on critical behaviour in generalized kadomtsev petviashvili equations an asymptotic description of the formation of dispersive shock waves in solutions to the generalized kadomtsev petviashvili kp equation is conjectured the asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the painleve i hierarchy several examples are discussed numerically to provide strong evidence for the validity of the conjecture the numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the subcritical kp equations while in the supercritical kp equations a blow up occurs after the formation of the dispersive shock waves

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

numerical study of blow up and dispersive shocks in solutions to generalized korteweg de vries equations abstract we present a detailed numerical study of solutions to general korteweg de vries equations with critical and supercritical nonlinearity both in the context of dispersive shocks and blow up we study the stability of solitons and show that they are unstable against being radiated away and blow up in the l 2 critical case the blow up mechanism by martel merle and raphael can be numerically identified in the limit of small dispersion it is shown that a dispersive shock always appears before an eventual blow up in the latter case always the first soliton to appear will blow up it is shown that the same type of blow up as for the perturbations of the soliton can be observed which indicates that the theory by martel merle and raphael is also applicable to initial data with a mass much larger than the soliton mass we study the scaling of the blow up time t in dependence of the small dispersion parameter and find an exponential dependence t and that there is a minimal blow up time t 0 greater than the critical time of the corresponding hopf solution for 0 to study the cases with blow up in detail we apply the first dynamic rescaling for generalized korteweg de vries equations this allows to identify the type of the singularity

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

a numerical approach to blow up issues for dispersive perturbations of burgers equation abstract we provide a detailed numerical study of various issues pertaining to the dynamics of the burgers equation perturbed by a weak dispersive term blow up in finite time versus global existence nature of the blow up existence for long times and the decomposition of the initial data into solitary waves plus radiation we numerically construct solitary waves for fractional korteweg de vries equations

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

an optimization algorithm for solving the rich vehicle routing problem based on variable neighborhood search and tabu search metaheuristics this paper presents a novel optimization algorithm that consists of metaheuristic processes to solve the problem of the capillary distribution of goods in major urban areas taking into consideration the features encountered in real life time windows capacity constraints compatibility between orders and vehicles maximum number of orders per vehicle orders that depend on the pickup and delivery and not returning to the depot with the intention of reducing the wide variety of constraints and complexities known as the rich vehicle routing problem this algorithm proposes feasible alternatives in order to achieve the main objective of this research work the reduction of costs by minimizing distances and reducing the number of vehicles used as long as the service quality to customers is optimum and a load balance among vehicles is maintained

on the convergence of a damped secant method with modified right hand side vector we present a convergence analysis for a damped secant method with modified right hand side vector in order to approximate a locally unique solution of a nonlinear equation in a banach spaces setting in the special case when the method is defined on r i our method provides computable error estimates based on the initial data such estimates were not given in relevant studies such as herceg et al 1996 krejic 2002 numerical examples further validating the theoretical results are also presented in this study

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

a computational algebraic investigation of the decomposition method for time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat or wave equations is investigated numerically using computational algebraic techniques

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

numerical study of fractional nonlinear schrodinger equations using a fourier spectral method we provide a detailed numerical investigation of dispersive schrodinger type equations involving a fractional laplacian in an one dimensional case by an appropriate choice of the dispersive exponent both mass and energy sub and supercritical regimes can be identified this allows us to study the possibility of finite time blow up versus global existence the nature of the blow up the stability and instability of nonlinear ground states and the long time dynamics of solutions the latter is also studied in a semiclassical setting moreover we numerically construct ground state solutions of the fractional nonlinear schrodinger equation

the numerical solution of sixth order boundary value problems by the modified decomposition method a fast and accurate algorithm is developed for the solution of sixth order boundary value problems bvps with two point boundary conditions a modified form of the adomian decomposition method is applied to construct the numerical solution for such problems the scheme is tested on one linear and two nonlinear problems the results demonstrate reliability and efficiency of the algorithm developed

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

convergence of adomian s decomposition method periodic temperatures abstract the convergence of adomian s decomposition method 1 as applied to periodic temperature fields in heat conductors is investigated

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

numerical solution of the small dispersion limit of korteweg de vries and whitham equations the cauchy problem for the korteweg de vries kdv equation with small dispersion of order 2 1 is characterized by the appearance of a zone of rapid modulated oscillations of wavelength of order these oscillations are approximately described by the elliptic solution of kdv where the amplitude wave number and frequency are not constant but evolve according to the whitham equations in this manuscript we give a quantitative analysis of the discrepancy between the numerical solution of the kdv equation in the small dispersion limit and the corresponding approximate solution for values of between 10 1 and 10 3 the numerical results are compatible with a difference of order close to the center within the interior of the whitham oscillatory zone of order 1 3 at the left boundary outside the whitham zone and of order at the right boundary outside the whitham zone 2007 wiley periodicals inc

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

fully discrete galerkin methods for the korteweg de vries equation abstract fully discrete schemes for the numerical simulation of solutions of the periodic initial value problem for the korteweg de vries equation are introduced implemented and tested of special interest are stable schemes featuring rates of convergence of order higher than two in both the spatial and temporal variable a careful analysis of the relative and absolute efficiency of these schemes is carried out and one of the schemes is applied to demonstrate that solutions of certain generalized korteweg de vries equations apparently may develop singularities in finite time

numerical study of shock formation in the dispersionless kadomtsev petviashvili equation and dispersive regularizations abstract the formation of singularities in solutions to the dispersionless kadomtsev petviashvili dkp equation is studied numerically for different classes of initial data the asymptotic behavior of the fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity the approach is first tested in detail in 1 1 dimensions for the known case of the hopf equation where it is shown that the break up of the solution can be identified with prescribed accuracy for dissipative regularizations of this shock formation as the burgers equation and for dispersive regularizations as the korteweg de vries equation the fourier coefficients indicate as expected global regularity of the solutions the kadomtsev petviashvili kp equation can be seen as a dispersive regularization of the dkp equation the behavior of kp solutions for small dispersion parameter 1 near a break up of corresponding dkp solutions is studied it is found that the difference between kp and dkp solutions for the same initial data at the critical point scales roughly as 2 7 as for the korteweg de vries equation

newton like method with modification of the right hand side vector this paper proposes a new newton like method which defines new iterates using a linear system with the same coefficient matrix in each iterate while the correction is performed on the right hand side vector of the newton system in this way a method is obtained which is less costly than the newton method and faster than the fixed newton method local convergence is proved for nonsingular systems the influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented relevant numerical examples are used to demonstrate the advantages of the proposed method

third order iterative methods without using any frechet derivative a modification of classical third order methods is proposed the main advantage of these methods is they do not need to evaluate any frechet derivative a convergence theorem in banach spaces just assuming the second divided difference is bounded and a punctual condition is analyzed finally some numerical results are presented