Symbolic Computation Techniques Research Articles

Numerical eigenanalysis of continuum geodesic acoustic mode by ideal magnetohydrodynamic model

In this study, we have developed a new eigenvalue code to solve the three-variables ideal magnetohydrodynamic (MHD) equation in flux coordinate system with the help of a symbolic vector analysis module and an automatic numerical discretization module, which were developed with the use of symbolic computation technique and were greatly facilitated the development of toroidal full MHD eigenvalue code SCELT. It has been used to solve for unstable modes and the present work is dedicated to the more difficult challenge of MHD continua, which ordinarily come with spatial singularities. With this new code, we were able to perform numerical calculations to determine the continuum spectrum of geodesic acoustic modes (GAMs), as well as obtain eigenfunctions for both the perturbed plasma displacement and perturbed magnetic field, incorporating a sufficient number of components. Through our global calculations, we have provided numerical evidence of singularities in the eigenfunctions, characterized by the type s−s0−1 and Ins−s0 . Moreover, we have examined the spatial variations of GAM fluctuations in regions both near and far from the singular surface. Here s represents a flux function that characterizes the magnetic flux surfaces and s=s0 defines the singular surface. Additionally, we have investigated the effects of finite aspect ratio, non-circular cross sections as well as the asymmetry on the frequency and eigenfunction of continuum GAMs through numerical analysis.

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Abstract The (3+1)-dimensional Geng equation is an extended version of the KdV model that describes the wave dynamics behavior of shallow water waves in complex applications. In this study, we discuss the (3+1)-dimensional Geng equation using the bilinear neural network method. By incorporating specific activation functions into the neural network model, new test functions are constructed. Using symbolic computational techniques and selecting appropriate parameters, we systematically obtain new meaningful exact solutions of some (3+1)-dimensional Geng equations, including dark lump solutions, three kinds of interaction solutions, and bright and dark soliton solutions. Furthermore, the results are visualized through diagrams of different categories, which intuitively demonstrate the evolution process and physical characteristics of the waves.

A Symbolic Algorithm for Solving Doubly Bordered k-Tridiagonal Interval Linear Systems

Doubly bordered k-tridiagonal interval linear systems play a crucial role in various mathematical and engineering applications where uncertainty is inherent in the system’s parameters. In this paper, we propose a novel symbolic algorithm for solving such systems efficiently. Our approach combines symbolic computation techniques with interval arithmetic to provide rigorous solutions in the form of tight interval enclosures. By exploiting the tridiagonal structure and employing a divide-and-conquer strategy, our algorithm achieves significantly reduced computational complexity compared to existing numerical methods. We also present theoretical analysis and provide numerical experiments to demonstrate the effectiveness and accuracy of our algorithm. The proposed symbolic algorithm offers a valuable tool for handling doubly bordered k-tridiagonal interval linear systems and opens up possibilities for addressing uncertainty in real-world problems with improved efficiency and reliability.

A symbolic approach to multiple Hurwitz zeta values at non-positive integers

Abstract In this article, we give another method to calculate the values of multiple Hurwitz zeta function at non-positive integers by means of Raabe’s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.

Application of the automatic code generation and symbolic computation techniques to numerical solution of PDE problems

A software package for the numerical solution of systems of linear and nonlinear partial differential equations by grid methods is presented, which combines symbolic computations and automatic code generation techniques. Discretization of differential equations is carried out by the methods of finite volumes and finite differences on two-dimensional and three-dimensional regular rectangular grids. The solution of the systems of nonlinear algebraic equations obtained as a result of discretization is performed by Newton’s method with employment of third-party SLAE solvers. The generated code can exploit shared and distributed memory parallelisms and take advantage of heterogeneous computations. The results of the numerical solutions of two sample CFD problems of the fluid flow generation in a cylindrical cavity filled with electrically conductive liquid driven the by traveling and rotating magnetic fields are presented.

SCELT (symbolic computation aided eigenvalue and linear code for tokamaks): A full MHD eigenvalue code in toroidal geometry developed with the use of a symbolic computation technique

In this work, we report the construction of an eigenvalue computer code largely in C++ language, SCELT, by using the symbolic computation technique for the first time to solve the linearized single fluid magnetohydrodynamic (MHD) eigenvalue problem in toroidal geometry. A symbolic vector analysis module is developed to function the automatic derivation of the tedious linearized full MHD equations in the magnetic flux coordinate system. Furthermore, another module is developed to implement the automatic numerical discretization. These two modules dramatically reduce the human workload and obviate the possibility of a mistake during code development. The tools provide a means of constructing matrices from differential operations and can be used for (generalized) linear problems, such as source driven and eigenvalue problems. Demo uses of both the symbolic vector analysis module and automatic numerical discretization module, such as the Poisson equation and tokamak equilibrium equation, are presented to demonstrate their advantages and potential broad applications. The full MHD eigenvalue code developed with these two modules is verified by the internal kink mode and tearing mode tests. Program summaryProgram Title: SCELT (Symbolic Computation aided Eigenvalue and Linear code for Tokamaks)CPC Library link to program files:https://doi.org/10.17632/35h3xmc28k.1Licensing provisions: GNU General Public License 3.0Programming language: C++, MATLABExternal routines/libraries:• EIGEN (https://eigen.tuxfamily.org/index.php?title=Main_Page)Nature of problem: The eigenvalue full MHD model determines a wide range of physical phenomena arising from the basic destabilizing forces, which are of considerable both fundamental and practical importance for magnetic fusion plasmas. From the viewpoint of physics, solving the model is equivalent to finding the spectrum of the system, and for realistic tokamak geometry, it becomes possible only with the help of computer simulation. From the viewpoint of mathematics, it is actually a problem of solving a set of partial differential equations in a general curvilinear coordinate system.Solution method: Different from the conventional method, a symbolic vector analysis module using the symbolic computation technique is developed to conduct the automatic expansion of the full MHD equations in the magnetic flux coordinate system. In addition, an automatic numerical discretization module is developed to automatically implement the numerical discretization. Facilitated by the two modules, the coefficients of the matrices of the eigenvalue equation will be automatically generated and passed to the linear eigenvalue solver (eig function of MATLAB). The eigenvalues and eigenvectors are resolved by the solver.Additional comments including restrictions and unusual features: As originally motivated by the development of an eigenvalue full MHD code, a symbolic vector analysis module by using symbolic computation technique and a closely related automatic numerical discretization module are developed and directly applied to physics code development for the first time at least in the fusion field. Many (generalized) linear physics problems can benefit from the standalone or integrated application of these two modules. The tokamak equilibrium and eigenvalue MHD problems presented in the paper can be considered two important and practical examples.

Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach

The paper investigates the multiple rogue wave solutions associated with the generalized Hirota–Satsuma–Ito (HSI) equation and the newly proposed extended (3 + 1)-dimensional Jimbo–Miwa (JM) equation with the help of a symbolic computation technique. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The obtained bilinear forms of the proposed equations successfully investigate the 1st, 2nd and 3rd-order rogue waves. The constructed solutions are verified by inserting them into original equations. The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves. Physical properties of these waves are governed by different parameters that are discussed in details.

On the relations between some well-known methods and the projective Riccati equations

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

Solving Stiff Systems by using Symbolic - Numerical Method

In this paper, an efficient symbolic-numerical procedure based on the power series method is presented for solving a system of differential equations. The basic idea is to substitute power series into the differential equations and to find a polynomial system of coefficients, where a powerful symbolic computation technique (i.e., Grobner basis) is used to solve the system. In fact, the proposed method is an excellent bridge between symbolic and numeric computation and specially, enables us to find the solution of linear and non-linear stiff systems. Numerical experiments were performed to justify our new approach.

Super-regular breathers for an inhomogenous optical fiber system

It is critical to excite pulse wave with higher amplitudes and higher frequencies in optical application. In this work, under investigation is an inhomogeneous fiber system described by a generalized variable coefficient Schrödinger equation. Analytical two-breather solution is obtained through Darboux transformation, a symbolic computation technique. The super-regular breathers can be generated via breathers colliding. The effects of amplitude and frequency amplifications can be observed as the spectral eigenvalue parameters are adjusted.

A symbolic approach to multiple zeta values at negative integers

A symbolic computation technique is used to derive closed-form expressions for an analytic continuation of the Euler–Zagier zeta function evaluated at negative integers. This continuation was recently proposed by Sadaoui. The approach presented here yields explicit contiguity identities, recurrences on the depth of the zeta values and their generating functions. Moreover, it allows to prove that the resulting multiple zeta values computed at negative integers coincide with those obtained by another analytic continuation technique that uses the Euler–MacLaurin summation formula.

A hybrid symbolic-numerical approach to the center-focus problem

We propose a new hybrid symbolic-numerical approach to the center-focus problem. The method allowed us to obtain center conditions for a three-dimensional system of differential equations, which was previously not possible using traditional, purely symbolic computational techniques.

A constructive algebraic analysis approach to Artstein's reduction of linear time-delay systems

Artstein's results show that a first-order linear differential system with delayed inputs is equivalent to a first-order linear differential system without delay under an invertible transformation which includes integral and time-delay operators. Within a constructive algebraic approach, we show how this reduction can be found again, generalized and interpreted as a particular isomorphism between modules defining the two above linear systems. Moreover, we prove that Artstein's reduction can be obtained in an automatic way by means of symbolic computation techniques, and thus can be implemented in dedicated computer algebra systems.

Computation of a Canonical Form for Linear 2D Systems

Symbolic computation techniques are used to obtain a canonical form for polynomial matrices arising from discrete 2D linear state-space systems. The canonical form can be regarded as an extension of the companion form often encountered in the theory of 1D linear systems. Using previous results obtained by Boudellioua and Quadrat (2010) on the reduction by equivalence to Smith form, the exact connection between the original polynomial matrix and the reduced canonical form is set out. An example is given to illustrate the computational aspects involved.

Toward Formal Design of Practical Cryptographic Hardware Based on Galois Field Arithmetic

This paper presents a formal method for designing cryptographic processor datapaths on the basis of arithmetic circuits over Galois fields (GFs). The proposed method describes GF arithmetic circuits in the form of hierarchical graph structures, where nodes represent sub-circuits whose functions are defined by arithmetic formulae over GFs, and edges represent data dependency between nodes. In this paper, we first introduce the application of graph representation to arithmetic circuits over extension fields of ${\mbi {GF}}({{\mbi {p}}^{\mbi {m}}})$ $({\mbi {p}} \geq {\bf 2})$ and composite fields, which are commonly used in the design of cryptographic processors. The newly proposed graph representation can be formally verified through symbolic computation techniques based on polynomial reduction and Grobner basis. We then demonstrate the capabilities of the proposed approach through an experimental design of a 128-bit AES (Advanced Encryption Standard) datapath including multiplicative inversion circuits over the composite field ${\mbi {GF}}{(((2^2)^2)^2})$ . The results show that the proposed method can describe such practical datapaths, as well as that complete verification of such a datapath can be carried out within a short period of time.

Solitons, Cnoidal Waves, Snoidal Waves and Other Solutions to Whitham-Broer-Kaup System

Using a computerized symbolic computation technique based on improved Jacobi elliptic function method, we find several solutions for Whitham-Broer-Kaup-Like (WBKL) system. These solutions contain hyperbolic, triangular solutions. When the parameters are taken as special values the solitary wave solutions can be obtained for other systems. The traveling wave solutions are also discussed that obtains solitary wave and singular soliton solution.

Seven mutually touching infinite cylinders

We solve a problem of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other. In fact, we exhibit two such sets of cylinders. Our approach is algebraic and uses symbolic and numerical computational techniques. We consider a system of polynomial equations describing the position of the axes of the cylinders in the 3 dimensional space. To have the same number of equations (namely 20) as the number of variables, the angle of the first two cylinders is fixed to 90 degrees, and a small family of direction vectors is left out of consideration. Homotopy continuation method has been applied to solve the system. The number of paths is about 121 billion, it is hopeless to follow them all. However, after checking 80 million paths, two solutions are found. Their validity, i.e., the existence of exact real solutions close to the approximate solutions at hand, was verified with the alphaCertified method as well as by the interval Krawczyk method.

Nonlinear transformations for the simplification of unconstrained nonlinear optimization problems

Formalization decisions in mathematical programming could significantly influence the complexity of the problem, and so the efficiency of the applied solver methods. This widely accepted statement induced investigations for the reformulation of optimization problems in the hope of getting easier to solve problem forms, e.g. in integer programming. These transformations usually go hand in hand with relaxation of some constraints and with the increase in the number of the variables. However, the quick evolution and the widespread use of computer algebra systems in the last few years motivated us to use symbolic computation techniques also in the field of global optimization. We are interested in potential simplifications generated by symbolic transformations in global optimization, and especially in automatic mechanisms producing equivalent expressions that possibly decrease the dimension of the problem. As it was pointed out by Csendes and Rapcsak (J Glob Optim 3(2):213–221, 1993), it is possible in some cases to simplify the unconstrained nonlinear objective function by nonlinear coordinate transformations. That means mostly symbolic replacement of redundant subexpressions expecting less computation, while the simplified task remains equivalent to the original in the sense that a conversion between the solutions of the two forms is possible. We present a proper implementation of the referred theoretical algorithm in a modern symbolic programming environment, and testing on some examples both from the original publications and from the set of standard global optimization test problems to illustrate the capabilities of the method.

Generating exact nonlinear ranking functions by symbolic-numeric hybrid method

This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments. The authors first transform the problem into a parameterized polynomial optimization problem, and obtain a numerical ranking function using polynomial sum-of-squares relaxation via semidefinite programming (SDP). A rational vector recovery algorithm is deployed to recover a rational polynomial from the numerical ranking function, and some symbolic computation techniques are used to certify that this polynomial is an exact ranking function of the loop programs. At last, the authors demonstrate on some polynomial loop programs from the literature that our algorithm successfully yields nonlinear ranking functions with rational coefficients.

Applications of Symbolic computation in MATLAB

This paper contains brief information about symbolic computation techniques. The location of symbolic computation within Computer Language classification is defined. MATLAB + MuPAD as combined method have been introduced. After some introductory information about MuPAD, some important applications of MATLAB + MuPAD are given. The method, transfer from MuPAD into MATLAB has been discussed as well. These two methods have been illustrated to engineering students on selected topics.