a study of convergence on the newton homotopy continuation method the traditional newton method is a well known and popular method for solving non linear equation it has high efficient in the convergence speed however we always need to guess the initial value in the iteration process good initial guess value can solve the equation quickly bad initial guess value usually will yield divergence homotopy continuation method is a kind of perturbation method it can guarantee the answer by a certain path if we choose the auxiliary homotopy function or call start system well this paper presents some useful rules for the choice of the auxiliary homotopy function to avoid the problem of divergence of traditional newton method

black box multigrid for nonsymmetric problems one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in previous work this defect was overcome and a code written in which one needed only to specify the logically rectangular positive definite matrix problem the code did everything else necessary to set up the auxiliary coarser problems to achieve a multigrid solution in this paper the previous work is extended to a single nonsymmetric equation

on maximum norm convergence of multigrid methods for two point boundary value problems multigrid methods applied to standard linear finite element discretizations of linear elliptic boundary value problems in two dimensions are considered in the multigrid method damped jacobi or damped gauss seidel is used as a smoother it is proven that the two grid method with nu pre smoothing iterations has a contraction number with respect to the maximum norm that is asymptotically bounded by c nu 1 2 ln h k 2 with h k a suitable mesh size parameter moreover it is shown that this bound is sharp in the sense that a factor ln h k is necessary

new iterative technique for solving nonlinear equations various problems of pure and applied sciences can be studied in the unified frame work of the nonlinear equations in this paper a new family of iterative methods for solving nonlinear equations is developed by using a new decomposition technique the convergence of the proposed methods is proved it is shown that the new family contains several well known iterative methods as special case for the implementation and performance of the new methods a nonlinear equation arising in the population model and another one arising in the motion of a particle on inclined plane is solved

homotopy perturbation method for a type of nonlinear coupled equations with parameters derivative abstract in this paper homotopy perturbation method is directly extended to investigate nonlinear coupled equations with parameters derivative and to derive their numerical solutions these nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations by choosing different values of the parameters in general formal numerical solutions as a result a very rapidly convergent series solution is obtained the efficiency and accuracy of the method are verified by using two famous examples coupled burgers and mkdv equations numerical solutions show that good results have been achieved

on the integrable perturbations of the camassa holm equation we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure we present an investigation of the nonlinear partial differential equations pde which are asymptotically representable as a linear combination of the equations from the camassa holm hierarchy for this purpose we use the infinitesimal transformations of dependent and independent variables of the original pde this approach is helpful for the analysis of the systems of the pde which can be asymptotically represented as the evolution equations of polynomial structure

some multi step iterative methods for solving nonlinear equations in this paper we develop some new iterative methods for solving nonlinear equations by using the techniques introduced in golbabai and javidi 2007 1 and rafiq and rafiullah 2008 20 we establish the convergence analysis of the proposed methods and then demonstrate their efficiency by taking some test problems

the hierarchical basis multigrid method for convection diffusion equations we make a theoretical study of the application of a standard hierarchical basis multigrid iteration to the convection diffusion equation discretized using an upwind finite element discretizations we show behavior that in some respects is similar to the symmetric positive definite case but in other respects is markedly different in particular we find the rate of convergence depends significantly on parameters which measure the strength of the upwinding and the size of the convection term numerical calculations illustrating some of these effects are given

on the behavior of the solution of the dissipative camassa holm equation with the arbitrary dispersion coefficient abstract in this paper we consider the dissipative camassa holm equation with arbitrary dispersion coefficient and compactly supported initial data we demonstrate the simple conditions on the initial data that lead to finite time blow up of the solution in finite time or guarantee that the solution exists globally also propagation speed for the equation under consideration is investigated

a new hybrid iteration method for solving algebraic equations abstract a new hybrid iteration method the hybrid s name was used by luo xing guo luo applied mathematics and computation 171 2 2005 1171 1183 has been proposed for solving a non linear algebraic equation f x 0 by using taylor s theorem in this paper we proposed a new hybrid iteration method and we show by one equation that this new hybrid iteration method is more quickly convergent than newton s method and hence than hybrid iteration method

a new iteration method for solving algebraic equations in this paper a nonlinear algebraic equation f x 0 by applying taylor s theorem is converted into a coupled iteration systeml x n 1 g x n 0andg x n f x n l x n wherel x ax 2 bx c the formula is of high convergence some examples are given

multigrid methods for convection diffusion problems discretized by a monotone scheme abstract we study multigrid mg methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection diffusion problems by an exponential fitting scheme the latter ensures the stability of the simplest possible coarse grid operators obtained from galerkin projections based on graph matching linear and nonlinear mg preconditioners are defined in the framework of algebraic multilevel iteration the option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two and three dimensional test problems

the solution of multipoint boundary value problems by the optimal homotopy asymptotic method in this work we solve multipoint boundary value problems using the optimal homotopy asymptotic method oham the proposed method is tested upon several problems from the literature and the results are compared with the available exact solution this method provides easy tools to control the convergence region of approximating solution series where ever necessary

a new iteration method with cubic convergence to solve nonlinear algebraic equations abstract in this paper a new iteration scheme is proposed to solve the roots of an algebraic equation f x 0 given an initial guess x0 the roots of the equation can be obtained using the following iteration scheme x n 1 x n f x n f 2 x n 2 f x n f x n f x n this iteration scheme has unique convergence characteristics different from the well known newton s method it is shown that this iteration method has cubic local convergence in the neighborhood of the root using this scheme real or complex roots for specific algebraic equations can be found because there are two iteration directions for a given initial guess two solutions can be found for certain algebraic equations with multiple roots examples are presented and compared with other methods

blow up of solutions for the dissipative dullin gottwald holm equation with arbitrary coefficients abstract we study the cauchy problem for the weakly dissipative dullin gottwald holm equation describing unidirectional propagation of surface waves in a shallow water regime u t 2 u x x t c 0 u x 3 u u x u x x x u 2 u x x 2 2 u x u x x u u x x x in the present paper we demonstrate the simple conditions on the initial data that lead to blow up of the solution in finite time

blow up phenomena for the weakly dissipative dullin gottwald holm equation in this paper we study the cauchy problem for the weakly dissipative dullin gottwald holm equation we establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time

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

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

on the inverse scattering approach and action angle variables for the dullin gottwald holm equation abstract the poisson brackets for the scattering data of the dullin gottwald holm equation are computed then the action angle variables are expressed in terms of the scattering data we also discuss on the inverse scattering problem for this equation

global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation this paper is concerned with the global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation we prove the global existence and uniqueness of solution to the viscous degasperis procesi equation for u0 l2 r the local well posedness for u0 l2 r has been proven based on the energy estimate the strong and weak limit behaviors of solution to the viscous degasperis procesi equation are obtained

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

new family of iterative methods for nonlinear equations in this paper we suggest and analyze a new family of iterative methods for solving nonlinear equations using the system of coupled equations coupled with decomposition technique several numerical examples are given to illustrate the efficiency and performance of the new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

new iterative schemes for nonlinear equations in this paper we suggest and analyze a new iterative method for solving nonlinear equations by using a new decomposition method we also discuss the convergence criteria of these iterative methods several numerical examples are given to illustrate the efficiency and performance of the new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems a convergence estimate is established for nonsymmetric multigrid algorithms for finite element methods for nonselfadjoint and indefinite elliptic problems it is shown without further regularity assumption beyond h 1 necessary to define the weak form that the v cycle algorithm converges with rate bounded by 1 cj 1 for sufficiently small coarse mesh size h 1 where j is the number of levels in the multigrid algorithm

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

some numerical methods for solving nonlinear equations by using decomposition technique in this paper we use the system of coupled equations involving an auxiliary function together with decomposition technique to suggest and analyze some new classes of iterative methods for solving nonlinear equations these new methods include the halley method and its variant forms as special cases various numerical examples are given to illustrate the efficiency and performance of the new methods these new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations

multigrid convergence for nonsymmetric indefinite variational problems and one smoothing step we prove that the multigrid method for linear systems arising from nonsymmetric and indefinite elliptic variational problems converges for all sufficiently small h with one smoothing step only previous proofs required that the number of smoothing steps be within certain bounds which were quite difficult to compute our theory is based on sharp energy norm convergence estimates for the symmetric definite case and lower order terms are treated as perturbations the resulting estimates are so sharp that we obtain even the convergence of the v cycle

a fourth order iterative method for solving nonlinear equations the present paper illustrates an iterative numerical method to solve nonlinear equations of the form f x 0 especially those containing the partial and non partial involvement of transcendental terms comparative analysis shows that the present method is faster than newton raphson method hybrid iteration method new hybrid iteration method and others cost is also found to be minimum than these methods the beauty in our method can be seen because of the optimization in important effecting factors i e lesser number of iteration steps lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods this work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative

new iterative methods for solving nonlinear equation by using homotopy perturbation method in this paper we suggest and analyze a new class of iterative methods for solving nonlinear equations by using the homotopy perturbation method convergence of their method is also considered here we also discuss the efficiency index and computational order of convergence of new methods several numerical examples are given to illustrate the efficiency and performance of these new methods these new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations

robust convergence of multilevel algorithms for convection diffusion equations a new approach is developed to analyze convergence of multilevel algorithms for convection diffusion equations this approach uses a multilevel recursion formula which can be applied to a variety of nonsymmetric problems here the recursion formula is applied to a robust multilevel algorithm for convection diffusion equations with convection in the x or y direction the multilevel algorithm uses semicoarsening line relaxation and prewavelets the convergence rate is proved to be less than 0 18 independent of the size of the convection term and the number of unknowns the assumptions allow the convection term to have a turning point so that an interior layer can appear in the solution of the convection diffusion equation the computational cost of the multilevel cycle is about o n log n independent of the size of the convection term where n is the number of unknowns it is proved that o log n multilevel cycles starting from the initial guess 0 lead to an o n 2 algebraic error with respect to the l infty norm independent of the size of the convection term

a third order newton type method for nonlinear equations based on modified homotopy perturbation method in this paper we present a new iterative method for solving nonlinear algebric equations by using modified homotopy perturbation method we also discuss the convergence criteria of the present method to assess its validity and accuracy the method is applied to solve several test problems

new iterative methods for nonlinear equations by modified hpm a numerical method based on modified homotopy perturbation method hpm is proposed for solving nonlinear algebric equations it is shown that the proposed method has third order convergence to assess its validity and accuracy the method is applied to solve several test problems

convergence analysis of a multigrid method for convection diffusion equations this paper is concerned with the convergence analysis of robust multigrid methods for convection diffusion problems we consider a finite difference discretization of a 2d model convection diffusion problem with constant coefficients and dirichlet boundary conditions for the approximate solution of this discrete problem a multigrid method based on semicoarsening matrix dependent prolongation and restriction and line smoothers is applied for a multigrid w cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion convection ratio for the contraction number of a multigrid v cycle a bound is proved which is uniform for a class of convection dominated problems the analysis is based on linear algebra arguments only

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

a fast method for the solution of the helmholtz equation in this paper we consider the numerical solution of the helmholtz equation arising from the study of the wave equation in the frequency domain the approach proposed here differs from those recently considered in the literature in that it is based on a decomposition that is exact when considered analytically so the only degradation in computational performance is due to discretization and roundoff errors in particular we make use of a multiplicative decomposition of the solution of the helmholtz equation into an analytical plane wave and a multiplier which is the solution of a complex valued advection diffusion reaction equation the use of fast multigrid methods for the solution of this equation is investigated numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies

global existence and blow up solutions for a nonlinear shallow water equation considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by dullin gottward and holm blow up can occur only in the form of wave breaking a wave breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow up rate is established the blow up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

on the well posedness problem and the scattering problem for the dullin gottwald holm equation in this paper we study the well posedness of the cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation the so called dgh equation which was derived by dullin gottwald and holm the issue of passing to the limit as the dispersive parameter tends to zero for the solution of the dgh equation is investigated and the convergence of solutions to the dgh equation as 2 0 is studied and the scattering data of the scattering problem for the equation can be explicitly expressed the new exact peaked solitary wave solutions are obtained in the dgh equation after giving the condition of existing peakon in the dgh equation it turns out to be nonlinearly stable for the peakon in the dgh equation

multigrid convergence for a singular perturbation problem abstract so far there has been no analysis of multigrid methods applied to singularly perturbed dirichlet boundary value problems only for periodic boundary conditions does the fourier transformation mode analysis apply and it is not obvious that the convergence results carry over to the dirichlet case since the eigenfunctions are quite different in the two cases in this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the dirichlet case

the limit behavior of the solutions to a class of nonlinear dispersive wave equations in this paper we study the limit behavior of the solutions to a class of nonlinear dispersive wave equations we also demonstrate that the solutions to eq 1 1 converge to the solution to the corresponding bbm equation as the parameter converges to zero and the convergence of solutions to eq 1 1 as 2 0 is studied in hs r s 32 given the discussion of the parameters in the nonlinear dispersive wave equation 1 1 one will obtain some conditions in which compacton and peakon occur

three step iterative methods for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

limit behavior of the global solutions to the degasperis procesi type equation abstract in this paper we study the degasperis procesi equation with a physically perturbation term a linear dispersion based on the global existence result we show that the solution of the degasperis procesi equation with linear dispersion tends to the solution of the corresponding degasperis procesi equation as the dispersive parameter goes to zero moreover we prove that smooth solutions of the equation have finite propagation speed they will have compact support if its initial data has this property

existence and uniqueness of low regularity solutions for the dullin gottwald holm equation we establish the local existence and uniqueness of solutions for the dullin gottwald holm equation with continuously differentiable periodic initial data the regularity conditions needed for the cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered

efficient method 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 by using decomposition technique in conjunction with the coupled system of equations we develop new iterative method for solving a system of nonlinear equations which has better efficiency index we prove the fifth order convergence of the proposed method we provide the comparison of efficiency index with some other well known methods for the implementation and performance of the new method we solve the kinematic synthesis problem for steering chandrasekhar integral equation and combustion problem for a temperature of 3000 c and compare the results with some existing methods

iterative methods with fourth order convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations involving only first derive of the function using a new decomposition technique which is due to noor m aslam noor numerical analysis and optimization lecture notes comsats institute of information technology islamabad pakistan 2006 m aslam noor new iterative schemes for nonlinear equations appl math comput in press and noor and noor m aslam noor k inayat noor some iterative schemes for nonlinear equations appl math comput in press we show that this new iterative method has fourth order convergence several numerical examples are given to illustrate the efficiency and performance of the new method

the improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids multigrid convergence rates degenerate on problems with stretched grids or anisotropic operators unless one uses line or plane relaxation for three dimensional problems only plane relaxation suffices in general while line and plane relaxation algorithms are efficient on sequential machines they are quite awkward and inefficient on parallel machines this paper presents a new multigrid algorithm based on the use of multiple coarse grids that eliminates the need for line or plane relaxation in anisotropic problems this algorithm is developed and the standard multigrid theory is extended to establish rapid convergence for this class of algorithms the new algorithm uses only point relaxation allowing easy and efficient parallel implementation yet achieves robustness and convergence rates comparable to line and plane relaxation multigrid algorithms the algorithm described here is a variant of mulder s multigrid algorithm w mulder j compact phys 83 1989 pp 303 323 for hyperbolic problem

newton like iterative methods for solving system of non linear equations abstract homotopy perturbation method hpm is applied to construct a new family of newton like iterative methods for solving system of non linear equations comparison of the result obtained by the present method with newton raphson method reveals that the accuracy and fast convergence of the new method

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

a new family of iterative methods for solving system of nonlinear algebric equations abstract homotopy perturbation method hpm is applied to construct a new iterative method for solving system of nonlinear algebric equations comparison of the result obtained by the present method with that obtained by revised adomian decomposition method hossein jafari varsha daftardar gejji appl math comput 175 2006 1 7 reveals that the accuracy and fast convergence of the new method

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

on modified newton methods for solving a non linear algebraic equations abstract in this paper we present a new modified newton method for solving a non linear algebraic equation by using the well known nedzhibov s method d k r babajee m z dauhoo an analysis of the properties of the variants of newton s method with third order convergence applied mathematics and computation 183 1 2006 659 684 we proposed a new iteration method and we show by one equation that this new iteration method is more quickly convergence than newton s method

analysis and comparison of geometric and algebraic multigrid for convection diffusion equations the discrete convection diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes estimates of error reduction rates for both geometric multigrid gmg and algebraic multigrid amg are established on uniform rectangular meshes for a model problem our analysis shows that gmg with line gauss seidel smoothing and bilinear interpolation converges if h gg epsilon 2 3 and amg with the same smoother converges more rapidly than gmg if the interpolation constant beta in the approximation assumption of amg satisfies beta ll frac h sqrt epsilon alpha hs 1mm mbox where alpha big begin smallmatrix 1 h sqrt varepsilon 2 h ge sqrt varepsilon end smallmatrix on unstructured triangular meshes the performance of gmg and amg both as solvers and as preconditioners for gmres are evaluated numerical results show that gmres with amg preconditioning is a robust and reliable solver on both type of meshes

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

higher order iterative schemes for nonlinear equations using decomposition technique in this paper we suggest and analyze some unique recurrence relations which can generate different classes of iterative methods for solving nonlinear equations using the system of coupled equations together with decomposition technique by using an auxiliary function various numerical examples are given to illustrate the efficiency and performance of the newly suggested methods these new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations

blow up phenomena for the weakly dissipative dullin gottwald holm equation revisited in this paper we investigate the cauchy problem for the weakly dissipative dullin gottwald holm equation we first establish the local well posedness result by kato s semigroup theory then we obtain the precise blow up scenario and present some blow up results for strong solutions to the equation finally we discuss the blow up rate of the wave breaking solutions this result complements the early one in the literature such as novruzov j math phys 54 092703 2013

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

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

wave breaking phenomenon for a modified two component dullin gottwald holm equation this paper is concerned with the wave breaking phenomenon for a modified two component dullin gottwald holm shallow water system we give sufficient conditions on the initial data to guarantee blow up of solutions in finite time

global weak solutions and wave breaking phenomena to the periodic degasperis procesi equation with strong dispersion abstract considered herein is the initial value problem for the periodic degasperis procesi equation with a strong dispersive term that is an approximation to the incompressible euler equation for shallow water waves the existence and uniqueness of global weak solutions are established moreover the periodic peakon and shockpeakon solutions of the equation are constructed

new exact periodic wave solutions for the dullin gottwald holm equation abstract in this paper the dullin gottwald holm equation is studied using semi inverse method and integral bifurcation method new periodic waves such as peakon like periodic wave compacton like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion the exact parametric representations of these waves are also presented

convergence analysis of a multigrid method for a convection dominated model problem the paper presents a convergence analysis of a multigrid solver for a system of linear algebraic equations resulting from the discretization of a convection diffusion problem using a finite element method we consider piecewise linear finite elements in combination with a streamline diffusion stabilization we analyze a multigrid method that is based on canonical intergrid transfer operators a direct discretization approach for the coarse grid operators and a smoother of line jacobi type a robust diffusion and h independent bound for the contraction number of the two grid method and the multigrid w cycle are proved for a special class of convection diffusion problems namely with neumann conditions on the outflow boundary dirichlet conditions on the rest of the boundary and a flow direction that is constant and aligned with gridlines our convergence analysis is based on modified smoothing and approximation properties the arithmetic complexity of one multigrid iteration is optimal up to a logarithmic term

he s parameter expanding methods for strongly nonlinear oscillators ji huan he s work on asymptotic techniques is briefly reviewed and his parameter expanding methods including modified lindstedt poincare method and bookkeeping parameter method are discussed in detail some remarkable virtues of the methods are exploited and their applications are illustrated

the analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems we prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems we provide results for the so called symmetric multigrid schemes we show that for the variable v script cycle and the w script cycle schemes multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns provided that the initial grid is sufficiently fine we show that the v script cycle algorithm also converges under appropriate assumptions on the coarsest grid but at a rate which may deteriorate as the number of levels increases this deterioration for the v script cycle may occur even in the case of full elliptic regularity finally the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory

iterative methods to nonlinear equations homotopy perturbation method is applied to construct a new iterative method for solving nonlinear algebraic equations the convergence of the new scheme is proposed an at least the cubic order of convergence is established several examples are presented and compared to other methods showing the accuracy and fast convergence of the new method

a note on the new iteration method for solving algebraic equations in ji huan he a new iteration method for solving algebraic equations appl math comput 135 2003 81 84 a new iteration method for solving algebraic equations has been proposed in this paper we show for one equation which given by the author and some other equations that the new iteration method fails to obtain expected results hence in contrast to previous claims it is no more quickly convergent than newton s method and may in fact be failed