a series solution of the cauchy problem for the generalized d dimensional schrodinger equation with a power law nonlinearity in this paper the cauchy problem for generalized d dimensional schrodinger equation with a power law nonlinearity is studied three methods homotopy analysis method ham homotopy perturbation method hpm and adomian decomposition method adm are applied to obtain series pattern solutions of the mentioned cauchy problem the recurrent relations for solving the mentioned cauchy problem is introduced for some cases of given initial conditions we obtain the closed form of the exact solutions

on nonautonomous linear systems of differential and difference equations with r symmetric coefficient matrices abstract let c n n i denote the set of continuous n n matrices on an interval i we say that r c n n i is a nontrivial k involution if r p l 0 k 1 l i d l p 1 where e 2 i k d 0 d 1 d k 1 n and p p l 0 k 1 u l with u l c d l d l i we say that a c n n i is r symmetric if r t a t r 1 t a t t i and we show that if a is r symmetric then solving x a t x or x a t x f t reduces to solving k independent d l d l systems 0 l k 1 we consider the asymptotic behavior of the solutions in the case where i t 0 finally we sketch analogous results for linear systems of difference equations

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

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

a new analytic algorithm of lane emden type equations an reliable ease to use analytic algorithm is provided for lane emden type equation which models many phenomena in mathematical physics and astrophysics this algorithm logically contains the well known adomian decomposition method different from all other analytic techniques this algorithm itself provides us with a convenient way to adjust convergence regions even without pade technique some applications are given to show its validity

new application of adomian decomposition method on fokker planck equation adomian decomposition method has been used for finding the explicit and numerical solutions of the reduced fokker planck equation in a force field the obtained results are compared with those found by other theories such as lie group theory the behaviour of the approximate distribution functions are shown graphically the results of lie group method confirm the correctness of those obtained by adomian decomposition method

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

the inverse eigenproblem with a submatrix constraint and the associated approximation problem for r s symmetric matrices abstract let r r n n and s r n n be nontrivial involutions i e r r 1 i and s s 1 i a matrix a r n n is called r s symmetric if r a s a this paper presents a r s symmetric matrix solution to the inverse eigenproblem with a leading principal submatrix constraint the solvability condition of the constrained inverse eigenproblem is also derived the existence the uniqueness and the expression of the r s symmetric matrix solution to the best approximation problem of the constrained inverse eigenproblem are achieved respectively an algorithm is presented to compute the r s symmetric matrix solution to the best approximation problem two numerical examples are given to illustrate the effectiveness of our results

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

the left and right inverse eigenvalue problems of generalized reflexive and anti reflexive matrices let nxn complex matrices r and s be nontrivial generalized reflection matrices i e r r r 1 i n s s s 1 i n a complex matrix a with order n is said to be a generalized reflexive or anti reflexive matrix if ras a or ras a in this paper the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti reflexive matrices are derived and the general solutions are also given in addition the associated approximation solutions in the solution sets of the above problems are provided the results in present paper extend some recent conclusions

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

a decomposition method for solving the nonlinear klein gordon equation in this paper adomian s decomposition scheme is presented as an alternate method for solving the nonlinear klein gordon equation the method is demonstrated by several examples comparing the scheme with existing collocation finite difference and finite element techniques shows that the present approach is highly accurate and converges rapidly

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

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

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

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

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

a discrete adomian decomposition method for discrete nonlinear schrodinger equations we present a new discrete adomian decomposition method to approximate the theoretical solution of discrete nonlinear schrodinger equations the method is examined for plane waves and for single soliton waves in case of continuous semi discrete and fully discrete schrodinger equations several illustrative examples and mathematica program codes are presented

algorithms for solving the inverse problem associated with k a k a s 1 in previous papers the authors introduced and characterized a class of matrices called k s 1 potent also they established a method to construct these matrices the purpose of this paper is to solve the associated inverse problem several algorithms are developed in order to find all involutory matrices k satisfying k a s 1 k a for a given matrix a c n n and a given natural number s the cases s 0 and s 1 are separately studied since they produce different situations in addition some examples are presented showing the numerical performance of the methods

direct search for exact solutions to the nonlinear schrodinger equation a five dimensional symmetry algebra consisting of lie point symmetries is firstly computed for the nonlinear schrodinger equation which together with a reflection invariance generates two five parameter solution groups three ansatze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear schrodinger equation various examples of exact solutions with constant trigonometric function type exponential function type and rational function amplitude are given upon careful analysis a bifurcation phenomenon in the nonlinear schrodinger equation is clearly exhibited during the solution process

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

tridiagonal commuting matrices and fractionalizations of dct and dst matrices of types i iv v and viii in this paper we first establish new relationships in matrix forms among discrete fourier transform dft generalized dft gdft and various types of discrete cosine transform dct and discrete sine transform dst matrices two new independent tridiagonal commuting matrices for each of dct and dst matrices of types i iv v and viii are then derived from the existing commuting matrices of dft and gdft with these new commuting matrices the orthonormal sets of hermite like eigenvectors for dct and dst matrices can be determined and the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst are defined the relationships among the discrete fractional fourier transform dfrft fractional gdft and various types of dfrct and dfrst are developed to reduce computations for dfrft and fractional gdft

on the computation of inverses and determinants of a kind of special matrices abstract in this paper the inverse and determinant of a special kind of centrosymmetric matrices are investigated based on the partition property of a matrix with centrosymmetric structure and algorithms for the inverse and determinant proposed in chen and yu 2011 a computation algorithm for the inverse and determinant of a centrosymmetric matrix is finally developed

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

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

a system of real quaternion matrix equations with applications abstract let h be the real quaternion algebra and h n m denote the set of all n m matrices over h let p h n n and q h m m be involutions i e p 2 i q 2 i a matrix a h n m is said to be p q symmetric if a paq this paper studies the system of linear real quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 a 2 x 2 c 3 x 2 b 2 c 4 a 3 x 1 b 3 a 4 x 2 b 4 c c we present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied as applications we discuss the necessary and sufficient conditions for the system a a x c a xb b c b a c xb c c c to have a p q symmetric solution we also show an expression of the p q symmetric solution to the system when the solvability conditions are met moreover we provide an algorithm and a numerical example to illustrate our results the findings of this paper extend some known results in the literature

some nilpotent tridiagonal matrices with a special sign pattern abstract we prove that for each n 2 there is a nilpotent n n tridiagonal matrix satisfying a the super diagonal is positive b the sub diagonal is negative c the diagonal is zero except that the 1 1 position is negative and the n n position is positive

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

an analytical method for solving linear fredholm fuzzy integral equations of the second kind in this paper we use the parametric form of a fuzzy number and convert a linear fuzzy fredholm integral equation to two linear systems of integral equations of the second kind in the crisp case for fuzzy fredholm integral equations with kernels the sign of which is difficult to determine a new parametric form of the fuzzy fredholm integral equation is introduced we use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solutions of the linear fuzzy fredholm integral equation of the second kind the proposed method is illustrated by solving some examples using the ham it is possible to find the exact solution or the approximate solution of the problem in the form of a series

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

on green s function in higher order stochastic differential equations abstract in adomian s solution of linear or nonlinear deterministic or stochastic differential equations the decomposition of the linear part into l r can be made so as to simplify green s function and consequently the resulting computations

inverse eigenproblem for r symmetric matrices and their approximation let email protected c n x n be a nontrivial involution i e r r 1 i n we say that email protected c n x n is r symmetric if rgr g the set of all nxnr symmetric matrices is denoted by gsc n x n in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors x i i 1 m in c n and a set of complex numbers l i i 1 m find a matrix email protected gsc n x n such that x i i 1 m and l i i 1 m are respectively the eigenvalues and eigenvectors of a we then consider the following approximation problem given an nxn matrix email protected find email protected email protected s e such that email protected email protected email protected min a s email protected email protected email protected where s e is the solution set of iep and email protected email protected is the frobenius norm we provide an explicit formula for the best approximation solution email protected by means of the canonical correlation decomposition

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

highly symmetric generalized circulant permutation matrices abstract in this paper we investigate generalized circulant permutation matrices of composite order we give a complete characterization of the order and the structure of symmetric generalized k circulant permutation matrices in terms of circulant and retrocirculant block 0 1 matrices in which each block contains exactly one or two entries 1 in particular we prove that a generalized k circulant matrix a of composite order n km is symmetric if and only if either k m 1 or k 0 or k 1 mod m and we obtain three basic symmetric generalized k circulant permutation matrices from which all others are obtained via permutations of the blocks or by direct sums furthermore we extend the characterization of these matrices to centrosymmetric matrices

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

r s conjugate solution to a pair of linear matrix equations abstract let r and s be m m and n n nontrivial real symmetric involutions an m n complex matrix a is termed r s conjugate if a ras where a denotes the conjugate of a in this paper necessary and sufficient conditions are established for the existence of the r s conjugate solution to the system of matrix equations ax c and xb d the expression is also presented for such solution to this system in addition the explicit expression of this solution to the corresponding optimal approximation problem is obtained furthermore the least squares r s conjugate solution with least norm to this system mentioned above is considered the representation of such solution is also derived finally an algorithm and numerical examples are given

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

adomian decomposition method for solving fractional nonlinear differential equations in this article we have discussed a new application of adomian decomposition method on time fractional nonlinear fractional differential equations three models with fractional time derivative of order a 0 a 1 are considered and solved by means of adomian decomposition method the behaviour of adomian solutions and the effects of different values of a are investigated numerical examples are tested to illustrate the pertinent feature of the proposed algorithm

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

inverses determinants eigenvalues and eigenvectors of real symmetric toeplitz matrices with linearly increasing entries abstract we explicitly determine the skew symmetric eigenvectors and corresponding eigenvalues of the real symmetric toeplitz matrices t t a b n a b j k 1 j k n of order n 3 where a b r b 0 the matrix t is singular if and only if c a b n 1 2 in this case we also explicitly determine the symmetric eigenvectors and corresponding eigenvalues of t if t is regular we explicitly compute the inverse t 1 the determinant det t and the symmetric eigenvectors and corresponding eigenvalues of t are described in terms of the roots of the real self inversive polynomial p n z z n 1 z n z 1 z 1 if n is even and p n z z n 1 z n z 1 if n is odd 1 2 2 c n 1

a reliable technique for solving the weakly singular second kind volterra type integral equations abstract in this paper the adomian decomposition method and the phenomenon of the self canceling noise terms are used for solving the weakly singular volterra type linear and nonlinear integral equations the solution obtained is in the form of a convergent power series with elegantly computable terms comparing this scheme with many collocation type methods that use the nonpolynomial basis shows that the present approach is effective and powerful many test modeling problems from mathematical physics linear and nonlinear are discussed to illustrate the effectiveness and the performance of the decomposition method

a new approach to the cubic schrodinger equation an application of the decomposition technique abstract a decomposition method is implemented for solving the cubic schrodinger equation

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

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

a new modification of adomian decomposition method for solving a kind of evolution equation in this paper we propose a new modification of adomian decomposition method for solving a kind of evolution equation the modification will improve the convergence of the series solution the validity of the modified technique is verified through illustrative examples

onf subnormal subgroups andf residuals of finite soluble groups all groups that we consider are finite and soluble recall that a formation is a class of groups which is closed under homomorphic images and subdirect products hence if f is a formation and g is a group which is a direct product of the subgroups a and b then g is in f if and only if a and b lie in f more generally doerk and w x hawkes 4 iv 1 18 proved that if g is a group such that g s a b then g s a b where g is the f residual of g that is the smallest normal subgroup of g with quotient in f the main purpose of this paper is the development of this result by means of the concept of f subnormal subgroup suppose that f is a saturated formation a maximal subgroup m of a group g is called f normal in g if grcore m g f a subgroup h of g g is called f subnormal in g if either h s g or there exists a chain h s h f h f f h s g such that h is an f normal maximal 0 1 n i subgroup of h for 0 f i n it is clear that if f s n the saturated iq1 formation of all nilpotent groups the f subnormal subgroups of g are exactly the subnormal subgroups of g let f be a subgroup closed saturated formation containing n it is rather easy to see that if f is closed under the product of normal subgroups then g s a b f for every pair of subnormal subgroups a and b such that g s ab this result does not remain true if a and b are

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

multilevel matrices with involutory symmetries and skew symmetries let n n1 n2 nk where k 1 and n1 nk are integers 1 for 1 i k let pi j 1i 1nj and qi j i 1knj and suppose that ui cni ni is a nontrivial involution i e ui ui 1 ini let ri ipi ui iqi 1 i k and denote r r1 rk if 0 1 l 2k 1 let i 1kli 2i 1 be its binary expansion we say that a cn n is r symmetric if riari 1 li a 1 i k thus we are considering matrices with k levels of block structure and an involutory symmetry or skew symmetry at each level we characterize the class of all r symmetric matrices and study their properties the theory divides into two parts corresponding to 0 and 0 problems involving an r 0 symmetric matrix split into the corresponding problems for 2k 1 matrices with orders summing to n while problems involving an r symmetric matrix with 0 split into the corresponding problems for 2k 1 1 matrices with orders summing to n the latter is also true of a b c where b is r 0 symmetric and c is r symmetric with 0

finite iterative hermitian r conjugate solutions of the generalized coupled sylvester conjugate matrix equations abstract in this paper an iterative algorithm for solving a generalized coupled sylvester conjugate matrix equations over hermitian r conjugate matrices given by a 1 v b 1 c 1 w d 1 e 1 v f 1 g 1 and a 2 v b 2 c 2 w d 2 e 2 v f 2 g 2 is presented when these two matrix equations are consistent the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round off error for any initial arbitrary hermitian r conjugate solution matrices v 1 w 1 some lemmas and theorems are stated and proved where the iterative solutions are obtained a numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results

asymptotic spectral properties of totally symmetric multilevel toeplitz matrices abstract let n n 1 n k be a multiindex and n j 1 k n j we say that n if n i 1 i k if r r 1 r k and s s 1 s k let r s r 1 s 1 s 1 s k we say that a multilevel toeplitz matrix of the form t n t r s r s 1 is totally symmetric let q k be the k fold cartesian product of q with itself and let t r r be the fourier coefficients of a function f f 1 k in l 2 q k that is even in each variable 1 k so that t n is totally symmetric for every n we associate the multiindex n with 2 k multiindices m n p 0 p 2 k 1 such that lim n m n p n 2 k 0 p 2 k 1 and p 0 2 k 1 m n p n and show that the singular values of t n separate naturally into 2 k sets s n 0 s n 2 k 1 with cardinalities m n 0 m n 2 k 1 such that the singular values in each set s n p are associated with singular vectors exhibiting a particular type of symmetry our main result is that the singular values in s n p and the singular values of t m n p are absolutely equally distributed with respect to the class g of functions bounded and uniformly continuous on r as n 0 p 2 k 1 if f is real valued then an analogous result holds for the eigenvalues and eigenvectors of t n

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

nonlinear stochastic differential delay equations abstract the methods used by adomian and his co workers to solve linear and nonlinear stochastic differential equations will be demonstrated to be applicable to differential equations deterministic or stochastic involving delays constant time dependent or random

coupled differential equations and coupled boundary conditions abstract it is demonstrated that the high accuracy for approximations requiring only a few terms which is typical of the decomposition method for nonlinear stochastic operator equations or special cases linear or deterministic holds for coupled equations and coupled boundary conditions as well

a numerical solution and an exact explicit solution of the nls equation we consider traveling wave solutions of the nonlinear schrodinger nls for short equation in this paper by considering the decomposition scheme we first obtain the exact solutions of the nls equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique the numerical solutions are compared with the known analytical solutions its remarkable accuracy is finally demonstrated in the study of some initial values of the nls equation

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

dft commuting matrix with arbitrary or infinite order second derivative approximation recently candan introduced higher order dft commuting matrices whose eigenvectors are better approximations to the continuous hermite gaussian functions hgfs however the highest order 2k of the o h2k ntimesn dft commuting matrices proposed by candan is restricted by 2k 1 les n in this paper we remove this order upper bound restriction by developing two methods to construct arbitrary order dft commuting matrices computer experimental results show that the hermite gaussian like hgl eigenvectors of the new proposed dft commuting matrices outperform those of candan in addition the hgl eigenvectors of the infinite order dft commuting matrix are shown to be the same as those of the n2 dft commuting matrix recently discovered in the literature

a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices in this paper a class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions we present the general solution of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices in real number field in addition in corresponding solution set of the constrained inverse eigenproblem for skew symmetric and centrosymmetric matrices the explicit expression of the nearest matrix to a given matrix in the frobenius norm is obtained

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

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

hermitian hermitian r symmetric and hermitian r skew symmetric procrustes problems let r cn n be a nontrivial unitary involution i e r r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i hermitian matrices ii hermitian r symmetric matrices iii hermitian r skew symmetric matrices given z w cn m we characterize the matrices a in s that minimize az w frobenius norm and given an arbitrary e cn n we find the unique matrix among the minimizers of az w in s that minimizes a e we also obtain necessary and sufficient conditions for existence of a s such that az w and assuming that the conditions are satisfied characterize the set of all such a