interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

consistent kdv expansion method and its applications to the kp equation abstract this paper proposed a novel method named as the consistent kdv equation expansion ckdve method for solving nonlinear differential equations with the help of the classical nonlinear kdv equation if a system satisfies the ckdve then it can be defined as ckdve solvability especially the kadomtsev petviashvili kp equation is also verified as a typical case of the ckdve solvability moreover the detailed discussion illustrates the effectiveness and convenience of the proposed method and presents the possibility of constructing relationships among many different partial differential equation models in the related field

new exact periodic solitary wave solutions for the 3 1 dimensional generalized kp and bkp equations the extended homoclinic test approach ehta is a straightforward and efficient method to solve nonlinear partial differential equations in this paper new exact solutions including kinky periodic solitary wave solutions periodic soliton solutions and periodic kink solutions for the 3 1 dimensional generalized kadomtsev petviashvili kp and b type kadomtsev petviashvili bkp equations are obtained moreover the strangely mechanical features of these wave solutions are investigated these solutions enrich the variety of the dynamics of higher dimensional nonlinear wave field

analytical solutions to a generalized growth equation abstract systems of the form x i i n j 1 x gij j i n j 1 x hij j i 1 n occur in the analysis of biological networks they also include as special cases the known growth laws and probability functions famous differential equations like those of bessel chebyshev and laguerre and solutions to important physical problems these systems have no known analytical solution however an important subclass comprising many of the special cases mentioned above is solved

s system modelling of endemic infections abstract infectious diseases often exhibit complex dynamics that can only be understood when analyzed with mathematical models these models are typically nonlinear which makes it difficult to compare their mutual merits it is shown how models for infectious diseases can be objectively compared upon representing them as special cases of a canonical form for nonlinear functions and differential equations called an s system besides comparison and classification this form allows natural generalizations of existing models and the objective evaluation of model extensions a variety of methods facilitate the analysis of infectious disease models in s system form

a computational stochastic methodology for the design of random meta materials under geometric constraints we present a computational stochastic methodology for generating and optimizing random meta material mm configurations with nonoverlapping geometric constraints subject to various types of covari

statistical densities cumulatives quantiles and power obtained by s system differential equations abstract a comprehensive approach for the analysis of statistical univariate distributions is advanced the method is based on recasting a distribution in a canonical set of first order nonlinear differential equations called an s system in which the derivative of each variable equals a difference of products of power law functions the recasting techniques are illustrated with some examples from central and noncentral probability distributions the density and the cumulative because of their close relationship can be simultaneously evaluated within one s system a simple inversion procedure permits the calculation of quantiles when in addition a noncentral distribution and its central counterpart are represented in one s system the power function and the inverse power function are obtainable given a set of initial conditions any differential equation solver can generate the desired solution particularly convenient and efficient is the program essyns designed specifically to solve and analyze s

finding multiple roots of nonlinear algebraic equations using s system methodology abstract the solution of nonlinear algebraic equations is a well known problem in many fields of science and engineering several types of numerical methods exist each with their advantages and disadvantages s system methodology provides a novel approach to this problem the result as judged by extensive numerical tests is an effective method for finding the positive real roots of nonlinear algebraic equations the motivation for this method begins with the observation that any system of nonlinear equations composed of elementary functions can be recast in a canonical nonlinear form known as an s system when the derivatives of these ordinary first order nonlinear differential equations are zero the resulting nonlinear algebraic equations can be represented as an underdetermined set of linear algebraic equations in the logarithms of the variables these linear equations together with a set of simple nonlinear constraints determine the multiple steady state solutions of the original system that are positive real the numerical solution is obtained by approximating the constraints as s system equations in steady state to yield a determined set of linear algebraic equations which then can be solved iteratively this method has been implemented and the experience to date suggests that the method is robust and efficient the rate of convergence to the solutions is quadratic a combinatorial method for selecting initial conditions has led to the identification of several and in many cases all of the positive real solutions for the set of problems tested it does so without resorting to analysis in the complex domain and without having to make random or problem specific provisions for initializing the procedure there is an obvious parallel implementation of the algorithm and there are additional generalizations and efficiencies yet to be realized

analyzing the combined multi waves polynomial solutions in a two layer liquid medium abstract in this work we investigate the behavior of mixed wave solutions in two layer liquid with dispersive waveguide these combined multi wave solutions were obtained in polynomial types for the 3 1 dimensional yu toda sasa fukuyama ytsf equation with variable coefficients by virtue of the generalized unified method and symbolic computations we formally derive polynomial solutions for this equation the obtained solutions include multi soliton solutions multi periodic solutions and multi elliptic solutions furthermore the physical insight and the movement role of the waves in the two layers are discussed and analyzed graphically for different selections of the arbitrary functions of the obtained solutions

new traveling waves solutions to generalized kaup kupershmidt and ito equations in this paper we consider a special fifth order kdv equation with constant coefficients and we obtain traveling wave solutions for it using the projective riccati equation method by mean of a scaling exact solutions to general kaup kupershmidt kk equation are obtained as a particular case exact solutions to standard kk equation can be derived using the same method we obtain exact solutions to standard ito equation by mean of scaling new exact solutions to general ito equation are formally derived

breather type soliton and two soliton solutions for modified korteweg de vries equation in this letter different kinds of solutions including breather type soliton and two soliton solutions are obtained for the modified korteweg de vries m kdv equation by using bilinear form the extended homoclinic test approach and dependent variable transformations moreover we point out that the author did not obtain so called periodic two soliton solutions in w long in press 1

exact three wave solution for higher dimensional kdv type equation a new periodic type of three wave solutions including periodic two solitary solution doubly periodic solitary solution and breather type of two solitary solution for the 1 2 dimensional and 1 3 dimensional kdv type equations are obtained using hirota s bilinear form and generalized three wave type of ansatz approach

on stability and trajectory boundedness of lotka volterra systems with polytopic uncertainty a new approach is proposed to study the stability property of the feasible equilibrium point and the trajectory boundedness of the lotka volterra lv dynamics in the presence of polytopic uncertainties stability analysis of general lv systems is not straightforward in comparison with its cooperative counterpart it is shown that under specific conditions the trajectory of the cooperative counterpart of the lv system can be considered as an upper bound for the trajectory of the corresponding dynamics consequently the stability of the cooperative counterpart of the lv system leads to boundedness of the trajectory the obtained results are extended for the lv systems with polytopic uncertainties it is shown that under specific conditions there is a region which the trajectory of the system belongs to it as time tends to infinity also the stability of lv systems with polytopic uncertainty is investigated the obtained results are confirmed through some numerical application based examples

applications of hta and ehta to ytsf equation new approaches homoclinic test approach hta and extended homoclinic test approach ehta are proposed to seek solitary wave solution of high dimensional nonlinear wave system exact periodic solitary wave periodic soliton cross solitary wave and doubly periodic wave solutions for ytsf equation are obtained using hta and ehta respectively

exact periodic solitary wave solutions for the 2 1 dimensional boussinesq equation abstract exact periodic solitary wave solutions for the 2 1 dimensional boussinesq equation are obtained by using the extended ansatz function method detailed behavior of the propagation of the periodic solitary wave solutions for the 2 1 dimensional boussinesq equation is illustrated by using the method of figure analysis the result shows that it is entirely possible for the 2 1 dimensional integrable equations or non integrable equations that there exist periodic solitary waves in the different direction the propagation of the periodic solitary waves is actually phase shifts of solitons and the amplitudes of non singular periodic solitary waves depend on frequency and wave number of periodic wave

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

exact soliton solutions for the fifth order sawada kotera equation exact soliton solutions for the fifth order sawada kotera equation are obtained by using the hirota bilinear method these solutions include one soliton solutions periodic two soliton solutions and singular periodic soliton solutions the results show that there exist periodic two soliton solutions and singular periodic soliton solutions for the fifth order sawada kotera equation

fourth and fifth order scaled rungs kutta algorithms for treating dense output new runge kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step as well as at the end of each step

a parallel fast algorithm for computing the helmholtz integral operator in 3 d layered media in this paper we propose a parallel fast algorithm for computing the product of the discretized helmholtz integral operator in layered media and a vector in o n qn z 2n xn ylogn xn y operations here n xn yn z is the number of sources and n q is the number of quadrature points used in the evaluation of the sommerfeld integral in the definition of layered media green s function for problems in thin layer media n z o 1 such a product forms the key step of many iterative solvers such as the krylov subspace based gmres and bicgstab for linear systems arising from the integral equation methods for the helmholtz equations the fast solver is based on two important techniques which reduce the cost of quadrature summations in the sommerfeld contour integral for green s functions in 3 d layered media the first technique is the removal of surface pole effects along the real axis integration contour by identifying the pole locations with a discrete wavelet transform in the second technique we apply a window based high frequency filter to shorten the contour length as a result the integral operator for the 3 d layered media can be efficiently written as a sum of 2 d hankel cylindrical integral operators and the latter can be calculated by either a tree code or a 2 d wideband fast multipole method in a fast manner numerical results show the efficiency and parallelism of the proposed fast algorithm

exact periodic cross kink wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional potential ytsf equation abstract in this paper the 3 1 dimensional potential ytsf equation is investigated exact solutions with three wave form including periodic cross kink wave periodic two solitary wave and breather type of two solitary wave solutions are obtained using hirota s bilinear form and generalized three wave approach with the aid of symbolic computation moreover the properties for some new solutions are shown with some figures

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

new periodic solitary wave solutions for the benjiamin ono equation in this paper the new periodic solitary wave and doubly periodic solutions for 1 1 dimensional benjiamin ono equation are obtained using the bilinear method and extended homoclinic test approach these results demonstrate that the integrable system has richly dynamical behavior even if it is 1 1 dimensional

homoclinic breather and rogue wave solutions to maccari equation abstract in this work a new extended homoclinic test combining with breather period limit method bplm for seeking homoclinic breather and rogue wave solution of nonlinear evolution 2 1 d maccari equation is proposed two new families of homoclinic breather wave solution and rational homoclinic solution homoclinic rogue wave for these equations are obtained respectively moreover dynamical features of homoclinic breather and rogue wave solution are exhibited this result shows that both homoclinic breather and rogue wave can occur in the same nonlinear system

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

the breather type and periodic type soliton solutions for the 2 1 dimensional breaking soliton equation exact breather type and periodic type soliton solutions including the double breather type soliton solutions the breather type periodic soliton solutions and breather type two soliton solutions and the periodic type two soliton and three soliton solutions for the 2 1 dimensional breaking soliton equation are obtained using the extended three wave method etm the results show that the etm may provide us with a straightforward and effective mathematical tool for seeking multi wave solutions of higher dimensional nonlinear evolution equations

new multi soliton solutions of a 3 1 dimensional nonlinear evolution equation in this paper applying homoclinic test approach to a 3 1 dimensional nonlinear evolution equation new kink breather soliton solutions are obtained in addition some new multi soliton solutions are obtained via three wave method it is shown that the homoclinic test approach and the three wave method can be used for constructing multi soliton solutions of nonlinear evolution equations in mathematical physics

new kink multi soliton solutions for the 3 1 dimensional potential yu toda sasa fukuyama equation by means of transformation of unknown function the non integrable system of 3 1 dimensional potential yu toda sasa fukuyama ytsf equation is converted into the combined equation of differently two bilinear forms by the darvishi s idea some new kink multi soliton solutions are obtained by using homoclinic test approach and three wave method respectively

extend three wave method for the 1 2 dimensional ito equation abstract in this work extend three wave method etm is used to construct the novel multi wave solutions of the 1 2 dimensional ito equation as a result three soliton solution doubly periodic solitary wave solutions periodic two solitary wave solutions are obtained it is shown that the extend three wave method may provide us with a straightforward and effective mathematical tool for seeking multi wave solutions of higher dimensional nonlinear evolution equations

the ehta for nonlinear evolution equations abstract two types of important nonlinear evolution equations are investigated by using the extended homoclinic test approach ehta some exact soliton solutions including breather type of soliton periodic type of soliton and two soliton solutions are obtained these results show that the extended homoclinic test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations

a comparison of the monomial method and the s system method for solving systems of algebraic equations two numerical methods for solving systems of equations have recently been proposed a method based on monomial approximations the monomial method and a technique based on s system methodology the s system method the two methods have been shown to be fundamentally identical that is they are both equivalent to newton s method operating on a transformed version of the system of equations yet when applied specifically to algebraic systems of equations they have significant computational differences that may impact the relative computational efficiency of the two methods these computational differences are described a combinatorial strategy for locating many and sometimes all solutions to a system of nonlinear equations has also been suggested previously this paper further investigates the effectiveness of this strategy when applied to either of the two methods

n soliton solutions to the bogoyavlenskii schiff equation and a quest for the soliton solution in 3 1 dimensions we study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the hirota s bilinear form we explicitly discuss the bogoyavlenskii schiff equation in 2 1 dimensions its analytical proof of multisoliton solution and a new feature are given being guided by the strong symmetry we also propose a new equation in 3 1 dimensions

stability properties of interpolants for runge kutta methods in solving initial value problems for ordinary differential equations by means of a discrete method we often need to know the solution on a set of points that differs from the grid systems with driving equation dense output discontinuity problems delay equations etc the nonnodal approximations are generally obtained by local interpolation and different features can be requested from the interpolants according to the problem treated in this paper we introduce the concept of stable interpolant for runge kutta methods roughly speaking a stable interpolant maintains the stability properties of the runge kutta method itself the lack of such stability makes the continuous extension not reliable especially in solving stiff equations

homoclinic breather wave solutions and doubly periodic wave solutions for coupled kdv equations abstract in this paper we first find out a proper variable transformation by the ideas of painleve expansion then we apply the extended homoclinic test approach to obtain two cycle breathing places wave solutions of eq 1 which describe interactions between two physical waves and these special solutions can be applied to explain the structure of certain physical phenomena thus this method can be applied to the study of other similar nonlinear coupled system

new exact solutions to the 2 1 dimensional ito equation extended homoclinic test technique exact soliton solutions to the 2 1 dimensional ito equation are studied based on the idea of extended homoclinic test and bilinear method some explicit solutions such as triangle function solutions soliton solutions doubly periodic wave solutions and periodic solitary wave solutions are obtained it shows that the 2 1 dimensional ito equation has richer solutions besides the elastic interactions of the solutions and their corresponding physical meaning are discussed

an o nlogn hierarchical random compression method for kernel matrices by sampling partial matrix entries abstract in this paper we propose an o n log n hierarchical random compression method hrcm for kernel matrix compressing which only requires sampling o n log n entries of a matrix the hrcm combines the hierarchical framework of the h matrix and a randomized sampling technique of column and row spaces for far field interaction kernel matrix we show that a uniform column row sampling of a far field kernel matrix thus without the need and associated cost to pre compute a costly sampling distribution will give a low rank compression of such low rank matrix independent of the matrix size and only dependent on the separation of the source and target locations this far field random compression technique is then implemented at each level of the hierarchical decomposition for general kernel matrices resulting in an o n log n random compression method error and complexity analysis for the hrcm are included numerical results for electrostatic and low frequency helmholtz wave kernels have validated the efficiency and accuracy of the proposed method in comparison of direct o n 2 summations