Symbolic Computation System Research Articles

The Asymptotic Expansion Method via Symbolic Computation

This paper describes an algorithm for implementing a perturbation method based on an asymptotic expansion of the solution to a second‐order differential equation. We also introduce a new symbolic computation system which works with the so‐called modified quasipolynomials, as well as an implementation of the algorithm on it.

Exact travelling wave solutions for some nonlinear (N+1)-dimensional evolution equations

In this paper, we implement the tanh-coth function method to construct the travelling wave solutions for (N + 1)-dimensional nonlinear evolution equations. Four models, namely the (N + 1)-dimensional generalized Boussinesq equation, (N + 1)-dimensional sine-cosine-Gordon equation, (N + 1)-double sinh-Gordon equation and (N + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. These equations play a very important role in mathematical physics and engineering sciences. The implemented algorithm is quite efficient and is practically well suited for these problems. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations. Mathematical subject classification: 35K58, 35C06, 35A25.

General Solution of Two Generalized Form of Burgers Equation by Using the (<i>G</i><sup>'</sup>/<i>G</i>)-Expansion Method

In this work, the (G'/G)-expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations. Our work is motivated by the fact that the (G'/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Infinite sequence soliton-like exact solutions of Nizhnik-Novikov-Vesselov equation

To construct the infinite sequence soliton-like exact solutions of nonlinear evolution equations and develop the characteristics of constructivity and mechanization of the first kind of elliptic equation, new type of solutions and the corresponding Bcklund transformation of the equation are presented. Based on this, infinite sequence soliton-like exact solutions of Nizhnik-Novikov-Vesselov equation are obtained with the help of symbolic computation system Mathematica, which includes infinite sequence smooth soliton-like solutions, infinite sequence peak soliton-like solutions and infinite sequence compact soliton-like solutions.

A PARAMETERIZATION PROCESS: FROM A FUNCTORIAL POINT OF VIEW

The parameterization process used in the symbolic computation systems Kenzo and EAT is studied here as a general construction in a categorical framework. This parameterization process starts from a given specification and builds a parameterized specification by adding a parameter as a new variable to some operations. Given a model of the parameterized specification, each interpretation of the parameter, called an argument, provides a model of the given specification. Moreover, under some relevant terminality assumption, this correspondence between the arguments and the models of the given specification is a bijection. It is proved in this paper that the parameterization process is provided by a functor and the subsequent parameter passing process by a natural transformation. Various categorical notions are used, mainly adjoint functors, pushouts and lax colimits.

New soliton solutions and soliton evolvements for the (3+1)-dimensional Burgers system

With the help of the symbolic computation system Maple and the improved projective method and variable separation method, a new family of solitory wave solutions for (3+1)-dimensional Burgers system is derived. Based on the derived solution, some novel soliton structures and soliton evolvements are investigated.

Exact solutions and soliton excitations for the (2+1)-dimensional Bogoyavlenskii-Schiff system

With the help of the symbolic computation system Maple and using the improved projective method and the variable separation method, a series of excitations of the (2+1)-dimensional Bogoyavlenskii-Schiff system is derived. According to the derived solitary wave solution, we obtain some novel localized excitations of the system.

New type infinite sequence exact solutions of the second KdV equation with variable coefficients

To construct a number of new infinite sequence exact solutions of nonlinear evolution equations and to study the two characteristics of constructivity and mechanicalness of the first kind of elliptic equation, new types of solutions and the corresponding Bcklund transformation of the equation are presented. Then the second kind of KdV equation with variable coefficients is chosen as a practical example and three kinds of new infinite sequence exact solutions are obtained with the help of symbolic computation system Mathematica, where are included the smooth soliton-like solutions, the infinite sequence peak soliton solutions, and the infinite sequence compact soliton solutions. The method can be used to search for new infinite sequence exact solutions of other nonlinear evolution equations with variable coefficients.

The Improved Riccati Equation Method and Exact Solutions to mZK Equation

We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

Multiple solitoff solutions and the evolution of (2+1)-dimensional breaking soliton equation

With the help of the symbolic computation system Maple, an extended G'/G method and a variable separation method, new exact solutions of the (2+1)-dimensional breaking soliton equation are derived. With the derived solitary wave solutions, we obtain multi Solitoff localized structures and study the soliton evolution with time.

A method of constructing infinite sequence soliton-like solutions of nonlinear evolution equations

The auxiliary equation method is used to construct the finite new exact solutions of nonlinear evolution equations. To search for infinite sequence soliton-like exact solutions of nonlinear evolution equations, characteristics of constructivity and mechanization of auxiliary equation method are analyzed and summarized. Therefore, the quasi-Bcklund transformation between new solutions of a kind of auxiliary equation with Riccati equation is presented, then (2+1)-dimensional modified dispersive water-wave system is taken as an applicable example to find infinite sequence soliton-like new exact solutions by choosing two kinds of formal solutions of nonlinear evolution equations with the help of symbolic computation system Mathematica, where included are the infinite sequence smooth soliton-like solutions, compact soliton solutions and peak soliton-like solutions.

Symbolic Computation and the Extended Hyperbolic Function Method for Constructing Exact Traveling Solutions of Nonlinear PDEs

On the basis of the computer symbolic system Maple and the extended hyperbolic function method, we develop a more mathematically rigorous and systematic procedure for constructing exact solitary wave solutions and exact periodic traveling wave solutions in triangle form of various nonlinear partial differential equations that are with physical backgrounds. Compared with the existing methods, the proposed method gives new and more general solutions. More importantly, the method provides a straightforward and effective algorithm to obtain abundant explicit and exact particular solutions for large nonlinear mathematical physics equations. We apply the presented method to two variant Boussinesq equations and give a series of exact explicit traveling wave solutions that have some more general forms. So consequently, the efficiency and the generality of the proposed method are demonstrated.

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.

Constructing infinite sequence exact solutions of nonlinear evolution equations

To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding Bäcklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.

The extended auxiliary equation method for the KdV equation with variable coefficients

This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.

ANALYTICAL TREATMENT OF SOME PARTIAL DIFFERENTIAL EQUATIONS ARISING IN MATHEMATICAL PHYSICS BY USING THEExp-FUNCTION METHOD

The Exp -function method with the aid of symbolic computational system can be used to obtain the generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics. In this paper, we study the analytic treatment of the Zakharov–Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. Exact solutions with solitons and periodic structures are obtained.

A symbolic computation system for the calculus of moving surfaces

The calculus of moving surfaces (CMS) provides analytic tools for finding solutions to a wide range of problems with moving surfaces including fluid film dynamics, boundary variation problems, and shape optimization problems. The CMS is an extension of tensor calculus on stationary surfaces to moving surfaces. As with any analytic framework, the complexity of calculations grows rapidly with the order of approximation. This quickly causes problems to become complex enough that hand calculations become error prone or intractable. A symbolic computation system will alleviate these problems, allowing researchers to examine problems that have not been previously solvable. No symbolic calculus system is currently available that supports the CMS. We have developed a prototype symbolic computation system that can solve boundary variation problems with the help of the CMS. Our system has been used to solve a series of model problems of interest to applied mathematicians. The motivation is a boundary problem proposed by Grinfeld and Strang [2] in 2004. What is the series in 1/N for the simple Laplace eigenvalues λN on a regular polygon with N sides? In [2], the idea of expressing λN,n as a series in 1/N was put forth and in [3] the first several terms were computed using the calculus of moving surfaces. The prototype implementation was successfully used to find an error in the fourth term in the series expansion in [3] which was previously computed by hand. The CMS approach to this problem is essentially the same as for Poisson’s equation on polygons, which is used as the model problem in [1]. This problem involves Poisson’s equation ∇i∇u = 1 and is therefore simpler than the eigenvalue equation ∇i∇u = −λu that is at the heart of the model problem. The fundamental simplification is unrelated to the CMS: it comes from the fact that all solution variations un (where n is the order of the variation) satisfy Laplace’s equation in the interior. This makes it easy to solve for un and use the result in the next order of variation. The expression for E1 and the equation for u1 are obtained analytically. Higher order variations follow by direct application of the rules of CMS. The second order variation u2 is governed by the boundary value system

Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations (NEE(s)), by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptic equation is highly studied and new type solutions and Bäcklund transformation are obtained. Then (2+1)-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type.

NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials

NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting (NC) variables; • constructing and solving sum of Hermitian squares (with commutators) programs for polynomials in NC variables. It can be used in combination with semidefinite programming software, such as SeDuMi, SDPA or SDPT3, to solve these constructed programs. This paper provides an overview of the theoretical underpinning of the sum of Hermitian squares (with commutators) programs and provides a gentle introduction to the primary features of NCSOStools.

A generalized analytical solution for a nonlinear infiltration equation using the exp-function method

The infiltration problem is one of the most interesting issues considered by geotechnical and water engineers. Many researchers have studied the infiltration problem and have developed models that can be categorized by analytical and numerical concepts. For nonlinear infiltration simulation, however, analytical solutions are few due to the difficulties and complexities involved. The Richards equation is one of the most well-known equations to describe the behavior of unsaturated infiltration zones in soil; many other relations have been introduced based on this equation. The exp-function method is one of the most recent analytical approaches used for the solution of nonlinear Partial Differential (or algebraic) Equations (PDE). In this paper, the exp-function method, with the aid of symbolic computation systems, in particular Maple, has been applied to the Richards equation to evaluate its effectiveness and reliability, and to reach a more generalized solution of the problem. Free parameters can be determined using initial or boundary conditions and the soil water content at any given time and depth is determined in a semi-infinite and unsaturated porous medium. It is shown that the exp-function method applied here results in a more realistic solution and that the concept is very effective and convenient.