on the existence and multiplicity of positive periodic solutions for first order vector differential equation abstract in this paper we consider the existence and multiplicity of positive periodic solutions for first order vector differential equation x t f t x t 0 a e t 0 under the periodic boundary value condition x 0 x here is a positive constant and f 0 r n r n is a caratheodory function some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones

existence of solutions to an initial value problem for the differential equation f t x x 0 using barrier strips abstract under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x x 0 a where the function f t x p may be singular at x a

a multidisciplinary optimization approach for vibration reduction in helicopter rotor blades abstract the paper addresses the integration of blade dynamics aerodynamics structures and aeroelasticity in the design of helicopter rotors using a formal optimization technique the interaction of the disciplines is studied inside a closed loop optimization process the goal is to reduce vibratory shear forces at the blade root with constraints imposed on dynamic structural and aeroelastic design requirements both structural and aerodynamic design variables are used multiobjective formulation procedures are needed since more than one design objective is used a nonlinear programming technique and an approximate analysis procedure are used for optimization substantial reductions are obtained in the vibratory root forces and moments while satisfying the remaining design criteria the results of the optimization procedure using two multiobjective formulation procedures are compared with a baseline or reference design

nonlinear boundary value problems for first order impulsive functional differential equations this paper is concerned with solutions of nonlinear boundary value problems for a class of first order functional differential equations we introduce new concept of lower and upper solutions and present that the method of lower and upper solutions coupled with monotone iterative technique is still valid obviously we extend previous results

periodic boundary value problems for a class of functional differential equations in this paper we show that the method of upper and lower solutions coupled with the monotone iterative technique is valid to obtain constructive proofs of existence of solutions for nonlinear periodic boundary value problems of functional differential equations without assuming properties of monotonicity in the nonlinear part

conjugate direction particle swarm optimization solving systems of nonlinear equations solving systems of nonlinear equations is a difficult problem in numerical computation for most numerical methods such as the newton s method for solving systems of nonlinear equations their convergence and performance characteristics can be highly sensitive to the initial guess of the solution supplied to the methods however it is difficult to select a good initial guess for most systems of nonlinear equations aiming to solve these problems conjugate direction particle swarm optimization cdpso was put forward which introduced conjugate direction method into particle swarm optimization pso in order to improve pso and enable pso to effectively optimize high dimensional optimization problem in one optimization problem when after some iterations pso got trapped in local minima with local optimal solution x conjugate direction method was applied with x as a initial guess to optimize the problem to help pso overcome local minima by changing high dimension function optimization problem into low dimensional function optimization problem because pso is efficient in solving the low dimension function optimization problem pso can efficiently optimize high dimensional function optimization problem by this tactic since cdpso has the advantages of method of conjugate direction cd and particle swarm optimization pso it overcomes the inaccuracy of cd and pso for solving systems of nonlinear equations the numerical results showed that the approach was successful for solving systems of nonlinear equations

monotone method for second order delayed differential equations with boundary value conditions in this paper we investigate the existence of extremal solutions of boundary value problems for second order differential equations with delayed arguments we show that a lower and upper solution method has been applied successfully to such problems an one sided lipschitz condition with corresponding constants is important assumption in our research

on the solvability of a singular initial value problem x f t x x using barrier strips abstract in this paper conditions which guarantee the existence of solutions to x f t x x x 0 a are given

stationary distribution of a stochastic cholera model between communities linked by migration abstract in this paper we develop and study a stochastic cholera model between communities linked by migration firstly we show that there exists a unique global positive solution of the considered system with any positive initial value then by using the stochastic lyapunov analysis method we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model our result shows that the migration and the small white noise are not beneficial for eliminating cholera finally some examples together with numerical simulations are introduced to support the theoretical result

fractal boundaries of basin of attraction of newton raphson method in helicopter trim a key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim particularly when free wake models are used trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero an appropriate initial estimate of the trim state is needed for successful helicopter trim this study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations set of initial guess points from which the nonlinear equations converge it is illustrated that the three main rotor pitch controls of collective pitch longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution trajectories of the newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed it is found that the basins of attraction can have fractal boundaries

newton homotopy analysis method for nonlinear equations abstract in this paper we present an efficient numerical algorithm for solving nonlinear algebraic equations based on newton raphson method and homotopy analysis method also we compare homotopy analysis method with adomian s decomposition method and homotopy perturbation method some numerical illustrations are given to show the efficiency of algorithm

periodic boundary value problem for first order impulsive functional differential equations this paper discusses the periodic boundary value problem for a class of first order impulsive functional differential equations we establish several existence results under weaker hypotheses by using the leray schauder alternative the lower and upper solution method and the monotone iterative technique the corresponding results in the literature are improved or extended some examples are also given to illustrate the advantage of the results

positive solution for system of nonlinear first order pbvps on time scales abstract in this paper we are concerned with the following system of nonlinear first order periodic boundary value problems on time scale t x i t f i t x 1 t x 2 t x n t 0 t 0 t x i 0 x i t i 1 2 n where f i 0 t 0 n r is continuous and there exists a constant m i 0 such that m i x i f i t x 1 x 2 x n 0 for x 1 x 2 x n 0 n t 0 t some existence criteria of positive solution are established by using a fixed point theorem for operators on cone

a non iterative method for solving non linear equations abstract in this paper using a hyperbolic tangent function tanh x 0 we develop a non iterative method to estimate a root of an equation f x 0 the problem of finding root is transformed to evaluating an integral and thus we need not take account of choosing initial guess the larger the value of the better the approximation to the root alternatively we employ the signum function sgn x instead of the hyperbolic tangent function which results in an exact formula for the root availability of the present method is shown by some numerical examples for which the traditional newton s method is not appropriate

numerical methods for determining the stability and response of periodic systems with applications to helicopter rotor dynamics and aeroelasticity abstract this paper reviews and extends some recently developed numerical techniques for analyzing the stability linear response and nonlinear response of periodic systems which are governed by ordinary differential equations with periodic coefficients the main emphasis is on applications aimed at rotor blade aeroelastic stability and response calculations in forward flight first floquer theory is br4iefly reviewed and numerically efficient methods for evaluating the transition matrix at the end of one period are described next the numerical treatment of the linear response problem is discussed finally the numerical solution of the nonlinear response problem is treated using quasilinearization and periodic shooting applications illustrating numerical properties of the methods are described

asymptotic profiles of basic reproduction number for epidemic spreading in heterogeneous environment the effect of diffusion rates on the basic reproduction number of a general compartmental reaction diffusion epidemic model in a heterogeneous environment is considered it is shown that when the d

monotone method for first order functional differential equations we study periodic boundary value problems relative to a general class of first order functional differential equations for this class of problems we develop the monotone iterative technique our formulation is very general including delay differential equations functional differential equations with maxima and integro differential equations but the case where the operator defining the functional dependence is not necessarily lipschitzian is also considered

waveform relaxation methods for periodic differential functional systems the convergence of waveform relaxation methods applied to a periodic differential functional system is studied the continuous dependence is also discussed

on discontinuous first order implicit boundary value problems t 0 1 2 under various boundary conditions cf e g 1 5 8 13 29 31 32 thereis little known about the existence of solutions of 1 2 even in the casewhen f is continuous unless it is possible to solve the equation 1 2 forthe highest derivative and then to apply standard existence theory such asleray schauder theory one approach taken for instance in 8 16 is toreduce equation 1 2 to a differential inclusion of the formu

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

periodic boundary value problem for first order impulsive delay differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive delay differential equations is considered

on newton type methods with cubic convergence recently there has been some progress on newton type methods with cubic convergence that do not require the computation of second derivatives weerakoon and fernando appl math lett 13 2000 87 derived the newton method and a cubically convergent variant by rectangular and trapezoidal approximations to newton s theorem while frontini and sormani j comput appl math 156 2003 345 140 2003 419 derived further cubically convergent variants by using different approximations to newton s theorem homeier j comput appl math 157 2003 227 169 2004 161 independently derived one of the latter variants and extended it to the multivariate case here we show that one can modify the werrakoon fernando approach by using newton s theorem for the inverse function and derive a new class of cubically convergent newton type methods

periodic boundary value problems for impulsive integro differential equations of mixed type by developing a comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first order nonlinear impulsive integro differential equations of mixed type

nonlinear two point boundary value problems applications to a cholera epidemic model this paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two point boundary value problem when an empirically constructed candidate for an approxim

existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions abstract this paper is concerned with the existence and approximation of solutions for a class of first order functional differential equations with periodic boundary conditions we present a new comparison result and extend previous results

singular and nonsingular first order initial value problems abstract under barrier strip type arguments we investigate the existence of global solutions to the initial value problem x f t x x x 0 a where the scalar function f t x p may be singular at x a

spectral bound and reproduction number for infinite dimensional population structure and time heterogeneity spectral bounds of quasi positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations the sign of the spectral bound of the variational matrix at 0 decides whether at low density the population becomes extinct or grows another important threshold parameter is the reproduction number mathcal r which is the spectral radius of a related positive matrix as is well known the spectral bound and mathcal r 1 have the same sign provided that the matrices have a particular form the relation between spectral bound and reproduction number extends to models with infinite dimensional state space and then holds between the spectral bound of a resolvent positive closed linear operator and the spectral radius of a positive bounded linear operator we also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete time models we illustrate the general theory

periodic boundary value problems for second order impulsive integro differential equations by developing a new comparison result and using the monotone iterative technique we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for second order nonlinear impulsive integro differential equations of mixed type

on the cauchy problem for first order discontinuous ordinary differential equations abstract we prove the existence of extremal caratheodory solutions for the initial value problem x t f t x t for a e t 0 1 x 0 0 under weak conditions over f 0 1 r r

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

periodic boundary value problems for delay differential equations with impulses existence and approximation of solutions for an impulsive delay differential equation with periodic boundary value conditions are presented we use comparison principles and monotone iterative techniques

periodic boundary value problem for first order impulsive functional differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of minimal and maximal solutions of the periodic boundary value problem for first order impulsive functional differential equations is considered

an iterative method for functional differential equations a suitable periodic boundary conditions for a functional differential equations email protected t f t x x t are conditions of the form x 0 q x 2 p q in this paper we use the notion of upper and lower solutions coupled with monotone iterative method to prove the existence of solutions of this periodic boundary value problem

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

design of helicopter rotor blades for optimum dynamic characteristics abstract the mass and stiffness distributions for helicopter rotor blades are to be tailored in such a way to give a predetermined placement of blade natural frequencies the optimal design is pursued with respect of minimum weight sufficient inertia and reasonable dynamic characteristics the finite element technique will be used as a tool rotor types include hingeless articulated and teetering

a cholera epidemic model in a spatiotemporally heterogeneous environment abstract the transmission of cholera a water and food borne intestinal infection involves complex interactions among human hosts pathogens and the environment and is impacted by the spatial variation and seasonal fluctuation in this paper we propose a new deterministic model to investigate the spatiotemporal dynamics of cholera transmission the model employs a reaction convection diffusion system to represent the spatial movement of the hosts and pathogens and incorporates time periodic parameters to describe the seasonality of the disease transmission and bacterial growth rates using the theory of principle eigenvalues we define and analyze the basic reproduction number of this model based on which we establish the threshold type results for cholera transmission in a spatiotemporally heterogeneous environment

discontinuous galerkin finite element in time for solving periodic differential equations the dynamics of rotating systems can be modeled by ordinary differential equations with periodic time varying coefficients and forcing discontinuous galerkin dg methods which are known to be higher order accurate a stable and locally conservative are used to solve such equations in this work dg methods are shown to find the steady state periodic solution of a constant coefficient structural dynamics problem with periodic forcing and b helicopter rotor rigid blade flapping equation which is a parametric ordinary differential equation with periodic time varying coefficients and periodic forcing convergence of the numerical solution obtained is studied for both the above equations while h and p are refined separately it is found that the dg method works well for these problems and allows the development of an element level error estimate which can be used to monitor convergence and hence develop adaptive schemes it is clear from the results that even though higher order polynomials result in much faster convergence with reduced computational effort it also results in deterioration of the numerical performance for p 12 two adaptive strategies based on the element level displacement jump norm i e h and hp refinement are proposed and demonstrated on problems with higher harmonic forcing primary controls and parametric variations

newton s method on the complex exponential function we show that when newton s method is applied to the product of a polynomial and the exponential function in the complex plane the basins of attraction of roots have finite area

an existence result on noncompact intervals for second order functional differential inclusions in this paper we investigate the existence of mild solutions on an infinite interval for second order functional differential inclusions in banach spaces we shall make use of a theorem of ma which is an extension to multivalued maps on locally convex topological spaces of schaefer s theorem

new high order convergence iteration methods without employing derivatives for solving nonlinear equations abstract a family of new iteration methods without employing derivatives is proposed in this paper we have proved that these new methods are quadratic convergence their efficiency is demonstrated by numerical experiments the numerical experiments show that our algorithms are comparable to well known methods of newton and steffensen furthermore combining the new method with bisection method we construct another new high order iteration method with nice asymptotic convergence properties of the diameters b n a n

stationary distribution of a stochastic cholera model with imperfect vaccination abstract in this paper a stochastic cholera model with imperfect vaccination is proposed and investigated we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic lyapunov function our result may provide some new insights for elimination of cholera some numerical simulation is provided to demonstrate our main result

periodic boundary value problem for non lipschitzian impulsive functional differential equations we consider a class of first order impulsive functional differential equations where the functional dependence is not necessarily a lipschitzian function the new maximum principle improves and extends previous results and uniqueness of solution between a lower and an upper solution for a particular nonlinear problem is presented we give conditions for existence of extremal solutions in an interval delimited by a lower and an upper solution

positive solutions for first order periodic boundary value problem this paper discusses the existence and multiplicity of positive solutions for the following first order periodic boundary value problemx t f t x t 0 0

fractal basins of attraction associated with a damped newton s method an intriguing and unexpected result for students learning numerical analysis is that newton s method applied to the simple polynomial z 3 1 0 in the complex plane leads to intricately interwoven basins of attraction of the roots as an example of an interesting open question that may help to stimulate student interest in numerical analysis we investigate the question of whether a damping method which is designed to increase the likelihood of convergence for newton s method modifies the fractal structure of the basin boundaries the overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting

the julia set of newton s method for multiple root abstract in this paper we analyze the julia set of newton method for multiple roots construct the julia sets of standard relaxed and multiple roots through the experimental mathematics method we drew the following conclusion 1 the julia sets of above methods for f z z z 1 has time rotation symmetry and its center is origin 2 the multiple roots domain of attraction of these kinds of julia sets are sensitive to 3 there is not simple root domain of attraction in relax method because z the root of f z is neutral or repelling fixed point of f z 4 is not the fixed point of f z so multiple root s julia set consists of a great number of attraction domain of multiple and simple roots 5 the experimental errors and truncation of coefficients cause the minimal effect to multiple root the more to relax the maximal to standard newton method

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

discontinuous first order functional boundary value problems abstract in this paper we prove a new existence result for functional boundary value problems with first order functional differential equations under weak conditions on the nonlinear part and monotonicity but not continuity with respect to the functional variable discontinuities with respect to the unknown are allowed to occur over general classes of time dependent sets which are assumed to satisfy a kind of unviability or solubility condition

threshold dynamics of a diffusive siri model with nonlinear incidence rate abstract this work is concerned with the dynamics of a diffusive siri epidemic model with nonlinear incidence rate we first establish the well posedness of the model then we show the basic reproduction number r 0 is a threshold parameter for the stability of the model in fact the disease free equilibrium is globally asymptotically stable when r 0 1 while the phenomena of uniform persistence occurs when r 0 1 if r 0 1 the disease free equilibrium is globally asymptotically stable under some assumptions we also clarify the relationship between r 0 and the local basic reproduction number moreover we establish the local and global stability of the endemic equilibrium provided that all parameters of this model are constants

numerical investigation of the three dimensional development in boundary layer transition a numerical method for solving the complete navier stokes equations for incompressible flows is introduced that is applicable for investigating three dimensional transition phenomena in a spatially growing boundary layer results are discussed for a test case with small three dimensional disturbances for which detailed comparison to linear stability theory is possible the validity of our numerical model for investigating nonlinear transition phenomena is demonstrated by realistic spatial simulations of the experiments by kachanov and levchenko1 for a subharmonic resonance breakdown and of the experiments of klebanoff et al 2 for a fundamental resonance breakdown

existence of solutions of boundary value problems for differential equations with delayed arguments we study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem a monotone iterative technique is applied

helicopter trim analysis by shooting and finite element methods with optimally damped newton iterations helicopter trim settings of periodic initial state and control inputs are investigated for convergence of newton iteration in computing the settings sequentially and in parallel the trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta these three methods broadly represent two main approaches of trim analysis adaptation of initial value and finite element boundary value codes to periodic boundary conditions particularly for unstable and marginally stable systems in each method both the sequential and in parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped newton iteration with an optimally selected damping parameter the impact of damped newton iteration including earlier observed divergence problems in trim analysis is demonstrated by the maximum condition number of the jacobian matrices of the iterative scheme and by virtual elimination of divergence the advantages of the in parallel scheme over the conventional sequential scheme are also demonstrated

a multidisciplinary optimization using semi analytical sensitivity analysis procedure and multilevel decomposition abstract this paper addresses the development of a multidisciplinary optimization procedure using an efficient semi analytical sensitivity analysis technique and multilevel decomposition for the design of aerospace vehicles a semi analytical sensitivity analysis procedure is developed for aerodynamic design sensitivities accuracy and efficiency of the sensitivity analysis procedure is established through comparison of the results with those obtained using a finite difference technique the optimization problem with the integration of aerodynamics and structures is decomposed into two levels optimization is performed for improved aerodynamic performance at the first level and improved structural performance at the second level aerodynamic analysis is performed by solving the three dimensional parabolized navier stokes equations a nonlinear programming technique and an approximate analysis procedure are used for optimization the procedure developed is applied to design the wing of a high speed aircraft results obtained show significant improvements in the wing aerodynamic and structural performance when compared to a reference or baseline wing configuration the use of the semi analytical sensitivity technique provides significant computational savings

extremal solutions of measure differential equations abstract we investigate the existence of the least and greatest solutions to measure differential equations as well as the relation between the extremal solutions and lower or upper solutions along the way we prove a fairly general local existence theorem and an analogue of peano s uniqueness theorem for measure differential equations finally we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales

helicopter rotor dynamics by finite element time approximation abstract starting from a variational formulation based on hamilton s principle the paper exploits the finite element technique in the time domain in order to generate by an automated procedure a numerical approximation of the nonlinear periodic equations peculiar to the dynamics of helicopter rotors with this method a unified approach is developed which can be used to solve both the response and stability problem two simple examples are presented that show how the method can be used to study the trimmed response of a fully articulated rigid blade and the mechanical instability analysis of a four bladed rotor

monotone iterative method for first order functional difference equations with nonlinear boundary value conditions in this paper we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first order functional difference equations with nonlinear boundary value conditions an example is given to illustrate the results obtained

nonlinear relaxation quasi newton algorithm for the compressible navier stokes equations a highly efficient implicit method for the computation of steady two dimensional compressible navier stokes flowfields is presented the discretization of the governing equations is hybrid in nature with flux vector splitting utilized in the streamwise direction and central differences with flux limited artificial dissipation used for the transverse fluxes line jacobi relaxation is used to provide a suitable initial guess for a new nonlinear iteration strategy based on line gauss seidel sweeps the applicability of quasi newton methods as convergence accelerators for this and other line relaxation algorithms is discussed and efficient implementations of such techniques are presented convergence histories and comparisons with experimental data are presented for supersonic flow over a flat plate and for several high speed compression corner interactions results indicate a marked improvement in computational efficiency over more conventional upwind relaxation strategies particularly for flowfields containing large pockets of streamwise subsonic flow

approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions we prove that it is possible to approximate the solutions of a nonlinear periodic boundary value problem via monotone sequences starting at piecewise continuous upper and lower solutions in fact we present here two different iterative methods explicit expression of each iteration is obtained by solving some linear initial value problems

on monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations abstract in this paper we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations we obtain some new results relative to the lower solution and upper solution with

discontinuous functional differential equations with delayed or advanced arguments we provide new results on the existence of extremal solutions for discontinuous differential equations with a deviated argument which can be either delayed or advanced the boundary condition is allowed to be discontinuous and to depend functionally on the unknown solution

dynamics for an seirs epidemic model with time delay on a scale free network abstract a new seirs epidemic model with time delay on a scale free network is introduced the basic reproduction number r 0 is defined and it does not depend on time delay by analyzing the characteristic equation of equilibria and constructing suitable lyapunov functionals the stability of steady states is proved system does not occur hopf bifurcation around disease free equilibrium e 0 uniform immunization and targeted immunization are investigated numerical simulations and sensitivity analysis are performed results indicate that time delay plays a significant part in propagation process of epidemics on a scale free network

extremality and comparison results for discontinuous third order functional initial boundary value problems extremality and comparison results are derived for third order functional initial boundary value problems differential equations of problems may involve discontinuous nonlinearities

solving the nonlinear equations by the newton homotopy continuation method with adjustable auxiliary homotopy function in this paper the concept of adjustable auxiliary homotopy function for the newton homotopy continuation method is presented by means of adjusting the auxiliary function we can solve the nonlinear equations and guarantee the solutions exactly without divergence rather than the traditional numerical methods such as the newton raphson method and so on

advanced differential equations with nonlinear boundary conditions we investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions sufficient conditions when the problem has extremal solutions or a unique solution are formulated linear advanced differential inequalities are also discussed

extremality results for first order discontinuous functional differential equations abstract in this paper we derive extremality and comparison results for first order explicit and implicit functional differential equations equipped with functional initial conditions differential equations and initial conditions may involve discontinuities

continuous dependence of nonmonotonic discontinuous differential equations continuous dependence of solutions for a class of nonmonotonic discontinuous differential equations is studied first a local existence theorem due to z wu is extended to a larger class then a result concerning continuous dependence for this larger class is proven this employs a type of convergence similar to gihman s convergence criterion witch is defined to be for all a b and y formule the significance of gihman s convergence criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence finally examples are presented to motivate and clarify this continuous dependence result

estimating the hausdorff besicovitch dimension of boundary of basin of attraction in helicopter trim helicopter trim involves solution of nonlinear force equilibrium equations as in many nonlinear dynamic systems helicopter trim problem can show chaotic behavior this chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot this study focuses on the boundary of the basin of attraction obtained for a set of control inputs we analyze the boundary by considering it at different magnification levels the magnified views reveal intricate geometries it is also found that the basin boundary exhibits the characteristic of statistical self similarity which is an essential property of fractal geometries these results led the authors to investigate the fractal dimension of the basin boundary it is found that this dimension is indeed greater than the topological dimension from all the observations it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature c 2012 elsevier inc all rights reserved

existence of solutions to initial value problems for first order differential equations under sign assumptions we investigate the global existence of solutions of the initial value problem f t x x 0 x a a where the function f t x p may be singular at t a

newton s method for the navier stokes equations with finite element initial guess of stokes equations abstract it is shown that finite element solutions of stokes equations may be chosen as the initial guess for the quadratic convergence of newton s algorithm applied to navier stokes equations provided there are sufficiently small mesh size h and the moderate reynold s number we provide also a mixed convergence analysis in terms of iterations and finite error estimates of the initial guess with a regularity estimate and error analysis for each newton s step