a new extended tanh function method and its application to the dispersive long wave equations in 2 1 dimensions in this paper a new extended tanh function method is suggested to constructing multiple exact solutions for nonlinear partial differential equations the validity and reliability of the method is tested by its application to the dispersive long wave equations in 2 1 dimensions more new explicit exact solutions are found

the tanh and the sine cosine methods for the complex modified k dv and the generalized k dv equations the complex modified k dv cmk dv equation and the generalized k dv equation are investigated by using the tanh method and the sine cosine method a variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation the study reveals the power of the two schemes where each method complements the other

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

algorithm and mechanization to cauchy problem of parabolic equation the present paper discusses the algorithm and mechanization to the cauchy problem of parabolic equation the author uses the initial data to construct the solutions that satisfy the initial condition and obtain a brief algorithm to this problem thereafter the algorithm is used in maple to solve this cauchy problem automatically

dominance rules in combinatorial optimization problems abstract the aim of this paper is to study the concept of a dominance rule in the context of combinatorial optimization a dominance rule is established in order to reduce the solution space of a problem by adding new constraints to it either in a procedure that aims to reduce the domains of variables or directly in building interesting solutions dominance rules have been extensively used over the last 50 years surprisingly to our knowledge no detailed description of them can be found in the literature other than a few short formal descriptions in the context of enumerative methods we are therefore proposing an investigation into what dominance rules are we first provide a definition of a dominance rule with its different nuances next we analyze how dominance rules are generally formulated and what are the consequences of such formulations finally we enumerate the common characteristics of dominance rules encountered in the literature and in the usual process of solving combinatorial optimization problems

a straightforward method for finding implicit solitary wave solutions of nonlinear evolution and wave equations the authors present a straightforward method for finding implicit solutions for nonlinear evolution and wave equations the method is illustrated by finding implicit single solitary wave solutions for the harry dym korteweg de vries modified korteweg de vries boussinesq and the generalised korteweg de vries equations

new multiple soliton like solutions to the generalized 2 1 dimensional kp equation a new generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations pdes in a unified way the main idea of this method is to take full advantage of the riccati equation which has more new solutions more new multiple soliton like solutions are obtained for the the generalized 2 1 dimensional kp equation

generalized riccati equation expansion method and its application to the 3 1 dimensional jumbo miwa equation based on the computerized symbolic system maple and a riccati equation a generalized riccati equation expansion method for constructing soliton like solutions of non linear evolution equations nees is presented by a new general ansatz the proposed method is more powerful than most of the existing tanh methods the extended tanh function method the modified extended tanh function method and generalized hyperbolic function method by use of the method we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nees making use of the method we study the the 3 1 dimensional jumbo miwa equation and obtain rich new families of the exact solutions including the non travelling wave and coefficient functions soliton like solutions singular soliton like solutions periodic form solutions

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

exact solutions of the bogoyavlenskii equation using the multiple g g expansion method in this work exact traveling wave solutions of the bogoyavlenskii equation are studied by the g g expansion method and its variants the solutions obtained are general solutions which are in the form of hyperbolic trigonometric and rational functions and a variety of special solutions like kink shaped antikink shaped bell type soliton solutions etc can easily be derived from the general results under certain domain these methods are more effective and simple than other methods and a number of solutions can be obtained at the same time in order to visualize the underlying dynamics of the obtained solutions two and three dimensional plots are drawn

some notes on multisplltting methods and m step preconditioners for linear systems abstract to solve the real nonsingular linear system ax b on parallel and vector machines multisplitting methods and m step preconditioners are considered in this work in particular 1 we extend adams and ong s idea and theory of the m step additive preconditioners and ruggiero and galligani s method of the arithmetic mean and we generalize the corresponding convergence results 2 we determine suitable relaxed m step preconditioners and solve the problem of minimizing the related condition number with respect to the relaxation extrapolation parameter involved 3 finally we complete the theoretical solution of the problem of determining the optimum sor additive iterative method for 2 cyclic positive definite matrices

explicit exact solutions for new general two dimensional kdv type and two dimensional kdv burgers type equations with nonlinear terms of any order in this paper we have improved the tanh method by means of a proper transformation and a general ansatz applying the improved method and direct assumption method with symbolic computation we consider two kinds of equations general two dimensional kdv type equations with nonlinear terms

new families of exact solutions to the integrable dispersive long wave equations in 2 1 dimensional spaces the integrable dispersive long wave equations especially the higher dimensional ones are of current interest in both physics and mathematics obtained in this paper via a symbolic computation based method are new families of exact solutions to the 2 1 dimensional integrable dispersive long wave equations sample solutions from those families are presented solitary waves are merely a special case in one family

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

travelling wave solutions and proper solutions to the two dimensional burgers korteweg de vries equation in this paper we study the two dimensional burgers korteweg de vries 2d bkdv equation by analysing an equivalent two dimensional autonomous system which indicates that under some particular conditions the 2d bkdv equation has a unique bounded travelling wave solution then by using a direct method a travelling solitary wave solution to the 2d bkdv equation is expressed explicitly which appears to be more efficient than the existing methods proposed in the literature at the end of the paper the asymptotic behaviour of the proper solutions of the 2d bkdv equation is established by applying the qualitative theory of differential equations

an iterative method for large sparse linear systems on a vector computer in this paper we consider thearithmetic mean method for solving large sparse systems of linear equations this iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible l matrices with a strong diagonal dominance the method is very suitable for parallel implementation on a multiprocessor system such as the cray x mp some numerical experiments on systems resulting from the discretization by means of the usual 5 point difference formulae of an elliptic partial differential equation are presented

a new dominance rule to minimize total weighted tardiness with unequal release dates abstract we present a new dominance rule by considering the time dependent orderings between each pair of jobs for the single machine total weighted tardiness problem with release dates the proposed dominance rule provides a sufficient condition for local optimality therefore if any sequence violates the dominance rule then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged we introduce an algorithm based on the dominance rule which is compared to a number of competing heuristics for a set of randomly generated problems our computational results indicate that the proposed algorithm dominates the competing algorithms in all runs therefore it can improve the upper bounding scheme in any enumerative algorithm the proposed time dependent local dominance rule is also implemented in two local search algorithms to guide these algorithms to the areas that will most likely contain the good solutions

travelling wave solutions of two dimensional korteweg de vries burgers and kadomtsev petviashvili equations the travelling wave solutions of the two dimensional korteweg de vries burgers and kadomtsev petviashvili equations are studied from two complementary points of view the first one is an adaptation of the factorization technique that provides particular as well as general solutions the second one applies the painleve analysis to both equations throwing light on some aspects of the first method and giving an explanation to some restriction on the coefficients as well as the relation between factorizations and integrals of motion

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

a krylov multisplitting algorithm for solving linear systems of equations abstract we consider the practical implementation of krylov subspace methods conjugate gradients gmres etc for parallel computers in the case where the preconditioning matrix arises from a multisplitting we show that the algorithm can be efficiently implemented by dividing the work into tasks that generate search directions and a single task that minimizes over the resulting subspace each task is assigned to a subset of processors it is not necessary for the minimization task to send frequent information to the direction generating tasks and this leads to high utilization with a minimum of synchronization we study the convergence properties of various forms of the algorithm and present results of numerical examples on a sequential computer

algorithm and mechanization to cauchy problem of wave equation with linear damping and source terms this paper discusses the cauchy problem of wave equation with linear damping and source terms the author reduces the problem to a parabolic one and uses the corresponding initial value to construct the solutions of the resulting parabolic problem thereafter he obtains a brief algorithmization formula to this problem with mechanization in maple

on validity of m step multisplitting preconditioners for linear systems let ax b be a linear system where a is a symmetric positive definite spd matrix m step multisplitting preconditioners which include the preconditioners based on multisplittings obtained by incomplete cholesky factorizations r bru c corral a martinez j mas siam j matrix anal appl 16 1995 1210 1222 for the conjugate gradient method are studied the validity of the proposed m step multisplitting preconditioners when a is an spd matrix is proved our results improve and extend previous ones

on series expansions giving closed form solutions of korteweg de vries like equations korteweg devries kdv like equations with higher degree nonlinearity are solved by a direct algebraic technique due to hereman et al j phys a 19 1986 pp 607 628 for two kdv like equations one with fifth degree nonlinearity the other a combined kdv and mkdv equation for particular choices of the coefficients of the nonlinear terms the kink and antikink solutions found by dey are recovered furthermore soliton solutions of the combined kdv and mkdv equation are found for all values of the coefficients closed form solutions for the calogero degasperis fokas modified mkdv equation are also developed applications of the solutions of these equations in quantum field theory plasma physics and solid state physics are mentioned the hereman et al method is illustrated and slightly extended and this direct series method is briefly compared to hirota s method

solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma the direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized the class of solitary wave solutions is extended to analytic rather than rational functions of the real exponential solutions of the linearized equation expanding the solutions in an infinite series in these real exponentials an exact solution of the nonlinear pde is obtained whenever the series can be summed methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed the algorithm is now suited to solving nonlinear equations by any symbolic manipulation program this direct method is illustrated by constructing exact solutions of a generalized kdv equation the kuramoto sivashinski equation and a generalized fisher equation

travelling wave solutions to the korteweg de vries burgers equation the existence and certain qualitative properties of travelling wave solutions to the korteweg de vries burgers equation are established the limiting behaviour of these waves when e tends to zero and when tends to zero is examined together with a singular limit wherein both e and tend to zero

symmetries and invariant solutions of the two dimensional variable coefficient burgers equation we discuss symmetries and reductions of the two dimensional burgers equation with variable coefficient we classify one dimensional and two dimensional subalgebras of the burgers symmetry algebra which is infinite dimensional into conjugacy classes under the adjoint action of the symmetry group invariance under one dimensional subalgebras provides reductions to lower dimensional partial differential equations further reductions of these equations to second order ordinary differential equations are obtained through invariance under two dimensional subalgebras the reduced odes are then analysed and shown that they belong to the polynomial class of second order equations which can be linearized only for particular values of parameters figuring in the coefficient

models of parallel chaotic iteration methods abstract we consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations ax b in the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed in the second model any processor can update the global approximation which resides in the central processor at any time this model is a generalization of a sequential iterative scheme due to ostrowski called the free steering group jacobi iterative scheme and a chaotic relaxation point iterative scheme due to chazan and miranker we show that when a is a monotone matrix and all the splittings are weak regular both models lead to convergent schemes

an exact approach for single machine subproblems in shifting bottleneck procedures for job shops with total weighted tardiness objective abstract in this paper we propose an exact solution method for single machine scheduling problems typically arising from bottleneck based decomposition of weighted tardiness job shops the encountered subproblems are characterized by delayed precedence constraints multiple local due dates per operation and an objective function that is given by a weighted sum of maximum tardiness values the key concept for solving these subproblems to optimality is a dominance rule whose underlying concepts have been newly developed to cope with the given structural properties furthermore a simple lower bound and a dedicated constraint programming technique are presented the efficiency of the proposed method is demonstrated by means of single machine problems collected during a run of a shifting bottleneck procedure for job shops in different size and due date tightness configurations

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

new travelling wave solutions to the ostrovsky equation in this work new travelling wave solutions to the ostrovsky equation are studied by employing the improved tanh function method with this method the ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the riccati equation we will compare our solutions with those gained by the other methods

derivation and implicit solution of the harry dym equation and its connections with the korteweg de vries equation the harry dym equation which is related to the classical string problem is derived in three different ways an implicit cusp solitary wave solution is constructed via a simple direct method the existing connections between the harry dym and the korteweg de vries equations are uniformised and simplified and transformations between their respective solutions are carried out explicitly whenever possible physical insights are provided

the sine cosine and the tanh methods reliable tools for analytic treatment of nonlinear dispersive equations the sine cosine method and the tanh method are used for analytic treatment of nonlinear dispersive equations nonlinear variants of boussinesq equation are used as vehicles to show the strength of these methods solutions of distinct physical structures solitons compactons solitary patterns solutions and periodic solutions are formally derived the results show that the change in physical structures of the obtained solutions depends mainly on exponents and on the coefficients of the derivatives involved

convergence of relaxed parallel multisplitting methods abstract multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation is used we investigate two different variants of relaxed multisplitting methods if a is an h matrix these methods converge if the relaxation parameter is from an interval 0 0 with 0 1

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

domain wall solutions of kdv like equations with higher order nonlinearity the author considers certain nonlinear partial differential equations which are korteweg de vries kdv like equations with higher order nonlinearity it is shown that these have kink domain wall solutions for particular values of the coefficients of the nonlinear terms the solutions are compared with the standard known solution of the lambda phi 2n field theories some conservation laws for these systems of equations are also given

the structure function method for asymmetric solitary waves we present a method for the reconstruction of the waveform of asymmetric solitary waves which are observed in weakly dispersive nonlinear media starting from the morphology of the waveform we first give local differential laws which govern the waveform about its extrema and at infinity then we analytically extend these laws to find a differential equation of the generalized korteweg de vries family for the waveform holding over the whole real axis a multi valued singular functional dependence of the nonlinear term of this equation on the amplitude of the solitary wave results which was not considered before and which is intimately connected with the asymmetric nature of the solitary wave if cast in terms of a suitable combination of the wave amplitude a combination which we call the structure function of the solitary wave the differential equation reduces to a modified korteweg de vries equation with a polynomial nonlinear term which we solve by quadrature including up to sixth order nonlinearities in particular the fourth order accurate waveforms are given in terms of elementary functions and they are in excellent agreement with observations for a given maximum wave amplitude several waveforms are found each having a different width this width decreases as the maximum wave amplitude increases

applications of the combined tanh function method with symmetry method to the nonlinear evolution equations based on the symbolic computation we combined the tanh function method with the symmetry method to construct new type of solutions of the nonlinear evolution equations for the first time with the combined method some new types of solutions of the coupled 2 1 dimensional nonlinear system of schrodinger equations were obtained and their properties were studied by analyzing their figures

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

some new efficient methods to solve the n 1 ri ϵti scheduling problem abstract in this paper we prove a sufficient condition for local optimality to solve the n 1 r i t i scheduling problem which is known to be np hard we then define a new dominant subset of schedules on the basis of this condition and propose several new approximate algorithms to construct schedules belonging to this subset numerical experiments enable us to compare them with classical approximate algorithms

extended wave solutions for a nonlinear klein gordon zakharov system abstract the nonlinear klein gordon zakharov kgz system is used as a vehicle to employ the sine cosine method and the extended tanh method to construct formally exact wave solutions each method presents various solutions with distinct formal properties and physical structures which mainly include new periodic wave solutions traveling wave solutions and solitary solutions in addition as special cases some of new rational functions type solutions are developed and extended

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

the optimal solution to the problem of complex extrapolation of a first order scheme abstract given the linear stationary first order iterative scheme x m 1 tx m c for the solution of the linear complex system i t x c its extrapolated complex scheme x m 1 t x m c t 1 i t is considered the problem which is studied and solved is that of determining an optimum value for over the set of complex numbers such that the extrapolated scheme considered converges asymptotically as fast as possible

an application of the decompositionmethod for the two dimensional kdv burgers equation a relatively new decomposition method is presented to find the explicit and numerical solutions of the two dimensional korteweg de vries burgers equation tdkdvb for short for the initial conditions some numerical results are also given for this equation by using the decomposition method

travelling wave solutions to nonlinear evolution and wave equations we have studied a series of ansatze ordinary differential equations of the first order which correspond to the travelling and or solitary wave solutions of some nonlinear partial differential equations we have investigated the conditions under which the nonlinear partial differential equations have certain kinds of travelling and or solitary wave solutions as a consequence of applications we can take the trial procedures to obtain the travelling wave solutions which is a very efficient method for solving several classes of nonlinear partial differential equations

travelling wave solutions to the two dimensional kdv burgers equation travelling wave solutions for the two dimensional kdv burgers equation are obtained by using mathematica they are shown to be sums of a shock wave and solitary wave

algorithmization formula and mechanization to cauchy problem of high dimensional wave equation this paper discusses the constructive method and mechanization to the cauchy problem of high dimensional wave equations the author uses the initial value to construct the solutions that satisfy the initial conditions and obtain a brief algorithmization formula to this problem thereafter the algorithmization formula is used in maple to solve this cauchy problem automatically

dominance based heuristics for one machine total cost scheduling problems we study the one machine scheduling problem with release dates and we look at several objective functions including total weighted tardiness and total weighted completion time we describe dominance rules for these criteria as well as techniques for using these dominance rules to build heuristic solutions we use them to improve certain well known greedy heuristic algorithms from the literature finally we introduce a tabu search method with a neighborhood based on our dominance rules experiments show the effectiveness of our techniques in obtaining very good solutions for all studied criteria

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

symbolic computation and a uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations on the basis of the computer symbolic system maple and the tanh method the riccati equation method as well as all kinds of improved versions of these methods we present a further uniform direct ansatze method for constructing travelling wave solutions of nonlinear evolution equations compared with the existing methods the presented method can be used to construct more new general solutions and we give some examples to illustrate the key step of our method

on the convergence of the parallel multisplitting aor algorithm abstract we present a class of relaxed parallel multisplitting algorithms called the parallel multisplitting aor algorithm for solving large nonsingular systems of equations ax b this new algorithm is a generalization and improvement of the relaxed parallel multisplitting method linear algebra appl 119 141 152 1989 based on the new algorithm model we establish another algorithm called the relaxed parallel multisplitting aor algorithm the convergence of these algorithms is discussed under the condition that a is a monotone matrix we obtain corresponding convergence results these convergence conditions are convenient to verify

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

weighted tardiness single machine scheduling with proportional weights this paper considers arkin and roundy s single machine weighted tardiness scheduling model with tardiness penalties proportional to the processing times it presents a two stage decomposition mechanism that proves to be powerful in solving the problem completely or reducing it to a much smaller problem three types of orderings of adjacent jobs are derived that play a crucial role in problem decomposition the decomposition method solves 155 out of 320 test problems with job sizes ranging from 20 to 150 it reduces 163 unsolved problems to smaller subproblems with sizes not exceeding 25 jobs the job sizes of the remaining two unsolved subproblems are 30 and 45

computations of soliton solutions and periodic solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in the present paper the focusing branch of the genuinely nonlinear dispersive k n n equation is studied in one two and three dimensional spaces in the case of 0

painleve analysis and backlund transformation in the kuramoto sivashinsky equation the authors apply the painleve test to the kuramoto sivashinsky nonlinear partial differential equation which describes many interesting fluid motions the equation passes the test in a weak sense although the painleve analysis does not yield the general solution they perform the backlund transformation in order to obtain new particular solutions the authors find explicitly all the solutions of the set of equations which define the painleve backlund transformation they reduce to only one type namely the already known kink shaped solitary wave