusion equation the paper considers the large time solution of the nonlinear diffusion equation x u t x c x u u x 0 for initial data with compact support the asymptotic behaviour of x and c x as x defines a parameter space and the nature of the solution at large time depends crucially on the location therein the paper examines the nature and uniformity of the asymptotic solutions for x in a restricted region of the entire parameter space and in certain cases gives the leading error terms a new integral invariant of the solution is also given

the non classical solution of the inhomogeneous non linear diffusion equation a complete classification of the symmetry reductions of the inhomogeneous nonlinear diffusion inld equation using the non classical method leads to one family of symmetry generators the resulting ordinary differential equation is of second order and non linear new solutions are obtained for some physical processes corresponding to particular values of n and p m

high order finite difference schemes for the solution of second order bvps we introduce new methods in the class of boundary value methods bvms to solve boundary value problems bvps for a second order ode these formulae correspond to the high order generalizations of classical finite difference schemes for the first and second derivatives in this research we carry out the analysis of the conditioning and of the time reversal symmetry of the discrete solution for a linear convection diffusion ode problem we present numerical examples emphasizing the good convergence behavior of the new schemes finally we show how these methods can be applied in several space dimensions on a uniform mesh

a finite difference scheme overconvergent at certain mesh points abstract in this paper a quasilinear elliptic equation in two variables x and y is approximated by a semidiscrete scheme discrete with respect to the variable x this semidiscrete scheme has a truncation error o h2 at the odd mesh points 1h 3h 5h 2n 1 h and o h4 at the even mesh points 2h 4h 2n 2 h 2nh 1 it is proved that the semidiscrete scheme is o h4 globally convergent with the rate of error decaying 2 a full o h4 discretization is obtained by a tri point finite difference scheme related with the variable y the system of algebraic equations which arises from the fully discrete scheme has sparse block matrix and is solved by the gauss block elimination method numerical results illustrating the method are given

numerical steady state and hopf bifurcation analysis on the diffusive nicholson s blowflies equation1research partially supported by natural sciences and engineering research council of canada 1 for the dirichlet boundary value problem of the diffusive nicholson s blowflies equation it was shown in ref 17 that in a certain range of the parameter space there is a unique positive steady state solution in this paper we propose a scheme to compute this steady state numerically in addition we describe an iterative procedure to locate the critical values of the delay where a hopf bifurcation of time periodic solutions takes place near the steady state some numerical simulations of both schemes are given

artin group actions on derived categories of threefolds motivated by the enhanced gauge symmetry phenomenon of the physics literature and mirror symmetry this paper constructs an action of an artin group on the derived category of coherent sheaves of a smooth quasiprojective threefold containing a configuration of ruled surfaces described by a finite type dynkin diagram the action ex tends over deformations of the threefold via a compatible action of the corresponding re flection group on the base of its deformation space all finite type dynkin diagrams are realized

differential transform method for the system of two dimensional nonlinear volterra integro differential equations in this paper we develop the differential transform method dtm for solving a class of the system of two dimensional linear and nonlinear volterra integro differential equations of the second kind to this end we give some preliminary results of the differential transform and describe the method of this paper we also give some examples to demonstrate the accuracy of the presented method

submodels of the generalization of the leith model of the phenomenological theory of turbulence and of the model of nonlinear diffusion in the inhomogeneous media without absorption we study a nonlinear equation which is equivalent to an equation of generalization of the leith model of turbulence and to the equation of the model of nonlinear diffusion in an inhomogeneous media without absorption using this equation all submodels admitting continuous lie transformation groups acting on the set of solutions of the equations of these submodels are obtained for obtained submodels all invariant submodels are found all essentially distinct invariant solutions describing these invariant submodels are found explicitly or their finding is reduced to solving nonlinear integral equations the integral equations defining these solutions reveal new possibilities for analytical and numerical studies the presence of arbitrary constants in these equations allows one to apply them to the study of different boundary value problems we have proved the existence and uniqueness of the solution for some boundary value problems we have investigated the following boundary value problems 1 a distribution of front density turbulent kinetic energy in a framework of the generalizion of the leith model of wave turbulence for which either the spectrum and its wavenumber derivative or the spectrum and its time derivative are given at the initial moment of time at a fixed wavenumber 2 a nonlinear diffusion process in an inhomogeneous media without absorption for which either the concentration and its gradient or the concentration and its rate of change are given at the initial moment of time at a fixed point under certain additional conditions we have established the existence and uniqueness of the solutions to boundary value problems describing these processes

exact results for the nonlinear diffusion equations delta u delta t delta delta x u 4 3 delta u delta x and delta u delta t delta delta x u 2 3 delta u delta x the author exploits local and non local symmetries of the two equations of the title and of some related equations some new exact solutions are derived and some specific boundary value problems of physical significance are considered

group transformations and the nonlinear heat diffusion equation a nonlinear diffusion equation with a power law diffusion coefficient is studied self similar and partially invariant solutions are shown to be identical the theory of homology is introduced and used to generate classes of equations the solutions of which are related through backlund transformations an example of an application is given which generalizes the solution of storm j appl phys 22 1951 pp 940 951 on heat conduction in metals

numeric analytic solutions of mixed type systems of balance laws the aim of the present analysis is to apply two relatively recent methods reduced differential transform method rdtm and differential transform method dtm for the solution of balance law systems new generalized transformed formulas are derived the new approaches provided the solution in the form of a rapidly convergent series with easily computable components in the rdtm case and costly components for the dtm a comparison between the two methodologies shows that the rdtm is more effective efficient powerful and able to be applicable for large class of nonlinear partial differential equations than the dtm two test modeling problems are discussed to illustrate the effectiveness and performance of rdtm

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

a new homotopy perturbation method for solving systems of partial differential equations abstract in this paper a new homotopy perturbation method nhpm is introduced for obtaining solutions of systems of non linear partial differential equations theoretical considerations are discussed to illustrate the capability and reliability of the method three examples are provided comparison of the results of applying nhpm with those of applying hpm reveal the effectiveness and convenience of the new technique

a numerical solution of second order linear partial differential equations by differential transform two dimensional differential transform provides an effective tool to numerically solve partial differential equations in this paper we propose a general numerical solution of second order linear partial differential equations using two dimensional differential transform numerical examples justify the proposed scheme

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

fdm for multi dimensional nonlinear coupled system of parabolic and hyperbolic equations a finite difference method fdm for multi dimensional nonlinear coupled system of parabolic and hyperbolic equations is studied nonlinear coupled property is fully considered variant techniques of discrete functional analysis such as summation by parts and deductive reasoning are used to make prior estimates the fd scheme is uniquely solvable and unconditionally stable with the approximation property of the scheme the convergence is obtained with a rate of o h 2 dt 2 order

localization and gluing of topological amplitudes we develop a gluing algorithm for gromov witten invariants of toric calabi yau threefolds based on localization and gluing graphs the main building block of this algorithm is a generating function of cubic hodge integrals of special form we conjecture a precise relation between this generating function and the topological vertex at fractional framing

one phase inverse stefan problem solved by adomian decomposition method abstract in this paper the solution of one phase inverse stefan problem is presented the problem consists of the reconstruction of the function which describes the distribution of temperature on the boundary when the position of the moving interface is well known the proposed solution is based on the adomian decomposition method and the least square method

the existence of noise terms for systems of inhomogeneous differential and integral equations in this paper we develop a framework for the necessary condition for the existence of noise terms for systems of inhomogeneous differential and integral equations we show that the noise terms are conditional and are generated for inhomogeneous equations if specific criteria is justified we numerically test our approach for a variety of systems of inhomogeneous problems

comments on solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method tari et al a tari m y rahimi s shahmorad f talati solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method j comput appl math 228 2009 70 76 presented some fundamental properties of tdtm for the kernel functions in two dimensional volterra integral equations here we suggest simple proofs of those fundamental properties by using the basic properties of tdtm furthermore we present some fundamental properties of tdtm for the kernel functions of a quotient type in two dimensional volterra integral equations numerical illustrations are demonstrated to show the effectiveness of the tdtm for solving two dimensional volterra integral equations

a new method for a generalized hirota satsuma coupled kdv equation abstract in this paper differential transform method dtm which is one of the approximate methods is implemented for solving the nonlinear hirota satsuma coupled kdv partial differential equation a variety of initial value system is considered and the convergence of the method as applied to the hirota satsuma coupled kdv equation is illustrated numerically the obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient numerical examples are illustrated the pertinent features of the proposed algorithm

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

solving partial differential equations by two dimensional differential transform method using two dimensional differential transform to solve partial differential equations pde is proposed in this study first the theory of two dimensional differential transform is introduced second taking two dimensional differential transform of a pde problem a set of difference equations is derived doing some simple mathematical operations on these equations we can get a closed form series solution or an approximate solution quickly finally three pde problems with constant and variable coefficients are solved by the present method the calculated results are compared very well with those obtained by other analytical or approximate methods

enhanced gauge symmetry and braid group actions enhanced gauge symmetry appears in type ii string theory as well as f and m theory compactified on calabi yau manifolds containing exceptional divisors meeting in dynkin configurations it is shown that in many such cases at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety this braid group covers the weyl group of the enhanced symmetry algebra which itself acts on the deformation space of the variety in a compatible way extensions of this result are given for nontrivial b fields on k3 surfaces explaining physical restrictions on the b field as well as for elliptic fibrations the present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local a 2n configuration in a threefold having global 2 monodromy

a proof of a conjecture of marino vafa on hodge integrals where i c1 li is the first chern class of li and j cj e is the j th chern class of the hodge bundle hodge integrals arise in the calculations of gromov witten invariants by localization techniques 14 7 the explicit evaluation of hodge integrals is a difficult problem the hodge integrals involving only classes can be computed recursively by witten s conjecture 26 proven by kontsevich 13 algorithms of computing hodge integrals are described in 2 in 23 m marino and c vafa obtained a closed formula for a generating function of certain open gromov witten invariants some of which has been reduced to hodge integrals by localization techniques which are not fully clarified mathematically this leads to a conjectural formula of hodge integrals to state this formula we introduce some notation following 28 let

exact solutions to nonlinear diffusion equations obtained by the decomposition method in this paper we develop a framework to obtain exact solutions to the nonlinear diffusion equations by employing the adomian decomposition method exact solutions are obtained for some important physical processes of power law diffusivities the method is capable of greatly reducing the size of computational domain and is presented in a general way so that it can be used in more diffusion classes

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

the adomian decomposition method in turning point problems in this paper the decomposition method is applied to boundary value problems of ordinary differential equations with a parameter exhibiting turning points

two dimensional differential transform for partial differential equations the differential transform is a numerical method for solving differential equations in this paper we present the definition and operation of the two dimensional differential transform a distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations partial differential equation of parabolic hyperbolic elliptic and nonlinear types can be solved by the differential transform we demonstrate that the differential transform is a feasible tool for obtaining the analytic form solutions of linear and nonlinear partial differential equation

a new algorithm for calculating adomian polynomials for nonlinear operators in this paper a reliable technique for calculating adomian polynomials for nonlinear operators will be developed the new algorithm offers a promising approach for calculating adomian polynomials for all forms of nonlinearity the algorithm will be illustrated by studying suitable forms of nonlinearity a nonlinear evolution model will be investigated

new classes of similarity solutions of the inhomogeneous nonlinear diffusion equations the lie similarity method has been used to extend the similarity solutions of the one dimensional inhomogeneous nonlinear diffusion equations we determine the lie point symmetry vector fields and calculate the similarity ansatz then we discuss the resulting nonlinear ordinary differential equations exact solutions are found and their relations to some real physical situations are discussed

o h 4 locally overconvergent semidiscrete scheme for the equation u t u xx f t x u abstract in the equation ut uxx f t x u the second derivative uxx is approximated by the finite difference operator lh which has o h2 local truncation error at the points h and 1 h next to the ends of the interval 0 1 and o h4 at the interior mesh points 2h 3h 1 2h it is proved that the obtained semidiscrete scheme is o h4 globally convergent the semidiscrete scheme is solved by the pure implicit finite difference scheme combined with the method of iteration and the gauss elimination method for tridiagonal matrices the nonstationary liouville s equation ut uxx e u has been solved by the proposed algorithm

topological strings and integrable hierarchies we consider the topological b model on local calabi yau geometries we show how one can solve for the amplitudes by using w algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the calabi yau in the highly effective fermionic brane formulation this leads to a free fermion description of the amplitudes furthermore we argue that topological strings on calabi yau geometries provide a unifying picture connecting non critical super strings integrable hierarchies and various matrix models in particular we show how the ordinary matrix model the double scaling limit of matrix models and kontsevich like matrix model are all related and arise from studying branes in specific local calabi yau three folds we also show how a model topological string on p 1 and local toric threefolds and in particular the topological vertex can be realized and solved as b model topological string amplitudes on a calabi yau manifold

a new algorithm for calculating two dimensional differential transform of nonlinear functions a new algorithm for calculating the two dimensional differential transform of nonlinear functions is developed in this paper this new technique is illustrated by studying suitable forms of nonlinearity three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm the present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity this gives the technique much wider applicability

numerical modelling of linear and nonlinear diffusion equations by compact finite difference method in this work accurate solutions to linear and nonlinear diffusion equations were introduced a combination of a sixth order compact finite difference scheme in space and a low storage third order total variation diminishing runge kutta scheme in time have been used for treatment of these equations the computed results with the use of this technique have been compared with the exact solution to show the accuracy of it here the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation the present method is seen to be a very good alternative method to some existing techniques for realistic problems

several new exact solutions for a fast diffusion equation by the differential constraints of the linear determining equations in this paper we propose a new approach to develop several new exact solutions to a fast nonlinear diffusion equation this approach rests on the differential constraints obtained by the method of linear determining equations the method is capable of reducing the nonlinear diffusion equation to systems of nonlinear ordinary equations the proposed scheme is presented in a general way so that it can be used in other diffusion processes

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

similarity solutions to three dimensional nonlinear diffusion equations the similarity transformation method has been used to solve three dimensional nonlinear diffusion equations the general lie group is calculated exact solutions to the cylindrical and spherical symmetry cases are found and their relations to some real physical processes are discussed

the modified decomposition method and pade approximants for solving the thomas fermi equation in this paper we propose a new approach to develop a nonperturbative approximate solution for the thomas fermi equation this approach rests on the recently developed modification of the adomian decomposition method the initial slope of the thomas fermi potential y 0 is computed by converting the obtained series solution into several diagonal pade approximants the proposed scheme is presented in a general way so that it can be used in applied sciences

convenient analytic recurrence algorithms for the adomian polynomials abstract in this article we present four analytic recurrence algorithms for the multivariable adomian polynomials as special cases we deduce the four simplified results for the one variable adomian polynomials these algorithms are comprised of simple orderly and analytic recurrence formulas which do not require time intensive operations such as expanding regrouping parametrization and so on they are straightforward to implement in any symbolic software and are shown to be very efficient by our verification using mathematica 7 0 we emphasize that from the summation expressions a n k 1 n u n k for the multivariable adomian polynomials and an k 1 n f k u 0 c n k for the one variable adomian polynomials we obtain the recurrence formulas for the u n k and the c n k these provide a theoretical basis for developing new algorithmic approaches such as for parallel computing in particular the recurrence process of one particular algorithm for the one variable adomian polynomials does not involve the differentiation operation but significantly only the arithmetic operations of multiplication and addition are involved precisely c n 1 u n n 1 and c n k 1 n j 0 n k j 1 u j 1 c n 1 j k 1 2 k n we also discuss several other algorithms previously reported in the literature including the adomian rach recurrence algorithm 1 and this author s index recurrence algorithm 23 36

submodels of model of nonlinear diffusion in the inhomogeneous medium involving absorption we study the five parameter model describing the process of nonlinear diffusion in an inhomogeneous medium in the presence of absorption for which the differential equation of the model admits a continuous lie group of transformations acting on the set of its solutions we found six submodels of the original model of nonlinear diffusion with different symmetry properties of these six submodels the five submodels with transient absorption for which the absorption coefficient depends on time according to a power law represent the greatest interest with a mathematical point of view and with the point of view of physical applications for each of these nonlinear submodels we obtained formulas for producing new solutions that contain arbitrary constants and we found all invariant submodels all essentially distinct invariant solutions describing these invariant submodels are found in an explicit form or are reduced to finding the solution of nonlinear integral equations the presence of the arbitrary constants in the integral equations that determine these solutions provide new opportunities for analytical and numerical study of boundary value problems for the received submodels and thus for the original model of nonlinear diffusion for the received invariant submodels we studied diffusion processes for which at the initial moment of the time at a fixed point is specified as a concentration and its gradient or as a concentration and its velocity solving of boundary value problems describing these processes is reduced to the solving of nonlinear integral equations we established the existence and uniqueness of solutions of these boundary value problems under some additional conditions the obtained results can be used to study the diffusion of substances diffusion of conduction electrons and other particles diffusion of physical fields and propagation of heat in inhomogeneous medium and also to study a turbulence leith model differential approximation to wave turbulence