fft formulations of adaptive fourier decomposition adaptive fourier decomposition afd has been found to be among the most effective greedy algorithms afd shows an outstanding performance in signal analysis and system identification as compensation of effectiveness the computation complexity is great that is especially due to maximal selections of the parameters in this paper we explore the discretization of the 1 d afd integration via with discrete fourier transform dft incorporating fast fourier transform fft we show that the new algorithm called fft afd reduces the computational complexity from o mn2 to o mnlogn the latter being the same as fft through experiments we verify the effectiveness accuracy and robustness of the proposed algorithm the proposed fft based algorithm for afd lays a foundation for its practical applications

nystrom type methods for fredholm integral equations with weak singularities nystrom type methods are constructed and justified for a class of fredholm integral equations of the second kind with kernels which may have weak diagonal and boundary singularities the proposed approach is based on a suitable smoothing change of variables and product integration techniques global convergence estimates are derived and a collection of numerical results is given

smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems abstract we construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form d y t f t y t with caputo type fractional derivatives d y of order 0 using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed algorithms two numerical examples are given

a geometrical method towards first integrals for dynamical systems we develop a method based on darboux s and liouville s works to find first integrals and or invariant manifolds for a physically relevant class of dynamical systems without making any assumption on these elements forms we apply it to three dynamical systems lotka volterra lorenz and rikitake

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

the approximate solution of a class of fredholm integral equations with a weakly singular kernel a method for finding the numerical solution of a weakly singular fredholm integral equation of the second kind is presented the taylor series is used to remove singularity and legendre polynomials are used as a basis furthermore the legendre function of the second kind is used to remove singularity in the cauchy type integral equation the integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the cauchy principal value sense without using numerical quadratures four examples are given to show the accuracy of the method

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

a numerical approach to the solution of abel integral equations of the second kind with nonsmooth solution abstract in this paper we study the feasibility of numerical solutions analogous to those methods introduced by bownds 1979 1982 and bownds and wood 1976 1979 for volterra integral equations of the second kind with weakly singular kernels

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

support vector machine adapted tikhonov regularization method to solve dirichlet problem numerical solutions of partial differential equations are traditional topics that have been studied by many researchers during the last decade support vector machine svm has been widely used for approximation problems the contribution of this paper is two folds one is to combine the reproducing kernel svm method with the tikhonov regularization method called the svm tik methods in which the kernels k and k see below are newly developed in the paper they are respectively phrased as the svm tik k and svm tik k methods the second contribution is to use the two models svm tik k and svm tik k to solve the dirichlet problem the methods are meshless they produce sparse representations in the linear combination form of specific functions the k and k kernels the generalization bound result in learning theory is used to give an estimation of the approximation errors with the illustrative examples the sparseness and robustness properties as well as the effectiveness of the methods are presented the proposed methods are compared with currently the most commonly used finite difference method fdm showing promising results

piecewise homotopy perturbation method for solving linear and nonlinear weakly singular vie of second kind the purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular volterra integral equation of the second kind especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem for this purpose we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like through modifying homotopy perturbation method an algorithm is successfully established to solve the linear and nonlinear weakly singular volterra integral equation of the second kind and the convergence of the algorithm is proved strictly comparisons are made between our method and other methods and the results reveal that the modified homotopy perturbation is effective

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

sinc collocation approximation of non smooth solution of a nonlinear weakly singular volterra integral equation a numerical method based on sinc collocation approximation for a class of nonlinear weakly singular volterra integral equations of a second kind with non smooth solution is given the numerical method given here combines a sinc collocation method with an explicit iterative process that involves solving a nonlinear system of equations we provide an error analysis for the method it is shown that the approximate solution converges to the exact solution at the rate of mexp cm where m is the number of collocation points and c is some positive constant some numerical results for several test functions are given to confirm the accuracy and the ease of implementation of the method

galerkin methods for second kind integral equations with singularities this paper discusses the numerical solution of fredholm integral equations of the second kind which have weakly singular kernels and inhomogeneous terms global convergence estimates are derived for the galerkin and iterated galerkin methods using splines on arbitrary quasiuniform meshes as approximating subspaces it is observed that due to the singularities present in the solution being approximated the resulting convergence may be slow it is then shown that convergence will be improved greatly by the use of splines based on a mesh which has been suitably graded to accommodate these singularities in fact it is shown that under suitable conditions the galerkin method converges optimally and the iterated galerkin method is superconvergent numerical ilustrations are given

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

first integrals and symmetries of time dependent hamiltonian systems in this paper first integrals of one dimensional time dependent hamiltonians h q p t are studied first of all it is shown that the second first integral j q h t t dq hp valid for autonomous hamiltonians h q p can be extended to the case of nonautonomous hamiltonians h q p t provided hp resp t is replaced with ip where i q p t is a first integral of h q p t resp a new time t j q i t t dq ip in a second part we derive a technique leading to a first integral i q p t from a lie symmetry as an example this technique together with the expression of the second first integral j q i t is used to completely integrate a special case of the linear harmonic oscillator with a time dependent frequency as a result lewis first integral is recovered from a lie symmetry in a way that generalizes the recent work of shivamoggi and muilenberg phys lett a 154 1 1991

mechanical quadrature methods and their extrapolation for solving first kind abel integral equations this paper presents high accuracy mechanical quadrature methods for solving first kind abel integral equations to avoid the ill posedness of problem the first kind abel integral equation is transformed to the second kind volterra integral equation with a continuous kernel and a smooth right hand side term expressed by weakly singular integrals by using periodization method and modified trapezoidal integration rule not only high accuracy approximation of the kernel and the right hand side term can be easily computed but also two quadrature algorithms for solving first kind abel integral equations are proposed which have the high accuracy o h 2 and asymptotic expansion of the errors then by means of richardson extrapolation an approximation with higher accuracy order o h 3 is obtained moreover an a posteriori error estimate for the algorithms is derived some numerical results show the efficiency of our methods

lorenz integrable system moves a la poinsot a transformation is derived which takes the lorenz integrable system into the well known euler equations of a torque free rigid body about a fixed point i e the famous motion a la poinsot the proof is based on lie group analysis applied to two third order ordinary differential equations admitting the same two dimensional lie symmetry algebra lie s classification of two dimensional symmetry algebras in the plane is used if the same transformation is applied to the lorenz system with any values of the parameters then one obtains euler equations of a rigid body about a fixed point subjected to a torsion depending on time and angular velocity the numerical solution of this system yields a three dimensional picture which looks like a tornado the cross section of which has a butterfly shape thus lorenz s butterfly has been transformed into a tornado

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

on the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems in this paper we first give a necessary condition for general nonlinear systems to have rational first integrals then by using the so called kowalevsky exponents we present a criterion for nonexistence of rational first integrals for semiquasihomogeneous systems

high accuracy combination algorithm and a posteriori error estimation for solving the first kind abel integral equations abstract this paper presents high accuracy combination algorithm for solving the first kind abel integral equations to avoid solving ill posed problems we transform the first kind abel integral equation to the second kind volterra integral equation with a continuous kernel expressed by a weakly singular integral by using integration rules the approximation of this kernel can be easily computed then two quadrature algorithms for solving abel integral equations are proposed which possess accuracy order o h 1 0 h 2 moreover a posteriori error estimate for the algorithms is derived both theory and numerical examples show that it is effective and saves storage capacity and computational work

the convergence rate of semi supervised regression with quadratic loss abstract it is known that the semi supervised learning deals with learning algorithms with less labeled samples and more unlabeled samples one of the problems in this field is to show at what extent the performance depends upon the unlabeled number a kind of modified semi supervised regularized regression with quadratic loss is provided the convergence rate for the error estimate is given in expectation mean it is shown that the learning rate is controlled by the number of the unlabeled samples and the algorithm converges with the increasing of the unlabeled sample number

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

operational matrix approach for the solution of partial integro differential equation in this paper an effective numerical method is introduced for the treatment of volterra singular partial integro differential equations they are based on the operational and almost operational matrix of integration and differentiation of 2d shifted legendre polynomials the methods convert the singular partial integro differential equation in to a system of algebraic equations convergence analysis and error estimates are derived for the proposed method illustrative examples are included to demonstrate the validity and applicability of the technique

neural network methods for boundary value problems with irregular boundaries partial differential equations pdes with boundary conditions dirichlet or neumann defined on boundaries with simple geometry have been successfully treated using sigmoidal multilayer perceptrons in previous works the article deals with the case of complex boundary geometry where the boundary is determined by a number of points that belong to it and are closely located so as to offer a reasonable representation two networks are employed a multilayer perceptron and a radial basis function network the later is used to account for the exact satisfaction of the boundary conditions the method has been successfully tested on two dimensional and three dimensional pdes and has yielded accurate results

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

superconvergence of numerical solutions to volterra integral equations with singularities in this paper we discuss the beta polynomial discrete collocation method based on practical meshes for volterra integral equations with weakly singular kernels it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and by a simple cheap postprocessing of the original discrete collocation solution

existence of solutions for a class of nonlinear volterra singular integral equations in this paper we prove some results concerning the existence of solutions for a large class of nonlinear volterra singular integral equations in the space c 0 1 consisting of real functions defined and continuous on the interval 0 1 the main tool used in the proof is the concept of a measure of noncompactness we also present some examples of nonlinear singular integral equations of volterra type to show the efficiency of our results moreover we compare our theory with the approach depending on the use of the theory of volterra stieltjes integral equations we also show that the results of the paper are applicable in the study of the so called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems

lie symmetries and invariants of the lotka volterra system in this paper we use the lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3d lotka volterra system

lie symmetries for the reduced three wave interaction problem the authors show how the method of lie symmetries can be used to obtain first integrals and for the identification of completely or partially integrable cases of the reduced three wave interaction problem

necessary conditions for the existence of quasi polynomial invariants the quasi polynomial and lotka volterra systems abstract we show that any quasi polynomial invariant of a quasi polynomial dynamical system can be transformed into a quasi polynomial invariant of a homogeneous quadratic lotka volterra dynamical system we show how this quasi polynomial invariant can be decomposed in a simple manner this decomposition permits to conclude that the existence of polynomial semi invariants in lotka volterra systems is a necessary condition for the existence of quasi polynomial invariants we derive a method which allows to construct the necessary conditions for existence of semi invariants on lotka volterra dynamical systems applications are given

dynamical systems embedded into lie algebras analytical and geometrical information on certain dynamical systems x is obtained under the assumption that x is embedded into a certain real lie algebra

numerical solution of nonlinear weakly singular partial integro differential equation via operational matrices in this paper we propose and analyze an efficient matrix method based on shifted legendre polynomials for the solution of non linear volterra singular partial integro differential equations pides the operational matrices of integration differentiation and product are used to reduce the solution of volterra singular pides to the system of non linear algebraic equations some useful results concerning the convergence and error estimates associated to the suggested scheme are presented illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method

extrapolation method for solving weakly singular nonlinear volterra integral equations of the second kind abstract based on a new generalization of discrete gronwall inequality in l tao h yong a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equality of the second kind j math anal appl 282 2003 56 62 navot s quadrature rule for computing integrals with the end point singularity in i navot a further extension of euler maclaurin summation formula j math phys 41 1962 155 184 and a transformation in p baratella a palamara orsi a new approach to the numerical solution of weakly singular volterra integral equations j comput appl math 163 2004 401 418 a new quadrature method for solving nonlinear weakly singular volterra integral equations of the second kind is presented the convergence of the approximation solution and the asymptotic expansion of the error are proved so by means of the extrapolation technique we not only obtain a higher accuracy order of the approximation but also get a posteriori estimate of the error

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equation of the second kind this paper presents a new discrete gronwall inequality using the inequality we prove convergence and error estimate of the numerical solutions of the second weakly singular volterra integral equation where discrete equation is derived by novot s quadrature formula

algebraic structures and invariant manifolds of differential systems algebraic tools are applied to find integrability properties of odes bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations since all equations in this class are related to a canonical quadratic differential system the lotka volterra system these algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given their subalgebras are associated to nonlinear invariant manifolds in the phase space these manifolds are calculated explicitly more general algebraic invariant surfaces are also obtained by combining a theorem of walcher and the lotka volterra canonical form applications are given for lorenz model lotka may leonard and rikitake systems

lie symmetries and infinite dimensional lie algebras of certain nonlinear dissipative systems we point out that the dissipative force free duffing oscillator and holmes rand nonlinear oscillator admit infinite dimensional symmetry algebras for certain parametric choices for which the systems become integrable from the nature of the symmetry vector fields one can also write down the integrals of motion for the above systems in addition we point out that the other dissipative systems discussed in the literature also admit infinite dimensional lie algebras

a nystrom interpolant for some weakly singular nonlinear volterra integral equations we consider a second kind weakly singular nonlinear volterra hammerstein integral equation defined by a compact operator and derive a nystrom type interpolant of the solution based on gauss radau nodes we prove the convergence of the interpolant and derive convergence estimates for equations with nonlinearity of algebraic kind we improve the rate of convergence by using a smoothing transformation some numerical examples are given

numerical solution of abel s integral equation by using legendre wavelets abstract a numerical method for solving abel s integral equation as singular volterra integral equations is presented the method is based upon legendre wavelet approximations the properties of legendre wavelet are first presented these properties are then utilized to reduce the singular volterra integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

artificial neural networks for solving ordinary and partial differential equations we present a method to solve initial and boundary value problems using artificial neural networks a trial solution of the differential equation is written as a sum of two parts the first part satisfies the initial boundary conditions and contains no adjustable parameters the second part is constructed so as not to affect the initial boundary conditions this part involves a feedforward neural network containing adjustable parameters the weights hence by construction the initial boundary conditions are satisfied and the network is trained to satisfy the differential equation the applicability of this approach ranges from single ordinary differential equations ode to systems of coupled ode and also to partial differential equations pde in this article we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the galerkin finite element method for several cases of partial differential equations with the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed

the dirichlet to neumann map for a nonlinear diffusion convection equation a method for constructing the dirichlet to neumann map for a nonlinear diffusion convection equation is presented the problem is reduced to the solution of a nonlinear integral equation in one independent variable existence and uniqueness of the solution may be proven for small times via a contraction mapping technique

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence