an analytic solution to the thomas fermi equation a perturbative procedure due to bender et al here referred to as the bmps procedure j math phys 30 1447 1989 and useful in solving difficult nonlinear problems has been used here to solve the thomas fermi t f equation the present work attempts to balance the ease of the ensuing analysis with the use of an analytic zero order function that already contains a good deal of the nonlinearity of the t f equation the initial slope of the t f potential is computed with 0 35 error in a second order application of the theory

piecewise homotopy methods for nonlinear ordinary differential equations abstract piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations these methods are based on the introduction of an artificial or book keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere three piecewise adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size are presented and applied to eight nonlinear ordinary differential equations it is shown that piecewise adaptive homotopy perturbation methods predict essentially the same solutions as matlab s variable step variable order solvers and variable order transition matrix techniques provided that five term approximations of the decomposition method are applied to both the displacement and the velocity it is also shown that piecewise homotopy perturbation techniques that use three term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second order accurate time linearization technique when the same step is employed in both schemes

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

a boundary condition with memory for kirchhoff plates equations in this paper we study the stability of solutions for kirchhoff plates equations with a memory condition working at the boundary we show that such dissipation is strong enough to produce exponential decay to the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially and with the same rate

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

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

new harnack inequalities and existence theorems for radially symmetric solutions of elliptic pdes with sign changing or vanishing green s function abstract we consider the perturbed hammerstein integral equation y t 1 t h 1 1 y 2 t h 2 2 y 0 1 g t s f s y s d s where 1 and 2 are linear functionals realized as stieltjes integrals with associated signed stieltjes measures we demonstrate that by utilizing a nonstandard order cone together with an associated nonstandard open set existence of a positive solution to the integral equation can be guaranteed under relatively mild hypotheses on the various constituent functions and functionals in the case in which the green s function t s g t s is allowed to both vanish and change sign as an example we apply our results to radially symmetric elliptic pdes of the form u h x g u x x r 1 r 2 r u x x b r 2 0 u r 1 x u r x x b 1 h u where 0 r 1 r r 2 is a linear functional and h is a continuous function as part of this analysis we demonstrate a new methodology for constructing a suitable harnack like inequality associated to the boundary value problem

a kantorovich type analysis for a fast iterative method for solving nonlinear equations abstract we revisit a fast iterative method studied by us in i k argyros on a two point newton like method of convergent order two int j comput math 88 2 2005 219 234 to approximate solutions of nonlinear operator equations the method uses only divided differences of order one and two function evaluations per step this time we use a simpler kantorovich type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case moreover we show that in some cases our method compares favorably and can be used in cases where other methods using similar information cannot s amat s busquier v f candela a class of quasi newton generalized steffensen s methods on banach spaces j comput appl math 149 2 2002 397 406 d chen on the convergence of a class of generalized steffensen s iterative procedures and error analysis int j comput math 31 1989 195 203 numerical examples are provided to justify the theoretical results

exact controllability of the euler bernoulli equation with controls in the dirichlet and neumann boundary conditions a nonconservative case this paper considers the euler bernoulli problem 1 1a d with boundary controls g 1 g 2 in the dirichiet and neumann boundary conditions respectively several exact controllability results are shown including the following problem 1 1a d is exactly controllable in an arbitrarily short time t 0 in the space of maximal regularity h 1 omega times v v as in 1 4 i with boundary controls g 1 in l 2 sigma g 2 equiv 0 under some geometrical conditions on omega ii with boundary controls g 1 in l 2 sigma and g 2 in l 2 0 t h 1 gamma without geometrical conditions on omega a direct approach is given based on an operator model for problem 1 la d and on multiplier techniques an additional difficulty of the particular boundary conditions is due to the fact that in the natural norms for the solution arising in the application of multiplier techniques the corresponding homogeneous problem is not conservative this difficulty is overco

global exact controllability of a class of quasilinear hyperbolic systems in this paper we consider a distributed parameter boundary control system described by a reducible quasilinear hyperbolic system formula posed on the domain 0 x 1 with the nonlinear boundary conditions formula in the case that the system islinear degenerate we show that it isgloballyexactly boundary controllable

stabilization of wave and plate like equations with nonlinear dissipation on the boundary abstract in this paper we consider second order in time equations of petrovsky type wave and plate like with nonlinear dissipative boundary conditions under certain geometric conditions imposed on the domain the results on asymptotic behavior of the solutions are established

on a nonlinear hammerstein integral equation with a parameter abstract this paper following theories for asymptotically linear operators the schaefer fixed point theorem decomposition of operators and critical point theory is mainly concerned with the existence and multiplicity of solutions to a nonlinear hammerstein integral equation with a parameter the results show that when the nonlinearity satisfies certain conditions different parametric intervals lead to different existence results however in some cases only the sign of the parameter makes a contribution to the existence of solutions for the problem our results can be applied to some well known boundary value problems and some examples are given

a perturbative approach to the thomas fermi equation in terms of the density a fast and absolute converging sequence for the thomas fermi tf initial slope was obtained by applying a perturbative approach c m bender et al j math phys 30 1447 1989 to the tf equation written in terms of the electronic density by this procedure the truncated series does not alternate in contrast to a previous work b j laurenzi j math phys 31 2535 1990 based in the traditional tf equation

on the kirchhoff plates equations with thermal effects and memory boundary conditions in this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the kirchhoff plates equations with thermal effect and memory conditions working at the boundary we show that the dissipation produced by the memory effect not depend on the present values of temperature gradient that is we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially with the same rate

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

inverse observability estimates for second order hyperbolic equations with variable coefficients abstract we consider a general second order hyperbolic equation defined on an open bounded domain r n with variable coefficients in both the elliptic principal part and in the first order terms as well at first no boundary conditions b c are imposed our main result theorem 3 5 is a reconstruction or inverse estimate for solutions w under checkable conditions on the coefficients of the principal part the h1 l2 energy at time t t or at time t 0 is dominated by the l2 norms of the boundary traces w a and wt modulo an interior lower order term once homogeneous b c are imposed our results yield under a uniqueness theorem needed to absorb the lower order term continuous observability estimates for both the dirichlet and neumann case with an explicit sharp observability time hence by duality exact controllability results moreover no artificial geometrical conditions are imposed on the controlled part of the boundary in the neumann case in contrast with existing literature the first step of our method employs a riemann geometry approach to reduce the original variable coefficient principal part problem in r n to a problem on an appropriate riemann manifold determined by the coefficients of the principal part where the principal part is the laplacian in our second step we employ explicit carleman estimates at the differential level to take care of the variable first order energy level terms in our third step we employ micro local analysis yielding a sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the neumann case

exponential decay rates for the solutions of euler bernoulli equations with boundary dissipation occurring in the moments only abstract we shall derive uniform decay rates for the total energy of the solutions to the euler bernoulli equations wherein energy dissipation occurs on the boundary the main feature which distinguishes this paper from other related works is the fact that the boundary feedback is acting through the moments only the key technical element responsible for the proof of the main result is a new regularity estimate for the solutions to the nonhomogeneous schrodinger equation

determination of periodic solution for a u1 3 force by he s modified lindstedt poincare method abstract this paper applies he s modified lindstedt poincare method to determine the periodic solutions of oscillators in a u 1 3 force the result obtained and comparison with analytical solution provides confirmation for the validity of he s modified lindstedt poincare method

picard s iterative method for nonlinear advection reaction diffusion equations picard s iterative method for the solution of nonlinear advection reaction diffusion equations is formulated and its convergence proved the method is based on the introduction of a complete metric space and makes uses of a contractive mapping and banach s fixed point theory from picard s iterative method the variational iteration method is derived without making any use at all of lagrange multipliers and constrained variations some examples that illustrate the advantages and shortcomings of the iterative procedure presented here are shown

a modification of newton s method for nondifferentiable equations in this paper we study the problem of approximating a solution of a nonlinear equation where the operator involved is nondifferentiable having the newton method as origin we construct a uniparametric family of iterative processes to approximate a solution of the equation to finish we consider several test problems

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

riccati equations for hyperbolic partial differential equations with l 2 0 t l 2 γ dirichlet boundary terms this paper studies the quadratic optimal control problem for second order linear hyperbolic partial differential equations defined on a bounded domain omega subset r n with boundary gamma both the finite interval case 0 t t infty and the infinite interval case t infty regulator problem are considered the distinguishing feature of the paper which differentiates it from previous scarce literature on the subject is that the controls are only l 2 0 t l 2 gamma functions which act in the dirichlet b c and that the corresponding solutions are penalized in the l 2 0 t l 2 omega norm smoother controls particularly in space were taken in the few previous works on this subject the well posedness of this formulation stems from recent results by the authors about regularity of second order hyperbolic mixed problems l t 1 l t 3 under minimal assumptions the optimal control is synthesized in a pointwise feedback form through an operator which is s

decay of solutions of wave equations in a bounded region with boundary dissipation abstract an energy decay rate is obtained for solutions of wave type equations in a bounded region in rn whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface

limit cycles of non smooth oscillators abstract an artificial parameter linstedt poincare method based on the introduction of a linear stiffness term a change of independent variable and the expansion of both the solution and the frequency of oscillation in power series of the artificial parameter and the theory of generalized functions are used to determine the frequency of limit cycles of non smooth oscillators the method is applied to three non smooth oscillators and it is shown that the first order approximation coincides with those obtained by means of harmonic balance and parameter expansion techniques whereas higher order ones are given in terms of convergent series

an artificial parameter decomposition method for nonlinear oscillators applications to oscillators with odd nonlinearities abstract a method for obtaining series solutions of nonlinear second order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well known adomian s decomposition technique the method is formulated in both integral and differential forms for the determination of the limit cycle of oscillators with odd nonlinearities two differential forms and one integral form of the artificial parameter method are presented these versions are based on introducing a linear stiffness term with an unknown frequency and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator the three formulations provide identical results and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified linstedt poincare methods however the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation it is also shown that two and three level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined

the existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term abstract in this paper we consider a plate equation with a logarithmic nonlinearity in the presence of a frictional damping we prove for a suitably chosen initial data a global existence as well as a decay result

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

a non iterative derivative free method for nonlinear ordinary differential equations most methods for the determination of approximate solutions of nonlinear ordinary differential equations require that the nonlinearities be sufficiently differentiable with respect to the dependent variable and its derivatives in this paper a non iterative method for obtaining approximate solutions of nonlinear ordinary differential equations which does not require the derivatives of the nonlinearities is presented and its convergence is proved the method is also generalized for the determination of the periodic solutions of autonomous nonlinear ordinary differential equations by introducing as dependent variable the unknown frequency of oscillation the method does not require the presence of small parameters and can be used in a piecewise fashion

uniform exponential energy decay of wave equations in a bounded region with l2 0 l2 γ feedback control in the dirichlet boundary conditions etant donne un ouvert borne r n n 2 on considere l equation d onde w tt w dans o avec des conditions initiales w o x w o x l 2 w 1 o x w 1 x h 1 et des conditions aux limites non homogenes de type dirichlet w t u t sur o etant la frontiere de

iterative and non iterative methods for non linear volterra integro differential equations iterative and non iterative methods for the solution of nonlinear volterra integro differential equations are presented and their local convergence is proved the iterative methods provide a sequence solution and make use of fixed point theory whereas the non iterative ones result in series solutions and also make use of fixed point principles by means of integration by parts and use of certain integral identities it is shown that the initial conditions that appear in the iterative methods presented here can be eliminated and the resulting iterative technique is identical to the variational iteration method which is derived here without making any use at all of lagrange multipliers and constrained variations it is also shown that the formulation presented here can be applied to initial value problems in ordinary differential volterra s integral and integro differential pantograph and nonlinear and linear algebraic equations a technique for improving accelerating the convergence of the iterative methods presented here is also presented and results in a lipschitz constant that may be varied as the iteration progresses it is shown that this acceleration technique is related to preconditioning methods for the solution of linear algebraic equations it is also argued that the non iterative methods presented in this paper may not competitive with iterative ones because of possible cancellation errors if implemented numerically an analytical continuation procedure based on dividing the interval of integration into disjoint subintervals is also presented and its limitations are discussed

boundary controllers and observers for the linearized schrodinger equation we consider a problem of stabilization of the linearized schrodinger equation using boundary actuation and measurements we propose two different control designs first a simple proportional collocated boundary controller is shown to exponentially stabilize the system how ever the decay rate of the closed loop system cannot be prescribed the second full state feedback boundary control design achieves an arbitrary decay rate we formally view the schrodinger equa tion as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction advection diffusion equations the resulting controller is then sup plied with the backstepping observer to obtain an output feedback compensator the designs are illustrated with simulations

dynamical boundary control of two dimensional wave equations vibrating membrane on general domain dynamical boundary control of a vibrating membrane with edge mass and of general shape is mathematically formulated as an abstract evolutionary system the approximate controllability and the strong stabilization by linear boundary damping feedback are established by an approach based on lasalle s invariance principle the mathematical model and the reduced system can be used to investigate some optimization problems

the secant method and divided differences holder continuous we apply the secant method to solve non linear operator equations in banach spaces a semilocal convergence result is obtained where the first order divided difference of the non linear operator is holder continuous for that we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution and give an explicit expression for the a priori error bounds moreover we apply our results to the numerical solution of a non linear boundary value problem of second order

on the picard lindelof method for nonlinear second order differential equations we consider a second order nonlinear ordinary differential equation which satisfies a lipschitz continuity condition and show that the method of variation of parameters allows us to write the solution as a volterra integral equation that includes the initial values of the solution and its first order derivative we also show that this integral formulation can be written as an iterative method which converges uniformly to the unique solution of the problem by making use of an integral identity the initial conditions can be eliminated from the integral equation and a new integral formulation that depends on the second order derivative of the solution is obtained it is shown that this integral formulation is exactly the same as that of the variational iteration method but does not require a variational principle neither does it require constrained variations for nonlinear terms we also show that the two integral formulations can be written in differential form and correspond to a two level iterative method by writing the second order ordinary differential equation as a two equation system of first order ordinary differential equations three different iterative procedures are developed but only one of them coincides with that derived from the two integral equations for the solution

a new perturbative approach to nonlinear problems a recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the lagrangian such as 4 by 2 1 and then treating as a small parameter it is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the lane emden thomas fermi blasius and duffing equations

third order iterative methods under kantorovich conditions we study a class of third order iterative methods for nonlinear equations on banach spaces a characterization of the convergence under kantorovich type conditions and optimal estimates of the error are found though in general these methods are not very extended due to their computational costs we will show some examples in which they are competitive and even cheaper than other simpler methods we center our analysis in both analytic and computational aspects

existence of three periodic positive solutions for a class of integral equations with parameters abstract in this paper we establish the existence of three periodic positive solutions for a class of abstract integral equations by leggett williams fixed point theorem using the existence results for abstract integral equations the population models are also considered

existence of positive negative and sign changing periodic solutions for a class of integral equations in this paper the simultaneous existence of positive negative and sign changing periodic solutions for a class of integral equations of the form f x email protected x x w wk x y f y f email protected y dy email protected email protected is considered where w is a closed subset of r n with a periodic structure our main result is different from most existing results since they provide three constant sign periodic solutions only

a newton like method and its application abstract in this paper we prove an existence and uniqueness theorem for solving the operator equation f x g x 0 where f is a gateaux differentiable continuous operator while the operator g satisfies a lipschitz condition on an open convex subset of a banach space as corollaries a theorem of tapia on a weak newton s method and the classical convergence theorem for modified newton iterates are deduced an existence theorem for a generalized euler lagrange equation in the setting of sobolev space is obtained as a consequence of the main theorem we also obtain a class of gateaux differentiable operators which are nowhere frechet differentiable illustrative examples are also provided

on linstedt poincare techniques for the quintic duffing equation abstract an artificial parameter method for the quintic duffing equation is presented the method is based on the introduction of a linear stiffness term and a new dependent variable both of which are proportional to the unknown frequency of oscillation the introduction of an artificial parameter and the expansion of both the solution and the unknown frequency of oscillation in series of the artificial parameter and results in linear ordinary differential equations at each order in the parameter by imposing the nonsecularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation the method does not require that a small parameter be present in the governing equation and its results are compared with those of linstedt poincare modified linstedt poincare harmonic balance and galerkin techniques it is shown that these four techniques predict the same first order approximation to the frequency of oscillation as the artificial parameter method presented in this paper and the latter introduces higher order corrections at second order whereas similar corrections are introduced by the linstedt poincare and modified linstedt poincare methods at orders equal to and higher than three

nonlinear boundary feedback stabilization for a semilinear kirchhoff plate with dissipation acting only via moments limiting behavior abstract a semilinear kirchhoff plate with a nonlinear dissipation acting via moments only is considered it is shown that the plate is uniformly stabilizable with uniform energy decay rates with respect to the parameter 0 representing rotational forces assumed small moreover it is shown that the solutions of the kirchhoff plate converge to solutions of the semilinear euler bernoulli plate when 0 which is also uniformly stable in finite energy norm

dirichlet boundary stabilization of the wave equation with damping feedback of finite range abstract a closed loop system consisting of the wave equation with a feedback acting in the dirichlet boundary condition in the form of a nonlocal one dimensional range operator defined on the velocity vector damping is considered beside the well posedness question generation of a feedback c 0 semigroup the boundary feedback stabilization problem is studied while asymptotic decay to zero in the uniform operator norm can never occur for the considered class of closed loop systems however checkable sufficient conditions on the vectors in the boundary conditions are provided that guarantee asymptotic decay to zero in the strong norm of appropriate sobolev spaces these conditions include but are not limited to the standard case of dissipativity of the feedback system the starting point of this approach is a functional analytic model for the study of nonsmooth boundary input hyperbolic equations recently developed i lasiecka and r triggiani appl math optim 7 1 1981 35 93

a new perturbative approach to nonlinear partial differential equations this paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity for the case of the burgers equation ut uux uxx the general nonlinear equation ut u ux uxx is considered and expanded in powers of the coefficients of the series to sixth order in powers of is determined and pade summation is used to evaluate the perturbation series for large values of the numerical results are accurate and the method is very general it applies to other well studied partial differential equations such as the korteweg de vries equation ut uux uxxx