Symbolic Computation System Research Articles

COMPARATIVE STUDY OF WAVELET METHODS FOR SOLVING BERNOULLI’S EQUATIONS

A comparative study of two numerical techniques is presented for solving nonlinear differential equations of the Bernoulli’s type. Proposed techniques are based on the conversion of nonlinear differential equations into linear differential equations by substituting particular factor and utilization of Haar wavelet collocation method (HWCM) and Hermite wavelet collocation method (HeWCM) to these linear equations. Searching for numerical solutions of such equations has attracted a considerable amount of research work where computer symbolic systems facilitate the computational work.

Riemann–Hilbert approach of the Newell-type long-wave–short-wave equation via the temporal-part spectral analysis

In this paper, the Riemann–Hilbert approach is systematically established for the Newell-type long-wave–short-wave equation. Firstly, we start spectral analysis from the t-part of the Lax pair other than the x-part to obtain enough analytic spectral functions formulating the desired Riemann–Hilbert problems. The associated Riemann–Hilbert problems are determined by the t-part of the Lax pair with the x-part playing only an auxiliary role. Secondly, the zero structures of the Riemann–Hilbert problems are obtained, from which Riemann–Hilbert problems corresponding to the reflectionless cases are explicitly solved. Then, based on the solutions of the Riemann–Hilbert problems, a general form of multi-soliton solutions is derived for the long-wave–short-wave equation. Thirdly, some interesting soliton dynamics of the obtained multi-soliton solutions are explored and then demonstrated via the symbolic computation system Mathematica. The Riemann–Hilbert approach proposed in this paper allows us to investigate the complicated spectral structure of the long-wave–short-wave equation and is very efficient to derive the general multi-soliton solutions of the equation.

NEW SOLITON SOLUTIONS OF NONLINEAR SPACE-TIME FRACTIONAL (2+1)- DIMENSIONAL ABLOW KAUP-NEWELL_SEGUR (AKNS) EQUATION BY STANDARD AND IMPROVED (G'/G) -EXPANSION METHOD

In this essay, we offer the standard and improved expansion method to seek right solutions of nonlinear fractional differential equation (2+1)-dimensional Ablow Kaup-Newell_Segur (AKNS) with the fractional complex transform. The fractional derivative is defined in the sense of modified Riemann-Liouville derivative. The solutions of this equation are acquired through of the hyperbolic, trigonometrical and rational functions. It has been shown the expansion method award several novel results which are effectual and easy to calculate by the assist of a symbolic computation system.

Proceedings Tenth International Workshop on Computing with Terms and Graphs

This volume contains a selection of the papers presented at TERMGRAPH 2018, the tenth edition of the international workshop on computing with terms and graphs. Graphs, and graph transformation systems, are used in many areas within Computer Science: to represent data structures and algorithms, to define computation models, as a general modelling tool to study complex systems, etc. Research in this area addresses a range of theoretical and practical issues, including the modelling of first- and higher-order term rewriting by (acyclic or cyclic) graph rewriting, graphical frameworks such as interaction nets and sharing graphs (optimal reduction), rewrite calculi for the analysis of functional programs, graph reduction implementations of programming languages, graphical calculi modelling concurrent and mobile computations, object-oriented systems, graphs as a model of biological or chemical systems, and automated reasoning and symbolic computation systems working on shared structures. Previous editions of TERMGRAPH took place in Barcelona (2002), Rome (2004), Vienna (2006), Braga (2007), York (2009), Saarbrucken (2011), Rome (2013), Vienna (2014) and Eindhoven (2016). TERMGRAPH 2018 is affiliated with FSCD, which is part of FLOC.

SOLITARY WAVE SOLUTIONS TO THE TZITZÉICA TYPE EQUATIONS OBTAINED BY A NEW EFFICIENT APPROACH

The properties of Tzitzéica equations in nonlinear optics have received a great attention of many recent studies. In this work, the so-called generalized exponential rational function method (GERFM) has been applied for finding the analytical solution of two nonlinear partial differential equations type of equations, namely Tzitzéica-Dodd-Bullough and Tzitzéica equation. The proposed method provides a wide range of closed-form travelling solutions leading to a very effective and simply-applied method by means of a symbolic computation system. The method not only provides a general form of solutions with some free parameters but also shows potential application to other types of nonlinear partial differential equations.

A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The Riccati-Bernoulli Sub-ODE method appears to be easier and more convenient by means of a symbolic computation system.

On the Motion of a Symmetrical Vehicle with Omniwheels with Massive Rollers

The dynamics of a symmetrical vehicle with omniwheels, moving along a fixed, absolutely rough horizontal plane, is considered, making the following assumptions: the mass of each roller is nonzero, there is a point contact between the rollers and the plane, and there is no slip. The equations of motion composed with the use of the Maxima symbolic computation system, contain additional terms, proportional to the axial moment of inertia of the roller and depending on angles of rotation of the wheels. The mass of the rollers is taken into account in those phases of motion when there is no change of rollers at the contact. The mass of rollers is considered to be negligible when wheels change from one roller to another. It is shown that a set of motions, existing in the inertialess model (i.e., the model that does not take into account mass of rollers), disappears, as well as its linear first integral. The main types of motion for a symmetrical three-wheeled vehicle, obtained by a numerical integration of equations of motion, are compared with results obtained on the basis of the inertialess model.

Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

Abstract In this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

An integro-differential structure for Dirac distributions

We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota–Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems handling such objects. In particular, we show that the Green's function of regular boundary problems (for linear ordinary differential equations) can be expressed naturally in the new setting and that it is characterized by the corresponding distributional differential equation known from analysis.

Exact solutions of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by improved $${\textbf{tan}} \left( {\frac{{\boldsymbol{\phi}} \left( {\boldsymbol{\xi}} \right)}{{\textbf{2}}}} \right)$$tanϕξ2-expansion method

This paper carries out exact solutions of the perturbed nonlinear Schodinger’s equation withy Kerr law nonlinearity by using the improved tan $$\left( {\frac{\phi \left( \xi \right)}{2}} \right)$$ -expansion method. The exact solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be easier and faster by means of symbolic computational system and can be applied to the other nonlinear evolution equations in mathematical physics.

The study of lump solution and interaction phenomenon to $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional generalized fifth-order KdV equation

In this paper, based on the Hirota bilinear method, a kind of lump solutions and two classes of interaction solutions are discussed to the $$(2+1)$$ -dimensional generalized KdV equation with the aid of symbolic computation system Mathematica. Analyticity is naturally guaranteed by taking special choices of the involved parameters to achieve a positive constant term. Particularly, these solutions with special values of the included parameters are plotted, as illustrative example.

The new types of wave solutions of the Burger's equation and the Benjamin–Bona–Mahony equation

In this article, we suggest the two variable (G′/G, 1/G)-expansion method for extracting further general closed form wave solutions of two important nonlinear evolution equations (NLEEs) that model one-dimensional internal waves in deep water and the long surface gravity waves of small amplitude propagating uni-directionally. The method can be regarded as an extension of the(G′/G)-expansion method. The ansatz of this extension method to obtain the solution is based on homogeneous balance between the highest order dispersion terms and nonlinearity which is similar to the (G′/G) method whereas the auxiliary linear ordinary differential equation (LODE) and polynomial solution differs. We applied this method to find explicit form solutions to the Burger's and Benjamin–Bona–Mahony (BBM) equations to examine the effectiveness of the method and tested through mathematical computational software Maple. Some new exact travelling wave solutions in more general form of these two nonlinear equations are derived by this extended method. The method introduced here appears to be easier and faster comparatively by means of symbolic computation system.

Virtual Reading: The Prospero Project Redux

Virtual reading is proposed as a computational strategy for investigating the structure of literary texts. A computer ‘reads’ a text by moving a window N-words wide through the text from beginning to end and follows the trajectory that window traces through a high-dimensional semantic space computed for the language used in the text. That space is created by using contemporary corpus-based machine learning techniques. Virtual reading is compared and contrasted with a 40 year old proposal grounded in the symbolic computation systems of the mid-1970s. High-dimensional mathematical spaces are contrasted with the standard spatial imagery employed in literary criticism (inside and outside the text, etc.). The “manual” descriptive skills of experienced literary critics, however, are essential to virtual reading, both for purposes of calibration and adjustment of the model, and for motivating low-dimensional projection of results. Examples considered: Heart of Darkness, Much Ado About Nothing, Othello, The Winter’s Tale.

Exact Soliton and Kink Solutions for New (3+1)-Dimensional Nonlinear Modified Equations of Wave Propagation

AbstractWe present exact solutions for new (3+1)-dimensional nonlinear equations of wave propagation and fluids. We use the tanh/sech method combined with a computer symbolic system for a reliable treatment of this work. We determine a variety of exact solutions for each equation that contains soliton, kink and periodic solutions. The constraints that guarantee the existence of soliton and kink solutions were examined and found to be related to the coefficients k, r and s of the space variables x, y, and z respectively.

New exact solutions of some nonlinear evolution equations of pseudoparabolic type

This paper aims to conduct an analytical study into some nonlinear models of pseudoparabolic type, including the Oskolkov, Oskolkov–Benjamin–Bona–Mahony–Burgers, and Benjamin–Bona–Mahony–Peregrine–Burgers equations. A number of new exact solutions for these pseudoparabolic type equations have been derived based on the modified Kudryashov method that its calculations are performed in a symbolic computation system known as Maple.

Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids

AbstractThe aim of this paper is to obtain the exact solutions of the strain wave equation applied for illustrating wave propagation in microstructured solids. The effective Kudryashov and functional variable methods along with the symbolic computation system have been used to accomplish the purpose.

SUB-MANIFOLD AND TRAVELING WAVE SOLUTIONS OF ITO'S 5TH-ORDER MKDV EQUATION

In this paper, we study Ito's 5th-order mKdV equation with the aid of symbolic computation system and by qualitative analysis of planar dynamical systems. We show that the corresponding higher-order ordinary differential equation of Ito's 5th-order mKdV equation, for some particular values of the parameter, possesses some sub-manifolds defined by planar dynamical systems. Some solitary wave solutions, kink and periodic wave solutions of the Ito's 5th-order mKdV equation for these particular values of the parameter are obtained by studying the bifurcation and solutions of the corresponding planar dynamical systems.

具任意次非线性项的Camassa-Holm方程的双孤子新解

The method combining the auxiliary equation, the function transformation and the variable separation solutions was proposed to construct the new 2-soliton and 2-period solutions to the arbitrary order nonlinear Camassa-Holm equation. Step 1, with 2 auxiliary equations, the function transformation and the variable separation solutions, the problem of solving the arbitrary order nonlinear Camassa-Holm equation was transformed to the problem of solving the nonlinear algebraic equations. Step 2, by means of symbolic computation system Mathematica, the solutions to the algebraic equations were obtained, and with the help of the relative conclusions on the auxiliary equation, the new 2-soliton and 2-period solutions were constructed.

A New Modification in Simple Equation Method and its applications on nonlinear equations of physical nature

Abstract The modification of the expansion method, namely, the New Modified Simple Equation Method is presented and subsequently implemented on nonlinear partial differential equations. The implementation of the new modification is demonstrated by solving Landau-Ginburg-Higgs equation and Cahn-Allen equation, as a result many new and more general multiple wave solution consisting of Hyperbolic function solutions, trigonometric function solutions and rational function solutions are obtained. Also double solitary wave solutions are found by substituting the different values for some parameters. The method appears to be straightforward and powerful by means of symbolic computational system.

Soliton Solutions for Nonlinear Evolution Equations with Symbolic Computation

The study of exact solutions of nonlinear evolution equations plays an important role in soliton theory. In this paper, we concentrate on proposing to a new Wronskian condition for the Levi equations by utilizing the Wronskian technique, which will lead to a way of constructing new double Wronskian representation of solutions for the Levi equations. Furthermore, some soliton solutions from the particular form of Wronskian determinant solutions are obtained with the aid of the symbolic computation system Maple.