a gauss newton approach to solving generalized inequalities generalized inequalities are systems of the form gx k 0 where g maps between normed linear spaces and k denotes the partial order induced by the closed convex cone k e g k r m1 0 rm2 in this paper a gauss newton type algorithm is presented for minimizing the distance function rhox mbox dist gx k inf vert gx k vert k in k the technique globalizes the well known newton methods for solving generalized inequalities and overcomes the difficulties associated with subgradient methods for the global minimization of

the essence of the homotopy analysis method the generalized taylor expansion including a secret auxiliary parameter h which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by liao the secret of h cannot be understood in the frame of the homotopy analysis method this is a serious shortcoming of liao s method we solve the problem through a detailed study of a simple example we show that the generalized taylor expansion is just the usual taylor s expansion at different point t 1 we prove that there is a relationship between h and t 1 which reveals the meaning of h and the essence of the homotopy analysis method as an important example we study the series solution of the blasius equation using the series expansion method at different points we obtain the same result with liao s solution given by the homotopy analysis method

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

a series solution of the foam drainage equation foaming occurs in many distillation and absorption processes the drainage of liquid foams involves the interplay of gravity surface tension and viscous forces in this paper we apply an analytic technique namely the homotopy analysis method ham to handle the foam drainage equation the powerful homotopy analysis method computes the solution in a rapidly convergent infinite series we also compare the numerical results obtained by hpm or adm and the same results obtained by using the analytic solutions obtained by ham

on the generalized wavelet galerkin method abstract in the frame of the traditional wavelet galerkin method based on the compactly supported wavelets it is important to calculate the so called connection coefficients that are some integrals whose integrands involve products of wavelets their derivatives as well as some known coefficients in considered differential equations however even for linear differential equations with non constant coefficient the computation of connect coefficients becomes rather time consuming and often even impossible in this paper we propose a generalized wavelet galerkin method based on the compactly supported wavelets which is computationally very efficient even for differential equations with non constant coefficients no matter linear or nonlinear problems some related mathematical theorems are proved based on which the basic ideas of the generalized wavelet galerkin method are described in details in addition some examples are used to illustrate its validity and high efficiency a nonlinear example shows that the generalized wavelet galerkin method is not only valid to solve nonlinear problems but also possesses the ability to find new solutions of multi solution problems this method can be widely applied to various types of both linear and nonlinear differential equations in science and engineering

on the application of the homotopy analysis method for a nonlocal mixed problem with bessel operator abstract in this research work we give analytic approximate solutions for a nonlocal initial boundary value problem where one of the classical conditions is replaced by an integral condition by employing the homotopy analysis method ham the series solution can be obtained by either using initial conditions partial t solution or by using the boundary conditions partial x solution this method allows us to control the convergence region of the series solution

the explicit series solution of sir and sis epidemic models in this paper the sir and sis epidemic models in biology are solved by means of an analytic technique for nonlinear problems namely the homotopy analysis method ham both of the sir and sis models are described by coupled nonlinear differential equations a one parameter family of explicit series solutions are obtained for both models this parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models our analytic results agree well with the numerical ones this analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology

on the sensitivity analysis of hoffman constants for systems of linear inequalities relying on a general variational method developed by the authors and lucchetti nonlinear anal to appear the origin of which goes back to ioffe trans amer math soc 251 1979 pp 61 69 we give a formula for the best hoffman constant sigma inf x notin p a b frac ax b infty d x p a b where p a b x ax le b is a nonempty polyhedron in mathbb r n we also sharpen some results of luo and tseng siam j matrix anal appl 15 1994 pp 636 659 by characterizing the continuity set of some hoffman constants and by pointing out their locally lipschitzian character we apply these results to the study of the behavior of the solution set of a linear program inf ax le b u scriptscriptstyle t x with respect to a b u

convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation absorption abstract we consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction and internal heat generation absorption using a similarity transformation the governing equations of the problem are reduced to a coupled third order nonlinear ordinary differential equations we first examine a number of special cases for which we may obtain exact solutions we then obtain analytical solutions by the homotopy analysis method and numerical solutions by a boundary value problem solver in order to further study the behavior of the nonlinear differential equations for various values of the physical parameters our numerical solutions are shown to agree with the available results in the literature we then employ the numerical results to bring out the effects of the suction parameter heat source sink parameter stretching parameter porosity parameter the prandtl number and the free convection parameter on the flow and heat transfer characteristics in the absence of suction and free convection our findings are in agreement with the corresponding numerical results of attia h a attia on the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation comput mater sci 38 2007 741 745

the homotopy analysis method for solving the fornberg whitham equation and comparison with adomian s decomposition method in this work an analytical technique namely the homotopy analysis method ham is applied to obtain an approximate analytical solution of the fornberg whitham equation a comparison is made between the ham results and the adomian s decomposition method adm and the homotopy perturbation method hpm the results reveal that ham is very simple and effective the ham contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of solution series

on the convergence properties of hildreth s quadratic programming algorithm we prove the linear convergence rate of hildreth s method for quadratic programming in both its sequential and simulateneous versions we give bounds on the asymptotic error constant and compare these bounds to those given by mandel for the cyclic relaxation method for solving linear inequalities

approximate homotopy symmetry and infinite series solutions to the perturbed mkdv equation abstract we devise a homotopy model for the perturbed mkdv equation with weak fourth order dispersion and weak dissipation and demonstrate that the related approximate equations lead to similarity reduction equations and similarity reduction solutions of different orders which are formally coincident respectively series reduction solutions for the perturbed mkdv equation are thus derived painleve ii type equations or hyperbolic secant function solutions are obtained for zero order similarity reduction equations higher order similarity reduction equations are linear variable coefficients ordinary differential equations

the effect of thermal radiation on the flow of a second grade fluid this paper reports the magnetohydrodynamic mhd flow and heat transfer characteristics of a second grade fluid in a channel analytic technique namely the homotopy analysis method ham is used to solve the momentum and energy equations the important findings in this paper are the effects of second grade parameter hartmann number reynolds number thermal radiation parameter prandtl and local eckert numbers on the velocity temperature skin friction coefficient and nusselt number

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

non similar series solution for boundary layer flow of a third order fluid over a stretching sheet this study deals with the boundary layer flow of a third order fluid induced by a stretching sheet by means of similarity transformation the governing non linear partial differential equation is reduced to a non linear ordinary differential equation the analytical solution of the problem is obtained using homotopy analysis method ham finally the solutions are also compared with the existing results in the literature

the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

homotopy analysis method for the sine gordon equation with initial conditions in this work approximate analytical solution of the sine gordon equation with initial conditions is obtained by the homotopy analysis method ham the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions it is shown that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

an approximate analytical solution of time fractional telegraph equation abstract in this article the powerful easy to use and effective approximate analytical mathematical tool like homotopy analysis method ham is used to solve the telegraph equation with fractional time derivative 1

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

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

axisymmetric flow due to a stretching sheet with partial slip the steady laminar axisymmetric flow of a newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet has been investigated the flow is governed by a third order non linear boundary value problem whose exact numerical solution has been obtained non iteratively in terms of the non dimensional slip parameter l a perturbation solution valid for small l and an asymptotic solution valid for large l have been derived finally a solution based upon he s homotopy perturbation method has been developed the latter being analytical is elegant but is sufficiently accurate for all values of l

convergence analysis of an iterative scheme for solving initial value problem for multidimensional partial differential equations existence and uniqueness of solutions of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations are proved using banach fixed point theorem an iterative scheme is derived and rigorous convergence analysis of this scheme and an error estimate are included as well several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme

an explicit analytic solution to the thomas fermi equation a new kind of analytic technique namely the homotopy analysis method is employed to give an explicit analytic solution of the thomas fermi equation and the related recurrence formulae of constant coefficients this solution can be regarded as the definition of the exact solution of the thomas fermi equation

numerical solutions of partial differential equations by discrete homotopy analysis method abstract this paper introduces a discrete homotopy analysis method dham to obtain approximate solutions of linear or nonlinear partial differential equations pdes the dham can take the many advantages of the continuous homotopy analysis method the proposed dham also contains the auxiliary parameter which provides a simple way to adjust and control the convergence region of solution series the convergence of the dham is proved under some reasonable hypotheses which provide the theoretical basis of the dham for solving nonlinear problems several examples including a simple diffusion equation and two dimensional burgers equations are given to investigate the features of the dham the numerical results obtained by this method have been compared with the exact solutions it is shown that they are in good agreement with each other

the influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method this paper provides an investigation regarding slip effects on thin film flow of a fourth grade fluid down a vertical cylinder the nonlinear problem that arises is solved for both exact and ham homotopy analysis method solutions exact and ham solutions are also compared finally we briefly describe the flow characteristics to include the effects of emerging parameters

heat transfer in unsteady axisymmetric second grade fluid this work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet the axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet mathematical analysis has been carried out for two heating processes namely i with prescribed surface temperature pst case and ii prescribed surface heat flux phf case the modelled non linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable the resulting non linear partial differential equations are solved analytically using homotopy analysis method ham the series solutions are developed and the convergence is properly discussed the series solutions and graphs of velocity and temperature are constructed particular attention is given to the variations of emerging parameters such as second grade parameter prandtl and eckert numbers

analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method abstract in this paper we develop a framework to obtain approximate solutions to systems of algebraic differential equations of fractional order by employing the homotopy analysis method ham the study highlights the efficiency of the method and its dependence on the auxiliary parameter numerical examples are examined to highlight the significant features of the ham method the method shows improvements over existing analytical techniques

on the mhd flow of a second grade fluid in a porous channel the steady flow of a second grade fluid in a porous channel is considered the constitutive equations are those used for a second grade fluid the fluid is electrically conducting in the presence of a uniform magnetic field applied in the transverse direction to the flow it is shown that an analytical solution is possible by employing a homotopy analysis method ham the convergence of the obtained solution is also taken into account assessment for the influence of various parameters of interest on the velocity is undertaken

analytical solutions to a generalized drinfel d sokolov equation related to dssh and kdv6 abstract analytical solutions to the generalized drinfel d sokolov gds equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 are obtained for various values of the model parameters in particular we provide perturbation solutions to illustrate the strong influence of the parameters 1 and 2 on the behavior of the solutions we then consider a miura type transform which reduces the gds equations into a sixth order nonlinear differential equation under the assumption that 1 under such a transform the gds reduces to the sixth order drinfel d sokolov satsuma hirota dssh equation also known as kdv6 in the very special case 1 2 the method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary 1 and 2 an error analysis of the obtained approximate analytical solutions is provided

trigonometric and hyperbolic type solutions to a generalized drinfel d sokolov equation abstract a class of trigonometric and hyperbolic type solutions to the generalized drinfel d sokolov gds equations u t 1 uu x 1 u xxx v x 0 and v t 2 uv x 2 v xxx 0 is obtained for the case in which 2 0 for various values of the other model parameters the method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter 2 in effect extending the exact solutions we do not assume traveling wave solution forms for the analytical solutions that is we solve the generalized drinfel d sokolov equations as pdes without resorting to transforming the system to odes an error analysis of the obtained approximate local analytical solutions is provided then we outline a general framework by which one many construct solutions in either sine cosine or sinh cosh basis we provide the general perturbation expansion via homotopy analysis and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors travelling solutions with time dependent amplitude are then discussed

new iterative technique for solving a system of nonlinear equations various problems of pure and applied sciences can be studied in the unified frame work of the system of nonlinear equations in this paper a new family of iterative methods for solving a system of nonlinear equations is developed by using a new decomposition technique the convergence of the new methods is proved efficiency index of the proposed methods is discussed and compared with some other well known methods the upper bounds of the error and the radius of convergence of the methods are also found for the implementation and performance of the new methods the combustion problem streering problem and van der pol equation are solved and the results are compared with some existing methods several new iterative methods are derived from the general iterative scheme using the ideas and techniques of this paper one may be able to suggest and investigate a wide class of iterative methods for solving the system of nonlinear equations this is another direction of future research

on the convergence of the iteration sequence in primal dual interior point methods recently numerous research efforts most of them concerned with superlinear convergence of the duality gap sequence to zero in the kojima mizuno yoshise primal dual interior point method for linear programming have as a primary assumption the convergence of the iteration sequence yet except for the case of nondegeneracy uniqueness of solution the convergence of the iteration sequence has been an important open question now for some time in this work we demonstrate that for general problems under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence except of course the assumption that the iteration sequence converges the iteration sequence converges hence we have not only established convergence of the iteration sequence for an important class of problems but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works

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

a modified spectral method for solving operator equations in this paper we introduce a modified spectral method for solving the linear operator equation l u f l d l h 1 h 2 where h 1 and h 2 are normed vector spaces with norms 1 and respectively and d l is the domain of l also for each h h 2 h 2 h h where is an inner product on h 2 in this method we make a new set n n 0 for h 1 using l and two sets in h 1 and h 2 then using the new set n n 0 we solve this linear operator equation we show that this method does not have some shortcomings of spectral method also we prove the stability and convergence of the new method after introducing the method we give some conditions that under them the nonlinear operator equation l u n u f can be solved some examples are considered to show the efficiency of method

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

global error minimization method for solving strongly nonlinear oscillator differential equations a modified variational approach called global error minimization gem method is developed for obtaining an approximate closed form analytical solution for nonlinear oscillator differential equations the proposed method converts the nonlinear differential equation to an equivalent minimization problem a trial solution is selected with unknown parameters next the gem method is used to solve the minimization problem and to obtain the unknown parameters this will yield the approximate analytical solution of the nonlinear ordinary differential equations odes this approach is simple accurate and straightforward to use in identifying the solution to illustrate the effectiveness and convenience of the suggested procedure a cubic duffing equation with strong nonlinearity is considered comparisons are made between results obtained by the proposed gem method the exact solution and results from five recently published methods for addressing duffing oscillators the maximal relative error for the frequency obtained by the gem method compared with the exact solution is 0 0004 which indicates the remarkable precision of the gem method

solving the klein gordon equation by means of the homotopy analysis method an analytic technique namely the homotopy analysis method is applied to solve the nonlinear travelling waves governed by the klein gordon equation the phase speed and the solution which are dependent on the amplitude a are given and valid in the whole region 0

adomian s decomposition method and homotopy perturbation method in solving nonlinear equations the adomian s decomposition method and the homotopy perturbation method are two powerful methods which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution by theoretical analysis of the two methods we show in the present paper that the two methods are equivalent in solving nonlinear equations

series solutions of nano boundary layer flows by means of the homotopy analysis method we present here a similar solution for the nano boundary layer with nonlinear navier boundary condition three types of flows are considered i the flow past a wedge ii the flow in a convergent channel iii the flow driven by an exponentially varying outer flows the resulting differential equations are solved by the homotopy analysis method different from the perturbation methods the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena numerical results are compared with the available exact results to demonstrate the validity of the present solution the effects of the slip length the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed

a homotopy analysis method for limit cycle of the van der pol oscillator with delayed amplitude limiting abstract in this work a powerful analytical method called liao s homotopy analysis method is used to study the limit cycle of a two dimensional nonlinear dynamical system namely the van der pol oscillator with delayed amplitude limiting it is shown that the solutions are valid for a wide range of variation of the system parameters comparison of the obtained solutions with those achieved by numerical solutions and by other perturbation techniques shows that the utilized method is effective and convenient to solve this type of problems with the desired order of approximation

an explicit solution of the large deformation of a cantilever beam under point load at the free tip the large deformation of a cantilever beam under point load at the free tip is investigated by an analytic method namely the homotopy analysis method ham the explicit analytic formulas for the rotation angle at the free tip are given which provide a convenient and straightforward approach to calculate the vertical and horizontal displacements of a cantilever beam with large deformation these explicit formulas are valid for most practical problems thus providing a useful reference for engineering applications the corresponding mathematica code is given in the appendix

solving the inverse problem of identifying an unknown source term in a parabolic equation the inverse problem of determining an unknown source term in a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition is considered the series pattern solution of the problem is obtained by using the weighted homotopy analysis method wham a description of the method for solving the problem and identifying the unknown source term is derived finally several examples are presented to illustrate this method

an improved optimal homotopy analysis algorithm for nonlinear differential equations abstract this paper presents an improved optimal homotopy analysis algorithm to deal with nonlinear first order differential equations we discuss the optimal selection of the auxiliary linear operator in the application of the homotopy analysis method that will accelerate the convergence of series solutions then a powerful algorithm that employed the optimal auxiliary linear operator of the homotopy analysis method is presented to facilitate the calculations numerical illustrations are examined to show the efficiency and the pertinent features of the proposed algorithm this algorithm could lead to a promising approach for many nonlinear problems in applied sciences

multivalued behavior for a two level system using homotopy analysis method abstract we use the homotopy analysis method ham to solve the system of equations modeling the two level system and extract results which will pinpoint to turbulent behavior we look at multivalued solutions as indicative of turbulence or turbulent like behavior we take different specific cases which result in multivalued velocities the solutions are in series form and application of ham ensures convergence in some region

a new analysis of fractional drinfeld sokolov wilson model with exponential memory abstract the key purpose of this study is to suggest a new fractional extension of nonlinear drinfeld sokolov wilson dsw equation with exponential memory the nonlinear dsw equation plays a great role in describing dispersive water waves the stability analysis is executed with the aid of fixed point theory the advantage of fhatm over the other existing techniques is that its solution contains an auxiliary parameter which plays a big role in controlling the convergence of the solution the outcomes of the study are presented in the form of graphs and tables the results achieved by the use of the suggested scheme unfold that the used computational algorithm is very accurate flexible effective and simple to perform to examine the fractional order mathematical models

sharp lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs the main purpose of this paper to give lipschitz constants for basic optimal solutions or vertices of solution sets and basic feasible solutions or vertices of feasible sets of linear programs with respect to right hand side perturbations the lipschitz constants are given in terms of norms of pseudoinverses of submatrices of the matrices involved and are sharp under very general assumptions there are two mathematical principles involved in deriving the lipschitz constants 1 the local upper lipschitz constant of a hausdorff lower semicontinuous mapping is equal to the lipschitz constant of the mapping and 2 the lipschitz constant of a finite set valued mapping can be inherited by its continuous submappings moreover it is proved that any lipschitz constant for basic feasible solutions can be used as an lipschitz constant for basic optimal solutions feasible solutions and optimal solutions

high order iterative methods without derivatives for solving nonlinear equations the new second order and third order iterative methods without derivatives are presented for solving nonlinear equations the iterative formulae based on the homotopy perturbation method are deduced and their convergences are provided finally some numerical experiments show the efficiency of the theoretical results for the above methods

a new iterative technique for a fractional model of nonlinear zakharov kuznetsov equations via sumudu transform abstract the main objective of this paper is to suggest a new computational technique namely new iterative sumudu transform method nistm to solve numerically nonlinear time fractional zakharov kuznetsov fzk equation in two dimensions we implemented the proposed technique on two test examples plotted the solution and compared the absolute error with the variational iterative technique vim and homotopy perturbation transform method hptm

series solutions of coupled van der pol equation by means of homotopy analysis method in this paper the homotopy analysis method ham is used to give series solutions of self exited oscillation systems governed by two van der pol equations which are coupled by a linear and a cubic term the frequency and amplitude of all possible periodic solutions are investigated it is found that there exist either in phase or out of phase periodic solutions only besides the in phase periodic oscillations are decoupled whose periods and amplitudes have nothing to do with the linear and cubic coupled terms however the out of phase periodic oscillations are strongly coupled whose period and amplitude can be controlled by the linear and cubic coupled terms

unsteady axisymmetric flow of a second grade fluid over a radially stretching sheet in this article we provide an analytic solution for the problem of unsteady axisymmetric flow of a second grade fluid over a radially stretching sheet the governing equations are first modeled and then converted into a non linear partial differential equation by utilizing similarity transformations the series solution is constructed by the homotopy analysis method ham the explicit expressions for the velocity and skin friction are developed and discussed for the sundry flow parameters it is worth mentioning that the present solution is valid for all values of the dimensionless time

analytical solutions of a coupled nonlinear system arising in a flow between stretching disks in this paper we consider the axi symmetric flow between two infinite stretching disks by using a similarity transformation we reduce the governing navier stokes equations to a system of nonlinear ordinary differential equations we first obtain analytical solutions via a four term perturbation method for small and large values of the reynolds number r also we apply the homotopy analysis method which may be used for all values of r to obtain analytical solutions these solutions converge over a larger range of values of the reynolds number than the perturbation solutions our results agree well with the numerical results of fang and zhang 22 furthermore we obtain the analytical solutions valid for moderate values of r by use of homotopy analysis

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als

on the convergence of the exponential multiplier method for convex programming in this paper we analyze the exponential method of multipliers for convex constrained minimization problems which operates like the usual augmented lagrangian method except that it uses an exponential penalty function in place of the usual quadratic we also analyze a dual counterpart the entropy minimization algorithm which operates like the proximal minimization algorithm except that it uses a logarithmic entropy proximal term in place of a quadratic we strengthen substantially the available convergence results for these methods and we derive the convergence rate of these methods when applied to linear programs

approximate analytical solutions of systems of pdes by homotopy analysis method in this paper the homotopy analysis method ham is applied to obtain series solutions to linear and nonlinear systems of first and second order partial differential equations pdes the ham solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions it is shown in particular that the solutions obtained by the variational iteration method vim are only special cases of the ham solutions

on approximate solutions of infinite systems of linear inequalities abstract this paper extends a j hoffman s results on approximate solutions of finite systems of linear inequalities to infinite systems of linear inequalities it is shown that for a given infinite system of linear inequalities satisfying certain conditions the euclidean distance from a vector x to the solution set of the system is equivalent to the biggest violation by x of the system thus if a vector x almost satisfies the system then x is close to a solution of the system

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

two dimensional legendre wavelet method for travelling wave solutions of time fractional generalized seventh order kdv equation in this paper a new method based on the legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve time fractional seventh order kdv skdv equation two dimensional legendre wavelet method is applied to compute the numerical solution of nonlinear time fractional skdv equation the approximate solutions of nonlinear time fractional skdv equation thus obtained by legendre wavelet method are compared with the exact solutions as well as homotopy analysis method ham the present scheme is very simple effective and convenient for obtaining numerical solution of the skdv equation

a comparative study on the analytic solutions of fractional coupled sine gordon equations by using two reliable methods new modified homotopy analysis method has been proposed the proposed novel method is well suited for present fractional s g equation it is obvious to see that the mham is more accurate easy and efficient than hptm fractional coupled s g equations have been first time solved by mham and hptm in this paper modified homotopy analysis method mham and homotopy perturbation transform method hptm have been implemented for solving time fractional coupled sine gordon equations we consider fractional coupled sine gordon equations which models one dimensional nonlinear wave processes in two component media the results obtained by modified homotopy analysis method mham and homotopy perturbation transform method hptm are then compared with the modified decomposition method mdm by using an initial value system the numerical solutions of coupled sine gordon equations have been represented graphically here we obtain the solution of fractional coupled sine gordon s g equations which is obtained by replacing the time derivatives with a fractional derivatives of order 1 2 and 1 2 the fractional derivatives here are described in caputo sense

on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations the variational iteration method and the homotopy analysis method as alternative methods have been widely used to handle linear and nonlinear models the main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently this paper deals with the numerical solutions of nonlinear fractional differential equations where the fractional derivatives are considered in caputo sense the main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations in these algorithms legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work the proposed algorithms are capable of reducing the size of calculations improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals numerical examples are examined to show the efficiency of the algorithms

analysis of reaction diffusion system via a new fractional derivative with non singular kernel abstract in this paper we obtain analytical solutions for the fractional cubic isothermal auto catalytic chemical system with caputo fabrizio and atangana baleanu fractional time derivatives in liouville caputo sense we utilize the q homotopy analysis transform method to compute the approximate solutions we find the optimal values of h so we assure the convergence of the approximate solutions finally we compare our results numerically with the finite difference method and excellent agreement is found

application of the optimal homotopy asymptotic method to squeezing flow in this paper axisymmetric flow of two dimensional incompressible fluids is studied the optimal homotopy asymptotic method oham is applied to derive a solution of the reduced fourth order nonlinear boundary value problem for comparison the same problem is also solved by the perturbation method pm the homotopy perturbation method hpm 3 and the homotopy analysis method ham 4 oham is parameter free and provides better accuracy at lower order of approximation moreover we can easily adjust and control the convergence region as a result it is concluded that the new technique oham shows fast convergence simplicity of application and efficiency

an analytic approach to solve multiple solutions of a strongly nonlinear problem based on a new kind of analytic method namely the homotopy analysis method an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using gelfand equation as an example its validity is verified by comparing the approximation series with the known exact solution and different from perturbation techniques this approach is independent upon any small large perturbation quantities so the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering

on extension of fenchel duality and its application by considering the epigraphs of conjugate functions we extend the fenchel duality applicable to a possibly infinite family of proper lower semicontinuous convex functions on a banach space applications are given in providing fuzzy kkt conditions for semi infinite programming

numeric analytic solutions of mixed type systems of balance laws the aim of the present analysis is to apply two relatively recent methods reduced differential transform method rdtm and differential transform method dtm for the solution of balance law systems new generalized transformed formulas are derived the new approaches provided the solution in the form of a rapidly convergent series with easily computable components in the rdtm case and costly components for the dtm a comparison between the two methodologies shows that the rdtm is more effective efficient powerful and able to be applicable for large class of nonlinear partial differential equations than the dtm two test modeling problems are discussed to illustrate the effectiveness and performance of rdtm

a modified numerical scheme and convergence analysis for fractional model of lienard s equation abstract the key purpose of the present work is to constitute a numerical algorithm based on fractional homotopy analysis transform method to study the fractional model of lienard s equations the lienard s equation describes the oscillating circuits the suggested scheme is a merger of homotopy analysis technique classical laplace transform and homotopy polynomials the uniqueness and convergence analysis of the solution are also discussed the numerical and graphical results elucidate that the suggested approach is very straightforward and accurate

homotopy analysis method for limit cycle flutter of airfoils an analytical approximate method for non linear problems namely the homotopy analysis method is for the first time employed to propose a new approach for the flutter system of a two dimensional airfoil with a cubic structural nonlinearity the frequencies and amplitudes of limit cycle flutters are described as maclaurin series of an embedding parameter a series of algebraic equations in the priori unknown coefficients of the series are derived all the equations are linear except the first one this provides us with a simple iteration scheme to seek the solutions to any desired accuracy numerical examples are presented to illustrate the validity and efficiency of the proposed approach the approximations of the frequencies and amplitudes of the limit cycles are obtained very accurately furthermore it is proved the frequencies are independent of the coefficient of the cubic nonlinearity while the amplitudes are in inverse proportion to the square root of the coefficient

optimal burn in procedure for mixed populations based on the device degradation process history abstract burn in is a method of elimination of initial failures infant mortality in the conventional burn in procedures to burn in an item means to subject it to a fixed time period of simulated use prior to actual operation then the items which failed during burn in are just scrapped and only those which survived the burn in procedure are considered to be of satisfactory quality thus when the items are subject to degradation phenomena those whose degradation levels at the end of burn in exceed a given failure threshold level are eliminated in this paper we consider a new burn in procedure for items subject to degradation phenomena and belonging to mixed populations composed of a weak and a strong subpopulation the new procedure is based on the whole history of the degradation process of an item periodically observed during the burn in and utilizes the information contained in the observed degradation process to assess whether the item belongs to the strong or weak subpopulation the problem of determining the optimal burn in parameters is considered and the properties of the optimal parameters are derived a numerical example is also provided to illustrate the theoretical results obtained in this paper

construction of newton like iteration methods for solving nonlinear equations in this paper we present a simple and yet powerful and easily applicable scheme in constructing the newton like iteration formulae for the computation of the solutions of nonlinear equations the new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations it provides a tool to develop new newton like iteration methods or to improve the existing iteration methods which contains the well known newton iteration formula in logic those all improve the newton method the orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of maple some numerical tests are given to support the theory developed in this paper

solution of the jeffery hamel flow problem by optimal homotopy asymptotic method the present article addresses jeffery hamel flow fluid flow between two rigid plane walls where the angle between them is email protected a new analytical method called the optimal homotopy asymptotic method oham is briefly introduced and then employed to solve the governing equation the validity of the homotopy asymptotic method is ascertained by comparing our results with numerical runge kutta method results the effects of the reynolds number re and the angle between the two walls email protected are highlighted in the proposed work the results reveal that the proposed analytical method can achieve good results in predicting the solutions of such problems

duality and infinite dimensional optimization the purpose is to establish various duality theorems including a generalization of the fenchel duality theorem of attouch and brezis to convex processes under relaxed interiority conditions and show the usefulness of the attouch and brezis type conditions in optimization the significance of the interiority conditions is that they provide easily verifiable conditions for a large class of infinite dimensional convex programs whereas the standard slater constraint qualification fails the results are applied to obtain duality theorems for general systems of inequalities and minimax results

generalizing homotopy analysis method to solve lotka volterra equation in the paper we generalized the homotopy analysis method to solve lotka volterra equation as a simple but typical example it illustrates the validity and the great potential of the homotopy analysis method to solve differential difference equations comparisons are made between the results of the proposed method and exact solutions the results reveal that the proposed method is valid for the lotka volterra equation

homotopy analysis method for higher order fractional integro differential equations in this paper we present the homotopy analysis method shortly ham for obtaining the numerical solutions of higher order fractional integro differential equations with boundary conditions the series solution is developed and the recurrence relations are given explicitly the initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions the comparison of the results obtained by the ham with the exact solutions is made the results reveal that the ham is very effective and simple the ham contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of series solution

an efficient numerical algorithm for the fractional drinfeld sokolov wilson equation abstract the fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique sumudu transform approach and homotopy polynomials to obtain the approximate solution of a nonlinear fractional drinfeld sokolov wilson equation the nonlinear drinfeld sokolov wilson equation naturally occurs in dispersive water waves the uniqueness and convergence analysis are shown for the suggested technique the convergence of the solution is fixed and managed by auxiliary parameter the numerical results are shown graphically results obtained by the application of the technique disclose that the suggested scheme is very accurate flexible effective and simple to use

compact structures in a class of nonlinearly dispersive equations with time fractional derivatives in this work variants of the kdv equation with time fractional derivatives which exhibits compactons solitons with compact support are investigated new solitary wave solutions with compact support are developed the homotopy perturbation method is employed to derive compact solutions for these variants the chosen initial solution or trial function plays a major role in changing the physical structure of the solution the proposed scheme can be used to construct compact and noncompact solutions for a wide class of nonlinear fractional evolution equations

method for constructing analytical solutions to the dym initial value problem we obtain approximate solutions to the dym equation and associated initial value problem for general initial data by way of an optimal homotopy analysis method by varying the auxiliary linear operator thereby varying the time evolution of approximate solutions we perform two independent homotopy analyses for the dym equation we obtain two separate approximate solutions each dependent on the respective convergence control parameter a method to control the residual error of each of these approximate solutions through the use of the embedded convergence control parameter is discussed treating each approximate solution individually we select the value of the convergence control parameter which minimizes a sum of squared residual error that is we use the so called optimal homotopy analysis method in order to pick the value of the convergence control parameter that leads to the most rapid convergence of the approximate solutions performing this error analysis with each approximate solution we directly compare the deviation of each type of solution to determine which time evolution linear operator in the homotopy provides more accurate approximations we illustrate such a comparison in a few explicit examples of initial data to determine which solution offers a better representation of the way solutions to the dym equation evolve in time with this we are able to determine the effect of a choice of linear operator on how accurately resulting solutions can approximate true solutions to the dym equation since we have optimally selected the convergence control parameter in each case our solutions are optimal in the sense that they minimize a global measure of the residual error over the domain subject to the choice of auxiliary linear operator used

implicit differential equation arising in the steady flow of a sisko fluid consideration is given to the thin film flow of a sisko fluid on a moving belt using the implicit function theorem an existence result for the solution of the resulting non linear differential equation is established also the homotopy analysis method is used to obtain approximate analytical solution of the problem for non integer values of the power index the numerical results thus obtained are presented graphically and the salient features of the solution are discussed for various values of the power index parameter the results obtained reveal many interesting behaviors that warrant further study of the equations related to non newtonian fluid phenomena especially the shear thinning phenomena shear thinning reduces the wall shear stress

a study on the convergence of homotopy analysis method in this study a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed first we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems then mainly we address the sufficient condition for convergence of the method the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution the analysis is illustrated by investigating the convergence results for some nonlinear differential equations the study highlights the power of the method