on a class of nonlinear dynamical systems the structure of a differential operator in the application of the decomposition method abstract a suitable expresion of a differential operator is deduced allowing the explicit calculation of the general term f h in the power series expansion of a function f x t to be utilized for solving nonlinear differential equations

polynomial nonlinearities in differential equations abstract adomian and his collaborators have found solutions for nonlinear stochastic or deterministic in the limiting case differential equations involving polynomial non linearities using the first author s a n polynomials and recently developed methods of calculating these polynomials it becomes very easy to write solutions for non linear terms such as y m for positive integers m the cases of negative integer m or even decimal powers will appear elsewhere

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

on the convergence region for decomposition solutions abstract the author s decomposition method using his a n polynomials for the nonlinearities has been shown to apply to wide classes of nonlinear or nonlinear stochastic operator equations providing a computable accurate solution which converges rapidly in computation the above is sufficient for a rapid test of convergence region

on the tricomi problem the tricomi equation is solved by the decomposition method 6 8 on the half space x y 0 and also for suitable boundary conditions given on the boundary of the hyperbolic elliptic domain

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

on the analytic solutions of the nonhomogeneous blasius problem in this article a totally analytic solution of the nonhomogeneous blasius problem is obtained using the homotopy analysis method ham this solution converges for 0 l c an analytical value of the critical value of l c was also obtained for the first time

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

nonlinear plasma response abstract the decomposition method g adomian stochastic systems academic press new york london 1983 g adomian stochastic systems ii in press and earlier work of adomian referenced therein allows solution without linearization of problems normally requiring the self consistent field technique the latter technique can provide a sufficiently accurate approximation in weakly coupled plasma but the decomposition method of adomian provides a superior and physically preferable methodology for the general case

intuitive approach to the approximate analytical solution for the blasius problem for the blasius problem we propose an approximate analytical solution in the form of a logarithm of the hyperbolic cosine function which satisfies the given boundary conditions and some known properties of the exact solution furthermore adding some hyperbolic tangent functions to this solution we obtain much more accurate approximate solution with the relative error less than 0 16 over the whole region the superiority of the proposed solutions is shown by comparison with the existing approximate analytical solution

nonlinear differential equations with negative power nonlinearities abstract differential equations involving the term y m where m is a positive integer are solved by the decomposition method 1 3 441 452

toeplitz operators on flows abstract let r act continuously on a compact hausdorff space x giving rise to a flow on x let c x and let t x denote the toeplitz operator on h 2 r determined by the function x on r defined by x t x t in this paper we investigate the relation between the spectral properties of t x the dynamical properties of the flow and the value distribution theory of the analysis proceeds by imbedding t x in a type ii factor and computing the real valued index of the operator a la connes our sharpest invertibility result asserts that if the flow is strictly ergodic and if the asymptotic cycle determined by the flow is injective on h 1 x z then t x is invertible if and only if does not vanish on x and determines the zero element in h 1 x z this generalizes the classical result of gohberg and krein and its extension to toeplitz operators with almost periodic symbols due to coburn douglas schaeffer and singer when is analytic in the sense that x belongs to h r for all x we relate the ii index of the toeplitz operator determined by with the density of the zeros of x in the upper half plane much of our efforts to achieve this result are devoted to generalizing to arbitrary flows the value distribution theory of analytic almost periodic functions developed by bohr jessen and tornehave and others

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

a new class of two step p stable tfpl methods for the numerical solution of second order ivps with oscillating solutions abstract a new class of two step linear symmetric methods is introduced for the numerical solution of second order initial value problems that have highly oscillatory solutions in this class for the first time in the literature we calculate the coefficients of the method by a combination of a trigonometrically fitted tf method and a vanishing of the phase lag and some of its derivatives vsdpl method and we construct a new class of methods that has the properties of tf methods and vsdpl methods which we call the tfpl method this method is of algebraic order 8 and has an important p stability property the main structure of the method is multiderivative and the combined phases were applied to expand the stability interval and to achieve p stability the advantage of the method in comparison with similar methods in terms of efficiency accuracy and stability is shown by its implementation in some important problems the undamped duffing equation etc

on the classification of commutator ideals abstract we compute the k theory of the commutator ideals associated with almost periodic toeplitz operators and use the k 0 group to classify these ideals

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

on the k theory of some c algebras of toeplitz and singular integral operators abstract we study the c algebras gj x r are i x r of singular integral operators and toeplitz operators respectively associated with a strictly ergodic flow x r we show that the commutator ideals of these algebras cgj x r and cl x r are simple and are closely related to the transformation group c algebra c x r we calculate the k theory of gj x r l x r and their commutator ideals the main results of this calculation corollary 3 8 4 and theorem 4 1 1 assert that c x r is contained in cgj x r and if j denotes the inclusion map then j k 0 c x r k 0 cgj x r is an order isomorphism and there is a short exact sequence 0 k 1 c r i i k 1 c r j i k 1 cgj x r 0 where i is the canonical imbedding of c r into c x r we show also that up to a change of scale there is a unique trace on each of the commutator ideals the key ingredient of our analysis is theorem 3 1 1 which asserts a bijective correspondence between silov representations of the algebra of analytic functions on the flow and c representations of gj x r and l x r this simultaneously generalizes coburn s theorem on the uniqueness of the c algebra generated by an isometry and douglas s theorem on the uniqueness of the c algebra generated by an isometric representation of a dense subsemigroup of r

transient behaviour of semilinear stochastic systems with random parameters abstract the transient solution of a class of nonautonomous stochastic differential equations with given random initial conditions is studied the considered evolution equation is characterized by the presence of a deterministic linear term and of a random nonlinear term whose parameters can be modeled by random processes of rice noise type analytical approximated expressions for the first and second order moments and for the probability density of the solution process are derived and are applied to determine the statistical properties of a class of stochastic nonlinear oscillators

on composite nonlinearities and the decomposition method abstract accurate convergent computable solutions using the decomposition method have been demonstrated in and papers for wide classes of nonlinear and or stochastic differential partial differential or algebraic equations it is shown specifically in this paper that composite nonlinearities of the form nx n0 n1 n2 x appearing in such equations where the ni are nonlinear operators can also be handled with the adomian an polynomials

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

classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data abstract this paper is all about the development and analysis of an epidemiological model related to the disease of diarrhea that occurred in ghana during 2008 2015 using real statistical data three new fractional order mathematical models have been developed on the basis of having information about existing classical model the new models are formulated with caputo caputo fabrizio caputo and the atangana baleanu caputo fractional order approaches while taking care of the dimensional analysis during the process of fractionalization besides existence and uniqueness for the solutions of the fractional order models under each case are proved with the help of fixed point theory whereas positivity and boundedness of models solution are also investigated steady states disease free and endemic equilibria points of the model and sensitivity of the basic reproductive number r 0 are also explored while many of the model s parameters are fixed the transmission rate of the disease has been estimated and so is the case with orders of the fractional models using minimum distance approach it has been found that the diarrhea model under investigation estimates the real statistical data well enough when considered with the atangana baleanu caputo fractional order operator which has non local and non singular kernel thus this fractional order operator of atangana baleanu in the present research study for the diarrhea model outperforms those having index law power law and stretched exponential kernels

on a class of nonlinear stochastic dynamical systems analysis of the transient behaviour abstract a class of nonlinear dynamical systems with random parameters and initial conditions is considered the transient time evolution of the mean value of the dependent variable is analyzed both by means of the stochastic averaging method and by adomian s decomposition method a proof of the equivalence of the decomposition method to the time series approximation of the dynamical response is provided an application is made to the van der pol equation and finally a comparison between the two previously discussed methods is given

transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data abstract this study proposes a new epidemiological fractional order mathematical model called mseir maternally derived immunity susceptible exposed infectious and recovered using three most widely used operators namely the classical caputo the caputo fabrizio cf and the atangana baleanu caputo abc during the process of fractionalization of the classical model the dimensional consistency has been taken care of and the experimental data for 20 weeks available in literature for the chickenpox outbreak in 2014 among school children of the shenzhen city of china has been employed in order to validate the fractional order model the use of fixed point theory helps to prove the existence and the uniqueness for the solutions of each fractional order model under consideration it is also proved that the model possesses a positively invariant region for a positive hyper octant r 4 for the fractional models disease free and endemic equilibria are found while computing basic reproduction number r 0 which helps to determine local asymptotic stability for the steady states furthermore three numerical methods recently made available in literature are used to carry out the numerical simulations for each operator under consideration an interesting feature called the norm is obtained based upon the statistical data in which the parameter for the transmission rate of the epidemic and the fractional order parameters in the models are obtained via least squares technique of optimization revealing the highest rate of performance for the abc fractional operator

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

a new four step p stable obrechkoff method with vanished phase lag and some of its derivatives for the numerical solution of radial schrodinger equation abstract in this paper we present a new four step implicit p stable obrechkoff method of tenth algebraic order for solving one dimensional second order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable schrodinger equation numerical stability and phase properties of the new methods are analyzed numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

a first approach in solving initial value problems in odes by elliptic fitting methods exponentially fitted and trigonometrically fitted methods have a long successful history in the solution of initial value problems but other functions might be considered in adapted methods specifically this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions the example considered is the undamped duffing equation where the forcing term is of autonomous type affected by a perturbation parameter the new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions the proposed elliptic fitting procedure has been tested on a variety of problems showing its good performance

application of fixed point method to obtain semi analytical solution to blasius flow and its variation the classical blasius flow and its variation sakiadis flow are investigated by the fixed point method fpm and the semi analytical solutions are obtained different from early perturbation series solutions the present semi analytical solutions are valid in the whole region without using matching procedure moreover these solutions can correctly provide the asymptotic property of the governing equations the comparison of the present results with numerical results is carried out which reveals that fpm is a convenient and effective tool for nonlinear problems

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

the optimal homotopy asymptotic method for solving blasius equation starting from the reality that many known methods fail in the attempt to obtain analytic solutions of blasius type equations in this work a new procedure namely optimal homotopy asymptotic method oham is proposed to obtain an explicit analytical solution of the blasius problem comparison with howarth s numerical solution as well as the obtained residual reveals that the proposed method is highly accurate this proves the validity and great potential of the proposed procedure oham as a new kind of powerful analytical tool for nonlinear problems

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

solving nonlinear differential equations with decimal power nonlinearities abstract the decomposition method stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press is applied to the solution of nonlinear differential equations of the form ly ny x t where l is a linear differential operator and ny is a nonlinear term of the form y with a decimal number