the variational iteration method for solving two forms of blasius equation on a half infinite domain the variational iteration method is applied for a reliable treatment of two forms of the third order nonlinear blasius equation which comes from boundary layer equations the study shows that the series solution is obtained without restrictions on the nonlinearity behavior the obtained series solution is combined with the diagonal pade approximants to handle the boundary condition at infinity for only one of these forms

new periodic and soliton solutions by application of exp function method for nonlinear evolution equations in this letter the kaup kupershmidt 2 1 dimensional potential kadomtsev petviashvili shortly pkp equations are presented and the exp function method is employed to compute an approximation to the solution of nonlinear differential equations governing the problem it has been attempted to show the capabilities and wide range applications of the exp function method this method can be used as an alternative to obtain analytic and approximate solution of different types of differential equations applied in engineering mathematics

a new multicomponent ckp hierarchy and solutions based on the eigenfunction symmetry constraints we first propose a new multicomponent c type subhierarchy for kadomtsev petviashvili mcckp hierarchy and its lax representation which provides an effective way to find two types of new ckp equation with self consistent sources along with their related lax representations also it admits reductions to k constrained ckp hierarchy and to a 1 1 dimensional soliton hierarchy with self consistent sources which gives rise to two types of new kaup kuperschmidt kk equation with self consistent sources and two types of new bidirectional kaup kuperschmidt equation with self consistent sources by using the solutions of the ckp and kk equations and their corresponding eigenfunctions n soliton solutions for the first type of ckp self consistent source ckpscs and of kkscs are constructed by means of method of variation in constant respectively

analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method abstract the tanh method is used to derive explicit travelling wave solutions for two fifth order integrable nonlinear evolution equations the fifth order kdv equation fkdv and the fifth order caudrey dodd gibbon cdg equation are used as vehicles for our analysis new families of solitary wave solutions and periodic solutions are formally obtained for these equations the presented framework reveals quite a number of useful features of the method used here

a note on a certain boundary layer equation abstract this paper deals with a differential equation of mixing layer the existence of a solution is established by using weyl technique

applying lehman s theorems to packing problems a 0 1 matrixa isideal if the polyhedronq a conv x qv a x 1 x 0 v denotes the column index set ofa is integral similarly a matrix isperfect ifp a conv x qv a x 1 x 0 is integral little is known about the relationship between these two classes of matrices we consider a transformation between the two classes which enables us to apply lehman s modified theorem about deletion minimal nonideal matrices to obtain new results about packing polyhedra this results in a polyhedral description for the stable set polytopes ofnear bipartite graphs the deletion of any neighbourhood produces a bipartite graph note that this class includes the complements of line graphs to date this is the only natural class besides the perfect graphs for which such a description is known for the graphs and their complements some remarks are also made on possible approaches to describing the stable set polyhedra of quasi line graphs and more generally claw free graphs these results also yield a new class oft perfect graphs

solution of the falkner skan equation by recursive evaluation of taylor coefficients we present a computational method for the solution of the third order boundary value problem characterized by the well known falkner skan equation on a semi infinite domain numerical treatments of this problem reported in the literature thus far are based on shooting and finite differences while maintaining the simplicity of the shooting approach the method presented in this paper uses a technique known as automatic differentiation which is neither numerical nor symbolic using automatic differentiation a taylor series solution is constructed for the initial value problems by calculating the taylor coefficients recursively the effectiveness of the method is illustrated by applying it successfully to various instances of the falkner skan equation

legendre wavelets method for solving differential equations of lane emden type a numerical method for solving the lane emden equations as singular initial value problems is presented using integral operator and convert lane emden equations to integral equations and then applying legendre wavelet approximations the properties of legendre wavelet are first presented these properties together with the gaussian integration method are then utilized to reduce the integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

an elementary introduction to the homotopy perturbation method this paper is an elementary introduction to the concepts of the homotopy perturbation method particular attention is paid to giving an intuitive grasp for the solution procedure throughout the paper

integer round up property for the chromatic number of some h perfect graphs we prove that every h perfect line graph and every t perfect claw free graph g has the integer round up property for the chromatic number for every non negative integral weight function c on the vertices of g the weighted chromatic number of g c can be obtained by rounding up its fractional relaxation as a corollary we obtain that the weighted chromatic number can be computed in polynomial time for these graphs finally we show a new example of a graph operation which preserves the integer round up property for the chromatic number and use it to provide a first example of a t perfect 3 colorable graph which does not have this property

box total dual integrality box integrality and equimodular matrices box totally dual integral box tdi polyhedra are polyhedra described by systems which yield strong min max relations we characterize them in several ways involving the notions of principal box integer polyhedra and equimodular matrices a polyhedron is box integer if its intersection with any integer box ell le x le u is integer we define principally box integer polyhedra to be the polyhedra p such that kp is box integer whenever kp is integer a rational r times n matrix is equimodular if it has full row rank and its nonzero r times r determinants all have the same absolute value a face defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron our main result is that the following statements are equivalent along our proof we show that a polyhedral cone is box tdi if and only if it is box integer and that these properties are carried over to its polar we illustrate these charaterizations by reviewing well known results about box tdi polyhedra we also provide several applications the first one is a new perspective on the equivalence between two results about binary clutters secondly we refute a conjecture of ding zang and zhao about box perfect graphs thirdly we discuss connections with an abstract class of polyhedra having the integer caratheodory property finally we characterize the box tdiness of the cone of conservative functions of a graph and provide a corresponding box tdi system

quantized representation of some nonlinear integrable evolution equations on the soliton sector the hirota algorithm for solving several integrable nonlinear evolution equations is suggestive of a simple construction of a quantized representation of these equations and their soliton solutions over a fock space of bosons or of fermions the classical nonlinear wave equation becomes a nonlinear equation for an operator the solution of this equation is constructed through an operator analog of the hirota transformation the classical n soliton solution is the expectation value of the solution operator in an n particle state in the fock space the effect of perturbations that modify soliton identity is demonstrated

backlund transformation of partial differential equations from the painleve gambier classification i kaup kupershmidt equation among the 50 nonlinear second order differential equations of painleve and gambier those which are linearizable provide a natural scheme for deriving the lax pair and the darboux transformation of a nonlinear partial differential equation when the order of the scattering problem is three this new method allows one to obtain from singularity analysis only the backlund transformation of the kaup kupershmidt equation a result which was missing until now

on claw free t perfect graphs a graph is called t perfect if its stable set polytope is defined by non negativity edge and odd cycle inequalities we characterise the class of all claw free t perfect graphs by forbidden t minors and show that they are 3 colourable moreover we determine the chromatic number of claw free h perfect graphs and give a polynomial time algorithm to compute an optimal colouring

on soliton solutions of the kaup kupershmidt equation ii anomalous n soliton solutions abstract in the second of two articles designated i physica d 137 2000 25 and ii exact soliton solutions of the kaup kupershmidt equation are derived using a simplified approach based in part on hirota s bilinear transform these solutions have not previously been obtained explicitly by analytic methods it is found that at each order the n soliton solution is characterised by an additional parameter which accounts for the anomalous nature of these solutions the first four solitons are constructed explicitly and in a novel departure from hirota s method an iterative procedure is presented for obtaining the general n soliton solution some possible alternative approaches to solving the kaup kupershmidt equation are considered

nonlocal symmetries and the kaup kupershmidt equation computational and geometric aspects of nonlocal infinitesimal symmetries of nonlinear partial differential equations are considered in particular the relation of nonlocal symmetries with classical generalized and internal symmetries is briefly discussed a nonlocal symmetry for the kaup kupershmidt equation is introduced and studied in some detail some explicit particular solutions are found with its help and a darboux like transformation is also obtained

a new algorithm for solving classical blasius equation in this paper a reliable algorithm is employed to investigate the classical blasius equation the algorithm is based mainly on applying the adomian decomposition method to the transformation of the blasius equation the results demonstrate reliability and efficiency of the proposed algorithm

the optimization of convergence for chebyshev polynomial methods in an unbounded domain by using the method of steepest descent i have compared the suitability of three different methods for solving problems in a semi infinite or infinite domain using chebyshev polynomials exponential mappings are uniformyly bad domain truncation and algebraic mapping both work well but each is superior for a different category of problems when the solution is an entire function then domain truncation is best it is always simpler to apply than algebraic mapping and always at least as accurate much more accurate for functions which decay very rapidly for singular functions on the other hand algebraic mapping is better because it is less sensitive to the mapping scale factor l permitting a better compromise in resolving both the singularity and the exponential decay for both types of problems i give simple explicit estimates of both the optimum choise of domain size or mapping factor and of the attainable accuracy for the model entire function exp azk on a semi infinite interval 0 the optimum domain size is l 0 896 na 1k and the smallest attainable error is roughly e n 0 896 k where n is the number of chebyshev polynomals used similar formulas for singular functions handled via algebraic mapping are given in 3 41 to 3 45 below

on bidirectional fifth order nonlinear evolution equations lax pairs and directionally dependent solitary waves in this paper lax pairs are constructed for two fifth order nonlinear evolution equations of boussinesq type which govern wave propagation in two opposite directions one of the equations is related to the well known sawada kotera sk equation and through its bilinear form is identified with the ramani equation the second equation about which very little seems to be known may be considered a bidirectional version of the kaup kupershmidt kk equation and is the main focus of this study the anomalous solitary wave of this latter equation is derived and is found to possess the remarkable property that its profile depends on the direction of propagation this type of directional dependence would appear to be quite unusual and to our knowledge has not been reported in the literature before now by taking an appropriate undirectional long wave limit it is shown that neither the ramani nor the bidirectional kaup kupershmidt bkk equation can be classified as truly boussinesq in character a distinction that is made precise in the study recursion formulas are given for generating an infinity of conserved densities for both equations these are used to obtain the first few conservation laws of the bkk and ramani equations explicitly not surprisingly they exhibit the same lacunary behavior as their unidirectional counterparts in conclusion a canonical interpretation of the n soliton solution of the bkk equation is proposed which provides a basis for constructing these anomalous solitons in a future work

a simple perturbation approach to blasius equation in this paper we couple the iteration method with the perturbation method to solve the well known blasius equation the obtained approximate analytic solutions are valid for the whole solution domain comparison with howarth s numerical solution reveals that the proposed method is of high accuracy the first iteration step leads to 6 8 accuracy and the second iteration step yields the 0 73 accuracy of initial slop

darboux and backlund transformations of the bidirectional sawada kotera equation abstract darboux and backlund transformations of the bidirectional sawada kotera equation are derived with the help of the resulting riccati equation as an application some explicit solutions of the bidirectional sawada kotera equation are obtained including rational solutions periodic solutions and soliton solutions

pseudospectral methods on a semi infinite interval with application to the hydrogen atom a comparison of the mapped fourier sine method with laguerre series and rational chebyshev expansions the fourier sine with mapping pseudospectral algorithm of fattal et al phys rev e 53 1996 1217 has been applied in several quantum physics problems here we compare it with pseudospectral methods using laguerre functions and rational chebyshev functions we show that laguerre and chebyshev expansions are better suited for solving problems in the interval r 0 for example the coulomb schrodinger equation than the fourier sinemapping scheme all three methods give similar accuracy for the hydrogen atom when the scaling parameter l is optimum but the laguerre and chebyshev methods are less sensitive to variations in l we introduce a new variant of rational chebyshev functions which has a more uniform spacing of grid points for large r and gives somewhat better results than the rational chebyshev functions of boyd j comp phys 70 1987 63

rational scaled generalized laguerre function collocation method for solving the blasius equation in this paper we propose a collocation method for solving the blasius equation the blasius equation is a third order nonlinear ordinary differential equation this approach is based on a rational scaled generalized laguerre function collocation method we also present the comparison of this work with some well known results and show that the present solution is accurate

nonlinear superposition formula for the kaup kupershmidt partial differential equation abstract the fifth order kaup kupershmidt kk equation is one of the solitonic equations related to the integrable cases of the henon heiles system as opposed to the sawada kotera equation which is its dual equation the construction of the n soliton solutions of kk is not an easy task in using a perturbation scheme or the hirota bilinear formalism from the backlund transformation obtained by singularity analysis considerations we here establish the permutability theorem for the kk equation this allows us to explain the anomalous structure of the n soliton solution empirically obtained by parker physica d 137 2000 25 33 34 48

a bidirectional kaup kupershmidt equation and directionally dependent solitons exact multisoliton solutions are obtained for the bidirectional kaup kupershmidt equation that was reported in a recent paper j math phys 42 2567 2001 this fills the gap that was left in the prior study these solutions are found to possess the same remarkable and unusual property as the solitary wave solution namely their wave profiles are directionally dependent in particular solitons are presented which describe multiple head on collisions between single and double humped solitary waves explicit expressions are given for the first few multisoliton solutions and a procedure is described for constucting the general n soliton by iteration on the solitons of lower order

on soliton solutions of the kaup kupershmidt equation i direct bilinearisation and solitary wave abstract in this first of two papers designated i and ii physica d 137 2000 34 a direct procedure is given for bilinearising the kaup kupershmidt and sawada kotera equations it is found that whereas the sawada kotera equation can be written in bilinear form in two distinct ways which are shown to be equivalent the kaup kupershmidt equation has a single bilinear representation this distinguishing feature which has previously received no explanation has consequences for obtaining the soliton solutions of these two closely related fifth order integrable equations the anomalous solitary wave of the kaup kupershmidt equation is reconstructed and a simplified approach to obtaining its n soliton solutions is proposed this provides the basis for constructing these solutions explicitly in ii

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

comments on a new algorithm for solving classical blasius equation by l wang abstract in a recent paper wang l wang a new algorithm for solving classical blasius equation appl math comput 157 2004 1 9 employed the adomian decomposition method adm to solve numerically the famous blasius equation the purpose of this note is to correct the numerical solution of wang and to present an improved numerical solution using the adm pade approach the numerical result compares reasonably well with the result obtained from a shooting method

the blasius function computations before computers the value of tricks undergraduate projects and open research problems the blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick semi infinite flat plate the blasius function is the solution to 2 f xxx f f xx 0 on x in 0 infty subject to f 0 f x 0 0 f x infty 1 we use this famous problem to illustrate several themes first although the flow solves a nonlinear partial differential equation pde toepfer successfully computed highly accurate numerical solutions in 1912 his secret was to combine a runge kutta method for integrating an ordinary differential equation ode initial value problem with some symmetry principles and similarity reductions which collapse the pde system to the ode shown above this shows that pde numerical studies were possible even in the precomputer age the truth both a hundred years ago and now is that mathematical theorems and insights are an arithmurgist s best friend and they can vastly reduce the computational burden second we show that special tricks applicable only to a given problem can be as useful as the broad general methods that are the fabric of most applied mathematics courses the importance of particularity in spite of these triumphs many properties of the blasius function f x are unknown we give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

some remarks on the fifth order kdv equations abstract in this paper we present the differential operators for the generalized fifth order kdv equation we give formal proofs on the hamiltonian properties including the skew adjointness and jacobi identity by using the prolongation method our results show that there are three third order hamiltonian operators which can be used to construct the hamiltonians however no fifth order operators are shown to pass the hamiltonian test although there are an infinite number of them and they are skew adjoint

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

numerical solutions of the classical blasius flat plate problem abstract this paper presents a numerical study of the non linear differential equation af ff 0 where a prime denotes differentiation with respect to the similarity variable and a is a parameter for a 1 and a 2 this equation is a form of the blasius relation for the flat plate flow in fluid mechanics several numerical solution are obtained using a runge kutta algorithm for high order initial value problems for 1 a 2

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

computing clique and chromatic number of circular perfect graphs in polynomial time a main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time grotschel et al in combinatorica 1 2 169 197 1981 perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one we consider a well studied superclass of perfect graphs satisfying this property the circular perfect graphs and show that for such graphs both clique and chromatic number are computable in polynomial time as well in addition we discuss the polynomial time computability of further graph parameters for certain subclasses of circular perfect graphs all the results strongly rely upon lovasz s theta function

extended soliton solutions for the kaup kupershmidt equation we extend the n soliton solutions of the kaup kupershmidt equation on a nonzero background decreasing as x 1 a 2 these new solutions describe the interaction of n solitary waves with a static bell shaped wave we give the conditions so that the backlund transformation relating those solutions and the n solitons of the sawada kotera equation will be satisfied

the sinc collocation method for solving the thomas fermi equation a numerical technique for solving nonlinear ordinary differential equations on a semi infinite interval is presented we solve the thomas fermi equation by the sinc collocation method that converges to the solution at an exponential rate this method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations this method is easy to implement and yields very accurate results

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods