comment on an aspect of a paper by g thompson g thompson j math phys 25 3474 1984 by way of example of a general method examined the existence of integrals of the form i ypx xpy 3 a x y px b x y py for the hamiltonian h 1 2 p2x p2y v x y the treatment was not complete a complete solution is given here as well as indications for further development to integrals of higher order in the momenta

the complex lorenz equations we have undertaken a study of the complex lorenz equations x x y y r z x ay z bz 12 x y xy where x and y are complex and z is real the complex parameters r and a are defined by r r1 ir2 a 1 ie and and b are real behaviour remarkably different from the real lorenz model occurs only the origin is a fixed point except for the special case e r2 0 we have been able to determine analytically two critical values of r1 namely r1c and r1c the origin is a stable fixed point for 0 r1c a hopf bifurcation to a limit cycle occurs we have an exact analytic solution for this limit cycle which is always stable if 1 then this limit is only stable in the region r1c rlc a transition to a finite amplitude oscillation about the limit cycle occurs the nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the stuart landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency this latter bifurcation is either a sub or super critical hopf like bifurcation to a doubly periodic motion the direction of bifurcation depending on the parameter values the nature of the bifurcation is complicated by the existence of a zero eigenvalue

painleve property and constants of the motion of the complex lorenz model the results of the painleve analysis and of a search for constants of the motion of the complex lorenz model are presented

an integrable sis model we provide a demonstration of the integrability of a classical model of an infectious disease which neither kills nor induces autoimmunity by means of the painleve analysis and use the lie theory of transformation groups to present an explicit solution

a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

the method of lines and invariant imbedding for elliptic and parabolic free boundary problems a combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension electrochemical machining and hele shaw flow in two dimensions and a stefan and ablation problem in three dimensions

first integrals and parametric solutions of third order odes admitting mathfrak sl 2 mathbb r a complete set of first integrals for any third order ordinary differential equation admitting a lie symmetry algebra isomorphic to is explicitly computed these first integrals are derived from two linearly independent solutions of a linear second order ode without additional integration the general solution in parametric form can be obtained by using the computed first integrals the study includes a parallel analysis of the four inequivalent realizations of and it is applied to several particular examples these include the generalized chazy equation as well as an example of an equation which admits the most complicated of the four inequivalent realizations

representations of one dimensional hamiltonians in terms of their invariants a general formalism for representing the hamiltonian of a system with one degree of freedom in terms of its invariants is developed those hamiltonians h q p t are derived for which any particular function i q p t is an invariant for each of those hamiltonians a function canonically conjugate to i q p t is derived which is also an invariant of h q p t the formalism is also presented for the case in which i q p t is expressed as a function of two canonically conjugate functions the formalism is illustrated by applying it to the case of a particle moving in a time dependent potential some earlier results are recovered and an invariant is found for a new potential lines for further study are outlined that may be fruitful for finding more examples of integrable systems

symmetry and singularity properties of second order ordinary differential equations of lie s type iii abstract we extend the work of abraham shrauner b abraham shrauner hidden symmetries and linearization of the modified painleve ince equation j math phys 34 1993 4809 4816 on the linearization of the modified painleve ince equation to a wider class of nonlinear second order ordinary differential equations invariant under the symmetries of time translation and self similarity in the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations

type ii hidden symmetries through weak symmetries for nonlinear partial differential equations abstract the type ii hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a lie point symmetry in b abraham shrauner k s govinder provenance of type ii hidden symmetries from nonlinear partial differential equations j nonlinear math phys 13 2006 612 622 abraham shrauner and govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries the lie point symmetries of a model equation and the two dimensional burgers equation and their descendants were used to identify the hidden symmetries in this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries we have considered the same models presented in b abraham shrauner k s govinder provenance of type ii hidden symmetries from nonlinear partial differential equations j nonlinear math phys 13 2006 612 622 as well as the wdvv equations of associativity in two dimensional topological field theory which reduces in the case of three fields to a single third order equation of monge ampere type we have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using lie symmetries however when is reduced by using nonclassical symmetries the reduced partial differential equation gains lie symmetries

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

lie subalgebras reduction of order and group invariant solutions one of the great utilities of lie symmetries of differential equations is in their use to reduce the order of ordinary differential equations and partial differential equations to ordinary differential equations this process is guided by the lie algebra of the symmetries admitted by the equation under study it is important in these reductions to use appropriate symmetries this will ensure that a sufficient number remain after the initial reduction to allow subsequent reductions by considering two and three dimensional subalgebras of the lie algebra of the symmetries of differential equations we indicate the fate of these symmetries under various reductions in doing so it is clear as to which symmetries need be prioritized in reduction

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

hidden and contact symmetries of ordinary differential equations the lie algebra for the maximal contact symmetries of third order ordinary differential equations odes is examined for type i and ii hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries ones invariant under the group associated with the ten dimensional maximal lie algebra may produce type i hidden symmetries for two parameter subgroups and type ii hidden symmetries for certain solvable non abelian three parameter subgroups in the third order odes when they are reduced in order a new class of type ii hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths two examples of odes invariant under subgroups of the ten parameter group under which y 0 is invariant demonstrate the new class of type ii hidden symmetries

the numerical evaluation of a class of logarithmically singular integral transforms abstract we consider the numerical evaluation of several logarithmically singular integral transforms which occur in the numerical solution of various types of integral equations with logarithmically singular kernels when the equations involve cauchy singular kernels as well the method is a substantial improvement in speed accuracy and ease of programming over the method introduced by fromme golberg and werth 5 convergence of the rule is proved under suitable differentiability conditions on the smooth part of the kernel and numerical results are presented substantiating this result

on the polynomial first integrals of certain second order differential equations it is shown that any first integral of type p2 x a polynomial of degree 2 in x of the differential equation x vx can be obtained from a pointlike gauge symmetry of the action al associated to l 1 2 x2 v t x the same result holds for any first integral of kind pn x when dynamical symmetries of al polynomials in x are allowed the neccessary and sufficient conditions that v t x must satisfy in order that x vx possesses a first integral of type pn x have been obtained these conditions reduce when n 2 to a condition obtained by leach the computational advantages and difficulties which appear in order to obtain first integrals for type pn x are also briefly discussed

a mesh independence principle for operator equations and their discretizations the mesh independence principle asserts that when newton s method is applied to a nonlinear equation between some banach spaces as well as to some finite dimensional discretization of that equation then the behavior of the discretized process is asymptotically the same as that for the original iteration and as a consequence the number of steps required by the two processes to converge to within a given tolerance is essentially the same so far this result has been proved only for certain classes of boundary value problems in this paper a proof is presented for a general class of operator equations and discretizations it covers the earlier results and extends them well beyond the cases that have been considered before

the real and complex lorenz equations in rotating fluids and lasers abstract the lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added this derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation the addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of lorenz equations both sets of equations are derived in examples in laser optics and baroclinic instability

autonomous self similar ordinary differential equations and the painleve connection we demonstrate an intimate connection between nonlinear higher order ordinary differential equations possessing the two symmetries of autonomy and self similarity and the leading order behaviour and resonances determined in the application of the painleve test similar behaviour is seen for systems of first order differential equations several examples illustrate the theory in an integrable case of the abc system the singularity analysis reveals a positive and a negative resonance and the method of leading order behaviour leads naturally to a laurent expansion containing both

on the equivalence of linear equations under nonlocal transformation abstract linear n th order n 3 ordinary differential equations have been shown to possess n 1 n 2 or n 4 lie point symmetries each class contains equations which are equivalent under point transformation by taking the example of third order equations we show that all linear equations are equivalent if the class of transformation is broadened to include nonlocal transformations and hence the representative of this class of equations is y n 0

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

painleve analysis yoshida s theorems and direct methods in the search for integrable hamiltonians treating as an example two dimensional similarity invariant hamiltonians with polynomial velocity dependent potentials the authors show how painleve analysis yoshida s theorems and direct methods can be combined in the search for integrable hamiltonians new integrable systems are found one of which has a polynomial second invariant of fifth order in the momenta most of the necessary calculations have been performed by applying various reduce packages developed for that purpose

first integrals via polynomial canonical transformations maharatna dutt and chattarji j math phys 20 2221 1979 discussed the use of time dependent canonical transformations for the determination of first integrals for time dependent hamiltonian systems one particular proposal that successive time dependent polynomial canonical transformations will enable first integrals to be found for a wider variety of time dependent polynomial hamiltonians than can be obtained using time dependent linear canonical transformations is shown to be not true for the paradigm which they selected it is suggested that their ansatz is ill founded in general

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

invariance of second order ordinary differential equations under two dimensional affine subalgebras of ermakov pinney lie algebra abstract using the only admissible rank two realisations of the lie algebra of the affine group in one dimension in terms of the lie algebra of lie symmetries of the ermakov pinney ep equation some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed one class includes the standard ep equation as a special case a new ep equation with a perturbed potential but admitting the same solution formula as ep itself arises the solution of the dissipative ep equation is also discussed

master partial differential equations for a type ii hidden symmetry an approach for determining a class of master partial differential equations from which type ii hidden point symmetries are inherited is presented as an example a model nonlinear partial differential equation pde reduced to a target pde by a lie symmetry gains a lie point symmetry that is not inherited hidden from the original pde on the other hand this type ii hidden symmetry is inherited from one or more of the class of master pdes the class of master pdes is determined by the hidden symmetry reverse method the reverse method is extended to determine symmetries of the master pdes that are not inherited we indicate why such methods are necessary to determine the genesis of type ii symmetries of pdes as opposed to those that arise in ordinary differential equations odes

a fifth order differential equation for charged perfect fluids we investigate the master nonlinear partial differential equation that governs the evolution of shear free spherically symmetric charged fluids we use an approach which has not been considered previously for the underlying equation in shear free spherically symmetric spacetimes we derive a fifth order purely differential equation that must be satisfied for the underlying equation to admit a lie point symmetry we then perform a comprehensive analysis of this equation utilising the lie symmetry analysis and direct integration this enables us to reduce the fifth order equation to quadratures earlier results are shown to be contained in our general treatment

approximating the singular integrals of cauchy type with weight function on the interval it is known that the solutions of characteristic singular integral equations sies are expressed in terms of singular integrals of cauchy type with weight functions w x 1 x n 1 x m where n 12 m 12 new quadrature formulas qfs are presented to approximate the singular integrals sis of cauchy type for all solutions of characteristic sie on the interval 1 1 linear spline interpolation modified discrete vortex method and product quadrature rule are utilized to construct the qfs estimation of errors are obtained in the classes of functions h a 1 1 a and c 1 1 1 it is found that the numerical results are very stable even for the cases of semi bounded and unbounded solutions of singular integral equation of the first kind

numerical solution of a free boundary problem in hypersonic flow theory nonequilibrium viscous shock layers abstract non equilibrium viscous shock layer flows produced near the stagnation point of a blunt body constitute a standard problem in hypersonic flow theory this flow configuration is usually modeled with the viscous shock layer vsl equations derived by writing the full navier stokes equations in boundary layer coordinates and performing an order of magnitude analysis the terms retained in this process assure that for moderate reynolds numbers the resulting equations are uniformly valid between the body surface and the shock which may be treated as a thin discontinuity the vsl equations are generally solved by starting with the thin layer approximation tvsl boundary conditions are specified at the body surface and the rankine hugoniot relations are imposed at the shock because the position of the shock is not known one has to solve a free boundary problem this paper presents a novel solution procedure for this situation a reduced coordinate is introduced and the free boundary problem is transformed into a nonlinear eigenvalue problem the new problem for an augmented set of variables is then solved with newton iterations and adaptive gridding the method is illustrated in the paper with a solution of the thin shock layer equations at the stagnation streamline the solution obtained on this line is then used as an initial condition for a two dimensional marching procedure the complete axisymmetric problem is solved by performing fully coupled newton iterations moreover we consider a new enlarged unknown function including the shock location the model includes detailed transport properties and complex kinetics for air dissociation and ionization however in order to focus on the numerical method and for the sake of simplicity thermodynamic equilibrium is assumed

solution of ordinary differential equations via nonlocal transformations the classical lie analysis provides a useful technique for the solution of ordinary differential equations via point symmetries transformations unfortunately the requirement that an nth order equation possesses at least an n dimensional solvable lie algebra of symmetries is only satisfied by a select number of equations we provide a class of second order ordinary differential equations with less than the required two lie point symmetries that can be solved via nonlocal transformations

nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order in this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations odes of arbitrary order related to linear odes through nonlocal relations the procedure makes use of the lie point symmetries of the linear odes and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear odes using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples we find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases we also extend this procedure to coupled higher order nonlinear odes with specific reference to second order nonlinear odes

exponential nonlocal symmetries and nonnormal reduction of order the conventional approach to double reduction of the order of an ordinary differential equation using lie symmetries is via the normal subgroups of point symmetries we show that provided that one is prepared to use nonlocal symmetries initial reduction by the nonnormal subgroup does not prevent the double reduction we further illustrate our results with the general third order equations invariant under the nonsolvable algebras sl 2 r of which the chazy equation is a noted example and so 3

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

type ii hidden symmetries of the linear 2d and 3d wave equations type ii hidden symmetries of the linear two dimensional and three dimensional wave equations are analysed these hidden symmetries are lie point symmetries that appear in addition to the inherited point symmetries when the number of independent and dependent variables of a partial differential equation is reduced by a lie point symmetry the provenance of these hidden symmetries of partial differential equations is identified to be the same as found recently for some nonlinear partial differential equations the appearance of type ii hidden symmetries depends not only on the lie symmetries used but on the order in which the symmetries are applied the presence of type ii hidden symmetries of partial differential equations complicates the prediction of symmetry reductions based on the lie algebra associated with the original lie point symmetries

the determination of nonlocal symmetries by the technique of reduction of order autonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an ansatz on the detailed structure of the symmetry because the coefficient function for the original independent variable appears only as its derivative in the reduced system symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

on the properties of a variant of the riccati system of equations a variant of the generalized riccati system of equations is considered it is shown that for 2n 3 the system admits a bilagrangian description and the dynamics has a node at the origin whereas for much smaller than a critical value the dynamics is periodic the origin being a centre it is found that the solution changes from being periodic to aperiodic at a critical point which is independent of the initial conditions this behaviour is explained by finding a scaling argument via which the phase trajectories corresponding to different initial conditions collapse onto a single universal orbit numerical evidence for the transition is shown further using a perturbative renormalization group argument it is conjectured that the oscillator exhibits isochronous oscillations the correctness of the conjecture is established numerically

first integrals and reduction of a class of nonlinear higher order ordinary differential equations abstract we propose a method for constructing first integrals of higher order ordinary differential equations in particular third fourth and fifth order equations of the form x n h x x n 1 x are considered the relation of the proposed method to local and nonlocal symmetries are discussed

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

the real and complex lorenz equations and their relevance to physical systems abstract we summarize some recently obtained results on real and complex lorenz equations and discuss their possible significance in relation to real fluid dynamical processes

a λ symmetry based method for the linearization and determination of first integrals of a family of second order ordinary differential equations we present a characterization of the second order ordinary differential equations odes that can be linearized by means of certain nonlocal transformations this characterization is given in terms of the coefficients of the equation and also determines the second order odes that admit symmetries and first integrals of some specific forms systematic methods to find these first integrals symmetries and the linearizing transformations are also derived

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

on the reduction methods for ordinary differential equations as tandard method based on the use of differential invariants of a lie group g e nables us to reduce any ordinary differential equation invariant under the action of g w e show that this method is applicable to vector fields more general than those associated with lie symmetries we characterize all such vector fields and study their relationship with nonlocal symmetries and symmetries govinder k s and leach p g l 1995 j phys a math gen 28 5349 59 muriel c and romero l 2001 ima j appl math 66 111 25

on the uniqueness of the schwarzian and linearisation by nonlocal contact transformation abstract we investigate the uniqueness of the schwarzian from a group theoretic point of view the kummer schwarz equation obtained from the schwarzian is also analysed for its group properties related equations arising in the study of geodesic curves on a surface of constant curvature are constructed these equations are generalised by reducing the symmetry imposed the unique generalisation is linearised under a nonlocal contact transformation

numerical evaluation of cauchy principal value integrals based on local spline approximation operators abstract in this paper a local spline approximation method defined for any function l 1 1 1 is applied to evaluate cauchy principal value integrals and convergence results with an error bound are established some comparisons with other methods and numerical examples are given

error bounds for spline based quadrature methods for strongly singular integrals abstract for the numerical evaluation of finite part integrals with singularities of order p 1 we give error bounds for quadrature methods based on spline approximation these bounds behave in the same way as the optimal ones the ideas of the proof are also useful for methods based on other approximation processes

symmetry lie algebras of nth order ordinary differential equations abstract we show that an nth n 3 order linear ordinary differential equation has exactly one of n 1 n 2 or n 4 the maximum point symmetries the lie algebras corresponding to the respective numbers of point symmetries are obtained then it is shown that a necessary and sufficient conditon for an nth n 3 order equation to be linearizable via a point transformation is that it must admit the n dimensional abelian algebra na1 a1 a1 a1 we discuss in detail the symmetry realizations of n 1 a1 s a1 finally we prove that an nth n 3 order equation q n h t q qn 1 cannot admit exactly an n 3 dimensional algebra of point symmetries which is a subalgebra of na1 gl 2 r

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

solvable structures and hidden symmetries we show that some examples in the literature of non standard symmetry reductions of ordinary differential equations can be understood using the concept of a solvable structure of one forms

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

hidden symmetries associated with the projective group of nonlinear first order ordinary differential equations hidden symmetries those not found by the classical lie group method for point symmetries are reported for nonlinear first order ordinary differential equations odes which arise frequently in physical problems these are for the special class of the eight nonabelian two parameter subgroups of the eight parameter projective group the first order odes can be transformed by non local transformations to new separable first order odes which then can be reduced to quadratures the first order odes include riccati equations and equations which in particular cases are of the form of abel s equation the procedure demonstrates the feasibility of integrating nonlinear odes that do not show any apparent lie group point symmetry applications to the vlasov characteristic equation and the reaction diffusion equation are given

generating functions and existence of contact symmetries of third order scalar ordinary differential equations analyzing the structure of generating functions associated to the contact symmetry condition of third order scalar ordinary differential equations constraints guaranteeing that the ode does not admit contact symmetries are obtained further inspecting the subalgebras generated by contact symmetries having a specific form an alternative procedure to generate third order equations with irreducible contact symmetry algebra sp 4 is developed

a group theoretic approach to a class of second order ordinary differential equations not possessing lie point symmetries we consider the solution of a class of second order ordinary differential equations not possessing lie point symmetries by group theoretic means the method involves increasing the order of these equations using homogeneity symmetry the solution of this new third order equation is then sought in the instances that it and the new reduced second order equation possess additional symmetries as a result the number of second order equations solvable by the lie theory of extended groups is increased

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

quadrature formula for approximating the singular integral of cauchy type with unbounded weight function on the edges new quadrature formulas qfs for evaluating the singular integral si of cauchy type with unbounded weight function on the edges is constructed the construction of the qfs is based on the modification of discrete vortices method mmdv and linear spline interpolation over the finite interval 1 1 it is proved that the constructed qfs converge for any singular point x not coinciding with the end points of the interval 1 1 numerical results are given to validate the accuracy of the qfs the error bounds are found to be of order o h a lnh and o h lnh in the classes of functions h a 1 1 and c 1 1 1 respectively

symmetry and singularity properties of a system of ordinary differential equations arising in the analysis of the nonlinear telegraph equations we investigate the symmetry and similarity properties of a system of equations arising in the analysis of the nonlinear telegraph equations the system of two equations can be decoupled and integrated although one of equations is linear in one of the dependent variables we are able to perform successfully a singularity analysis we are able to interpret the results of the singularity analysis in terms of the possibility of the existence of a subsidiary solution

symmetries and first integrals for dissipative systems the connection between symmetries time independent and time dependent first integrals for autonomous systems of first order ordinary differential equations is discussed several new results which have been obtained by applying a reduce package are reported

lie symmetries for lie systems a lie system is a nonautonomous system of first order differential equations admitting a superposition rule i e a map expressing its general solution in terms of a generic family of particular solutions and some constants using that a lie system can be considered as a curve in a finite dimensional lie algebra of vector fields a so called vessiot guldberg lie algebra we associate every lie system with a lie algebra of lie point symmetries induced by the vessiot guldberg lie algebra this enables us to derive lie symmetries of relevant physical systems described by first and higher order systems of differential equations by means of lie systems in an easier way than by standard methods a generalization of our results to partial differential equations is introduced among other applications lie symmetries for several new and known generalizations of the real riccati equation are studied

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method