Symbolic Computation System Research Articles

Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method

Abstract Based on the symbolic computation system Maple , Adomian decomposition method, developed for differential equations of integer order, is directly extended to derive explicit and numerical solutions of the fractional KdV–Burgers equation. The fractional derivatives are described in the Caputo sense. According to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional KdV–Burgers equation with time- and space-fractional derivatives. Finally, the solutions of our model equation are calculated in the form of convergent series with easily computable components.

New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov–Kuzentsov equation

In this paper, based on hyperbolic tanh-function method and homogeneous balance method and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov–Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.

Symbolic computation of exact solutions for the (2+1)-dimensional generalized stochastic KP equation

Abstract In this paper, the F -expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.

New exact solitary-wave solutions for the K(2,2,1) and K(3,3,1) equations

Abstract In this paper, K(2, 2, 1) equation: ut + (u2)x − (u2)xxx + u5x = 0 and K(3, 3, 1) equation: ut + (u3)x − (u3)xxx + u5x = 0, are investigated. New exact solitary solutions are developed by using the decomposition method and symbolic computation system, Maple.

Generalized algebraic method and new exact traveling wave solutions for (2 + 1)-dimensional dispersive long wave equation

With the help of the symbolic computation system Maple, a new generalized algebraic method to uniformly construct solutions in terms of special function of nonlinear partial differential equations is presented by means of a more general ansatz. As an application of the method, we choose a (2 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by the method proposed by Fan [E.G. Fan, Phys. Lett. A 300 (2002) 243] and find other new and more general solutions at the same time, which include polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic, and soliton solutions, Jacobi, and Weierstrass doubly periodic wave solutions.

Generalized Q– S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh–Rose systems

Based on symbolic computation system Maple and Lyapunov stability theory, a systematic, powerful and concrete scheme is extended to study the generalized Q– S (lag, anticipated and complete) synchronization between two identical modified Chua’s circuit with different initial values and between two different chaotic systems: Hindmarsh–Rose system and modified Chua’s circuit. Numerical simulations are used to verify the effectiveness of the obtained controller.

A unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations

A unified basic frame of rational expansion methods is presented, which leads to closed-form solutions of nonlinear evolution equations (NLEEs). The new unified algorithms are given to find exact rational formal polynomial solutions of NLEEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some physical significance NLEEs to illustrate the methods. As a consequence, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and tanh methods, but also find other new and more general solutions at the same time.

AN EFFICIENT METHOD FOR FINDING THE EXACT SOLUTION OF NONLINEAR EVOLUTION EQUATIONS

In this paper, based on the idea of the homogeneous balance method, the special truncated expansion method is improved. The Burgers-KdV equation is discussed and its many exact solutions are obtained with the computerized symbolic computation system Mathematica. Our method can be applied to finding exact solutions for other nonlinear partial differential equations too.

Symbolic computation of exact solutions expressible in rational formal hyperbolic and elliptic functions for nonlinear partial differential equations

Abstract With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.

Soliton-like and periodic form solutions to (2+1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE , we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.

Computing with locally effective matrices

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set ℤℕ × ℕ = {f: ℕ × ℕ → ℤ} ). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated.

Exact solutions for the coupled Klein–Gordon–Schrödinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrodinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

Abundant families of Jacobi elliptic function-like solutions for a generalized variable coefficients 2D KdV equation via the extended mapping method

Based on the symbolic computation system Mathematica and an extended mapping method, we study a generalized variable coefficients 2D KdV equation and obtain new families of exact solutions, which include Jacobi elliptic function-like, combined Jacobi elliptic function-like, soliton-like, kink-bell soliton-like and triangular function-like solutions. The solutions of the well-known cylindrical Kadomtsev–Petviashvili equation can be recovered as special case of the 2D KdV equation.

Some Features of the Computer-aided Derivation of the Motion Equations of a Package of Mechanical Systems and Their Decomposition

Computer-aided derivation of the motion equations of a package of mechanical systems was considered. Its mathematical support was based on the F.M. Kulakov method. The freeflying space robotic module for servicing the satellite stations was discussed as an example of the package of mechanical systems. The right side of the resulting mathematical model generated by the symbolic computer system such as Maple was represented as easy-to-use relations for decomposition of the object motion equations. These relations underlie decomposition of the mathematical model of motion of the space robotic module.

A modified hyperbolic function method and its application to the Toda lattice

In this paper, based on the computer symbolic system--Maple, we presented a modified hyperbolic function method to search for the exact solutions of nonlinear differential-difference equations (DDEs). The discrete Toda lattice is chosen to illustrate the method, and abundant new exact solutions are obtained, which include kink-shaped solitary wave solutions and other new types of exact solutions. This method can also be applied to many other nonlinear differential-difference equations (DDEs) in mathematics physics.

Exact solitary-wave solutions with compact support for the modified KdV equation

In this paper, the modified KdV equation, ut + u2ux + uxxx = 0, which exhibits compactons: solitons with compact support, is investigated. New solitary solutions with compact support are developed by using the decomposition method and symbolic computation system, Maple.

New exact solutions to the generalized coupled Hirota–Satsuma Kdv system☆

Abstract In this paper, the projective Riccati equations method is improved. The improved method can be applied to find more and new exact solutions to the nonlinear evolution equations with the aid of symbolic computation system like Maple. Some of the solutions are found for the first time.

Exact travelling wave solutions of nonlinear evolution equations in [formula omitted] and [formula omitted] dimensions

Based upon the symbolic computation system Maple, the projective Riccati equation method is improved to seek for travelling wave solutions in ( 1 + 1 ) and ( 2 + 1 ) -dimensional nonlinear evolution equations by introducing two new elementary bell-shaped and kink-shaped functions. The validity and reliability of the method is tested by applying it to the ( 2 + 1 ) -dimensional generalized of the shallow water wave equation, potential KdV equation and Burgers–KdV equation.

Dealing with algebraic expressions over a field in Coq using Maple

We describe an interface between the Coq proof assistant and the Maple symbolic computation system, which mainly consists in importing, in Coq, Maple computations regarding algebraic expressions over fields. These can either be pure computations, which do not require any validation, or computations used during proofs, which must be proved (to be correct) within Coq. These correctness proofs are completed automatically thanks to the tactic Field, which deals with equalities over fields. This tactic, which may generate side conditions (regarding the denominators) that must be proved by the user, has been implemented in a reflexive way, which ensures both efficiency and certification. The implementation of this interface is quite light and can be very easily extended to get other Maple functions (in addition to the four functions we have imported and used in the examples given here).

Exact Traveling Wave Solutions for the Modified Kawahara Equation

The mapping method is used with the aid of the symbolic computation system Mathematica for constructing exact solutions of the modified Kawahara equation. By this method the modified Kawahara equation is investigated and new exact traveling wave solutions are obtained. The solutions obtained in this paper include Jacobi elliptic solutions, combined Jacobi elliptic solutions, solitary wave solutions, periodic wave solutions, trigonometric solutions and rational solutions.