Use Of Symbolic Computation Research Articles

New Solitary Wave Patterns of the Fokas System in Fiber Optics

The Fokas system, which models wave dynamics using a single model of fiber optics, is the design under discussion in this study. Different types of solitary wave solutions are obtained by utilizing generalized Kudryashov (GKP) and modified Kudryashov procedures (MKP). These novel concepts make use of symbolic computations to come up with a dynamic and powerful mathematical approach for dealing with a variety of nonlinear wave situations. The results obtained in this paper are original and have the potential to be useful in mathematical physics.

Thinking about the oceanic shallow water via a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system

Currently, fluid mechanics has been paid attention to. Hereby, making use of symbolic computation, we investigate a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system describing, e.g., the dispersive long waves in the oceanic shallow water. As for, e.g., the horizontal velocity of the water wave and height of the deviation from the equilibrium position of the water, we work out (1) two sets of the hetero-Bäcklund transformations, each of which, from that system to a known linear partial differential equation, and (2) two sets of the similarity reductions, each of which, from that system to a known ordinary differential equation. Our hetero-Bäcklund transformations and similarity reductions depend on the coefficients in that system, as for, e.g., the oceanic shallow water.

Similarity reductions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid dynamics

Rather intriguing, the paper Chin. J. Phys. 73 (2021) 600-612 has studied a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics, while fluid dynamics has a wide range of applications, including those for geophysics, mechanical engineering, civil engineering, chemical engineering, astrophysics and biology. In this paper, taking into consideration certain nonlinear waves in fluid dynamics, we investigate a generalized variable-coefficient version of the aforementioned equation. Making use of symbolic computation, with respect to the amplitude or elevation of the relevant wave, we construct out two sets of the similarity reductions, which rely on the variable coefficients in the generalized equation.

Oceanic long-gravity-water-wave investigations on a variable-coefficient nonlinear dispersive-wave system

Oceanic water waves are actively investigated. One example is a variable-coefficient nonlinear dispersive-wave system modelling the long gravity water waves in a shallow oceanic environment. The system describes the surface velocity of the water wave and the wave elevation. Making use of symbolic computation, we perform the Painlevé analysis and work out two sets of the bilinear forms, two sets of the N-soliton solutions and one set of the similarity reductions for the aforementioned system, with N being a positive integer. We also graphically discuss those soliton solutions. What we accomplish should rely on the variable coefficients. This paper could be of some use for the future oceanic studies.

An Approximation of Minimax Control using Random Sampling and Symbolic Computation

Abstract We aim at the synthesis of an approximate minimax control for a system dynamic given in the form x(t) = f(x(t), u (t), d(t)) where x represents the state, u the control (or input) and d a disturbance (or perturbation). The disturbance d(t) is prescribed to a compact domain D on which it can take any value (bounded uncertainty). Our method makes use of symbolic computation to enclose the solutions corresponding to all possible disturbance d (-) ϵ D, together with a simple algorithm of random sampling to select the minimum control u⁎(•). We illustrate the interest of this approximate minimax control on an example of a biochemical reactor.

Almost Exact Computation of Eigenvalues in Approximate Differential Problems

Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

The contemporary powerful mathematical software enables a new approach to handling and manipulating complex mathematical expressions and other mathematical objects. Particularly, the use of symbolic computation leads to new contribution to constructing and analyzing numerical algorithms for solving very difficult problems in applied mathematics and other scientific disciplines. In this paper we are concerned with the problem of determining multiple zeros when the multiplicity is not known in advance, a task that is seldom considered in literature. By the use of computer algebra system Mathematica, we employ symbolic computation through several programs to construct and investigate algorithms which both determine a sought zero and its multiplicity. Applying a recurrent formula for generating iterative methods of higher order for solving nonlinear equations, we construct iterative methods that serve (i) for approximating a multiple zero of a given function f when the order of multiplicity is unknown and, simultaneously, (ii) for finding exact order of multiplicity. In particular, we state useful cubically convergent iterative sequences that find the exact multiplicity in a few iteration steps. Such approach, combined with a rapidly convergent method for multiple zeros, provides the construction of efficient composite algorithms for finding multiple zeros of very high accuracy. The properties of the proposed algorithms are illustrated by several numerical examples and basins of attraction.

Representations and Use of Symbolic Computations in the Theory of Heun Equations

A first-order 2 × 2 system equivalent to the Heun equation is obtained. A deformed Heun equation in symmetric form is presented. Series solutions of this equation are presented. A four-parameter subfamily of deformed confluent Heun equations whose solutions have integral representations is found.

On complete list of affine homogeneous surfaces of (ɛ, 0)-types in the space ℂ3

The problem of description of affine homogeneous real hypersurfaces in complex spaces is an important part of the problem of holomorphic classification of homogeneous manifolds, which has no complete solution till now, even in the 3-dimensional case. The method developed by the authors, based on affine canonical equations and techniques of matrix Lie algebras, allowed them earlier to obtain complete descriptions of two natural classes of affine homogeneous real hypersurfaces in the 3-dimensional complex space. In this paper, we present the complete description of one more class. The description includes examples already known and (obtained with the use of symbolic computations) Lie algebras corresponding to the other homogeneous manifolds belonging to the types under consideration. The main result of the paper is obtained by means of integration of these algebras.

Approximate End-Effector Tracking Control of Flexible Multibody Systems Using Singular Perturbations

In many cases, the design of a tracking controller can be significantly simplified by the use of a 2-degrees of freedom (DOF) control structure, including a feedforward control (i.e., the inversion of the nominal system dynamics). Unfortunately, the computation of this feedforward control is not easy if the system is nonminimum-phase. Important examples of such systems are flexible multibody systems, such as lightweight manipulators. There are several approaches to the numerical computation of the exact inversion of a flexible multibody system. In this paper, the singularly perturbed form of such mechanical systems is used to give a semianalytic solution to the tracking control design. The control makes the end-effector to even though not exactly, but approximately track a certain trajectory. Thereby, the control signal is computed as a series expansion in terms of an overall flexibility of the bodies of the multibody system. Due to the use of symbolic computations, the main calculations are independent of given parameters (e.g., the desired trajectories), such that the feedforward control can be calculated online. The effectiveness of this approach is shown by the simulation of a two-link flexible manipulator.

Symbolic Reduction for High-Speed Power System Simulation

High-speed simulations of power transmission systems, which often rely on solving nonlinear systems of equations, are an increasingly important tool for training, testing equipment, on-line control and situational awareness. Such simulations, however, suffer from two major problems: (1) they can be too computationally demanding to simulate large, complex systems within appropriate time constraints; and (2) they are difficult to develop and debug. Prior work has shown how symbolic computation can be used to help reduce both problems. In this paper, we: (1) review common concepts in power system simulations; (2) summarize prior use of symbolic computation in power system simulation; (3) explore the advantages and disadvantages achieved via symbolic techniques; (4) extend the techniques to solve linear systems via a priori symbolic LU decomposition; and (5) demonstrate the advantages of symbolic techniques on a transient event simulation of the IEEE 118-bus test power system, which runs in one-tenth the time of an equivalent traditional (sparse matrix) approach.

Introducing the Analysis of Bifurcation in Dynamical Systems by Symbolic Computation

The aim of this paper is to introduce a few topics about nonlinear systems that are usual in electrical engineering but are frequently disregarded in undergraduate courses. More precisely, the main subject of this paper is to present the analysis of bifurcations in dynamical systems through the use of symbolic computation. The necessary conditions for the occurrence of Hopf, saddle-node, transcritical or pitchfork bifurcations in first order state space nonlinear equations depending upon a vector of parameters are expressed in terms of symbolic computation. With symbolic computation, the relationship between the state variables and the parameters that play a crucial role in the occurrence of such phenomena can be established. Firstly, the symbolic computation is applied to a third order dynamic Lorenz system in order to familiarise the students with the technique. Then, the symbolic routines are used in the analysis of the simplified model of a power system, bringing new insights and a deeper understanding about the occurrence of these phenomena in physical systems.

Families of non-travelling wave solutions to a generalized variable coefficient two-dimensional KdV equation using symbolic computation

Abstract Making use of symbolic computation and the generalized Riccati equation expansion method, some exact non-travelling wave solutions for a generalized variable coefficients two-dimensional KdV equation are obtained. By means of some suitable selections of the arbitrary functions including in the obtained solutions, the results obtained by Elwakil et al. [see: Chaos, Solitons & Fractals 19 (2004) 1083] can be recovered. From our results, some exact solutions for the cylindrical Kadomatsev–Petviashvilli equation can be also derived.

Families of non-travelling wave solutions to a generalized variable coefficient two-dimensional KdV equation using symbolic computation

Making use of symbolic computation and the generalized Riccati equation expansion method, some exact non-travelling wave solutions for a generalized variable coefficients two-dimensional KdV equation are obtained. By means of some suitable selections of the arbitrary functions including in the obtained solutions, the results obtained by Elwakil et al. [see: Chaos, Solitons & Fractals 19 (2004) 1083] can be recovered. From our results, some exact solutions for the cylindrical Kadomatsev–Petviashvilli equation can be also derived.

Symbolic calculation of the moments of the time of ruin

Lin and Willmot [Insurance: Mathematics and Economics 27 (2000) 19] derived a set of defective renewal difference equations involving the moments of the time of ruin in the classical insurance risk model. In principal, this allows for evaluation of these moments. Even with their result, however, it becomes apparent that obtaining even the first two moments is complicated due to the recursive nature of the relation and sheer volume of algebra. To combat these difficulties and obtain higher order moments, we explore the use of symbolic computation as a tool to automate the algebraic detail. In particular, we present a Mathematica program that algebraically calculates the moments of the time of ruin.

A New One-Pass Transformation into Monadic Normal Form

We present a translation from the call-by-value lambda-calculus to monadic normal forms that includes short-cut boolean evaluation. The translation is higher-order, operates in one pass, duplicates no code, generates no chains of thunks, and is properly tail recursive. It makes a crucial use of symbolic computation at translation time.

New computational procedure to the pole placement problem by static output feedback

The authors report on a study of the pole placement problem via static output feedback for linear time-invariant multivariable systems from a geometric point of view. The main tool for this analysis is a novel formulation of the equations characterising the solution. This allows an explicit expression of the output feedback gain matrix K. A computational procedure to find the matrix K is given that does not require restrictive assumptions on the open- and closed-loop system. The algorithm makes use of symbolic computation, performed using the software MATHEMATICA(R).

Symbolic computation with Maple V for undergraduate electromagnetics

Although symbolic analysis has been successfully used as an educational tool for mathematics, circuits, signals and systems courses, its value in electrical engineering education is still a controversial issue. The paper presents the use of symbolic computation in electromagnetics, where the complexity of the involved mathematics often prevents the students from understanding the underlying physical aspects. The educational strategy the authors propose is complementary to the numerical approach. It is based on actual symbolic calculation of electromagnetic field, not just on field visualization. An example of a Maple V worksheet dedicated to the computation of magnetic field with the Biot-Savart-Laplace method (BSL) is presented. The worksheet contains a theoretical introduction, 8 solved problems (equations, text and plots) and 12 proposed projects. After going through several solved problems, the students are asked to make changes in the worksheet and to discuss the results. They are finally required to conceive their own worksheets for solving new proposed problems, using the BSL worksheet as a model. Their experience shows that the symbolic computation has an outstanding educational potential not only in mathematics, circuits and systems, but also in electromagnetics.

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar $\phi^3$ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension $d=4$, we obtain an integrodifferential Dyson-Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at $d=6$, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator-coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.