Symbolic Computation System Research Articles

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.

SymGrid-Par

short-paper SymGrid-Par: parallel orchestration of symbolic computation systems Share on Author: The SCIEnce project View Profile Authors Info & Claims ACM Communications in Computer AlgebraVolume 44Issue 3/4September/December 2010 pp 213–216https://doi.org/10.1145/1940475.1940523Published:28 January 2011 0citation496DownloadsMetricsTotal Citations0Total Downloads496Last 12 Months1Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

New infinite sequences exact solutions to sine-Gordon-type equations

In order to obtain infinite sequence exact solutions to the sine-Gordon-type equations, the auxiliary equations of triangular function type and hyperbolic function type, Bäcklund transformation and nonlinear superposition formula of the solutions are presented. And the method is used to construct new infinite sequence exact solutions to the sine-Gordon equation, mKdV-sine-Gordon equation, (n+1)- dimensional double sine-Gordon equation and the sinh-Gordon equation with the help of symbolic computation system Mathematica, which include infinite sequence triangular function solutions, the infinite sequence hyperbolic function solutions, infinite sequence Jacobi elliptic function solutions and infinite sequence computation solutions.

Hyperbolic, Trigonometric, and Rational Function Solutions of Hirota‐Ramani Equation via (G′/G)‐Expansion Method

The (G′/G)‐expansion method is proposed to construct the exact traveling solutions to Hirota‐Ramani equation: ut − uxxt + aux(1 − ut) = 0, where a ≠ 0. 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 peak solitary wave solutions ofDegasperis-Procesi equation

To construct infinite sequence peak solitary wave solutions to nonlinear evolution equations, Bäcklund transformation of Riccati equation and nonlinear superposition formula of the solutions are introduced, then Degasperis-Procesi equation is taken as an example and infinite sequence peak solitary wave solutions and periodic solutions of the equation are obtained with the help of symbolic computation system Mathematica.

A method for constructing infinite sequence complexiton solutions to nonlinear evolution equations

To seek new infinite sequence complexiton solutions to nonlinear evolution equations(NEE(s)), the formula of nonlinear superposition of the solutions and Bcklund transformation of Riccati equation are presented, and as an illusrative exapmle, the generalized Boussinesq equation is chosen to obtain new infinite sequence complexiton solutions with the aid of symbolic computation system Mathematica, which includes complexiton solutions of hyperbolic function, triangular function type with rational function and hyperbolic function with triangular function. The method is of significance to construct infinite sequence complexiton solutions to other NEEs.

Exact solutions and localized excitations of (3+1) -dimensional Jimbo-Miwa equation

With the help of the symbolic computation system Maple and a projective equation and a linear variable separation method, new exact solutions of the (3+1) -dimensional Jimbo-Miwa equation are derived. Based on the derived solitary wave solution, some novel localized excitations are investigated.

General Solution of Generalized (2+1)–Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –Expansion Method

In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. 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.

New infinite sequence exact solutions of nonlinear evolution equations with variable coefficients by the second kind of elliptic equation

In the paper, to construct new infinite sequence exact solutions of nonlinear evolution equations, several kinds of new solutions of the second kind of elliptic equation Bäcklund transformation are proposed. The KdV equation containing variable coefficients and forcible term, combined with (2+1)-dimensional and (3+1)-dimensional Zakharov-Kuznetsov equation with variable coefficients is taken as example to construct new infinite sequence exact solutions of these equations with the help of symbolic computation system Mathematica, which include infinite sequence compact soliton solutions of Jacobi elliptic function and triangular function, and infinite sequence peak soliton solutions.

Exact solutions and soliton structures of (2+1)-dimensionalZakharov-Kuznetsov equation

With the help of the symbolic computation system Maple and an improved Riccati equation mapping approach, a series of exact solutions of the (2+1)-dimensional Zakharov-Kuznetsov equation (ZK) is derived. Based on the derived solution, we obtain some special soliton structures.

New Explicit Solutions of (1 + 1)-Dimensional Variable-Coefficient Broer—Kaup System

By using the compatibility method, many explicit solutions of the (1 + 1)-dimensional variable-coefficient Broer-Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponential function, and Airy function. Some figures of the solutions are given by the symbolic computation system Maple.

The algebraic logic of kinship terminology structures

AbstractJones' proposed application of Optimality Theory assumes the primary kinship data are genealogical definitions of kin terms. This, however, ignores the fact that these definitions can be predicted from the computational, algebralike structural logic of kinship terminologies, as has been discussed and demonstrated in numerous publications. The richness of human kinship systems derives from the cultural knowledge embedded in kinship terminologies as symbolic computation systems, not the post hoc constraints devised by Jones.

New application to Riccati equation

To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov–Kuznetsov equation, Karamoto–Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.

The [formula omitted]-expansion method for Tzitzéica type nonlinear evolution equations

In this work, the ( G ′ G ) -expansion method is applied for constructing more general exact solutions of the three nonlinear evolution equations with physical interest namely, the Tzitzéica equation, the Dodd–Bullough–Mikhailov (DBM) equation and the Tzitzéica–Dodd–Bullough (TDB) equation. 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.

A series of new exact solutions of nonlinear evolution equations

AbstractWith the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Application of [formula omitted]-expansion method to travelling wave solutions of three nonlinear evolution equation

In this work, the G ′ G -expansion method is proposed for constructing more general exact solutions of the three nonlinear evolution equations arising in fluids science with physical interest namely, Vakhnenko–Parkes equation, generalized regularized long wave (RLW) equation and symmetric regularized long wave (SRLW) equation. 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.

A case-study in algebraic manipulation using mechanized reasoning tools

In this article, two different mechanized reasoning tools (Coq and Isabelle/HOL) are used in order to prove some basic but significant properties extracted from a symbolic computation system called Kenzo. In particular, we focused on a property called ‘cancellation theorem’, which can be applied to the proof of the idempotence property of a differential morphism. This result is used as a case-study to compare the capabilities and styles of the two proof assistants. The conclusion of our comparison is that both tools are adequate to solve the case-study presented in this article in a rather similar way but depending on their specific styles. This research is part of a more general project devoted to increase the reliability of computer algebra systems for calculations in algebraic topology.

Application of the [formula omitted]-expansion method to Kawahara type equations using symbolic computation

In this paper, Kawahara type equations are selected to illustrate the effectiveness and simplicity of the G ′ / G -expansion method. With the aid of a symbolic computation system, three types of more general traveling wave solutions (including hyperbolic functions, trigonometric functions and rational functions) with free parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters, involved in the traveling waves, as special values.

Homotopy perturbation technique for solving two-point boundary value problems – comparison with other methods

The two-point boundary value problems occur in a wide variety of problems in engineering and science. In this paper, we implement the homotopy perturbation method for solving the linear and nonlinear two-point boundary value problems. The main aim of this paper is to compare the performance of the homotopy perturbation method with extended Adomian decomposition method and shooting method. As a result, for the same number of terms, the homotopy perturbation method yields relatively more accurate results with rapid convergence than other methods. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

Potential problems with random parameters by the generalized perturbation-based stochastic finite element method

The main aim of this paper is to present the new version and the application of the generalized nth order stochastic perturbation technique to the solution of the field problems with random coefficients. The variational stochastic nth order principles are proposed here using Taylor expansions with random parameters together with the additional stochastic finite element method discretization of the problem; a derivation of up to the fourth probabilistic moments and the coefficients of the potential function are also provided. The formulation is dual in the sense that both Direct Differentiation Method and the Response Function Method in its local version are proposed and implemented using the symbolic computer system MAPLE to determine numerically the random system response and to make the additional visualization. This methodology is illustrated with two examples - 1D viscous irrotational and incompressible fluid flow and 2D torsion of the prismatic rectangular beam with random transverse modulus.