Computer Algebra System Maple Research Articles

A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays

In this paper, the Taylor collocation method has been used the integro functional equation with variable bounds. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. We have introduced the method to solve the functional integral equations with variable bounds. We have also improved error analysis for this method by using the residual function to estimate the absolute errors. To illustrate the pertinent features of the method numeric examples are presented and results are compared with the other methods. All numerical computations have been performed on the computer algebraic system Maple 15.

Design and implementation of maple packages for processing offsets and conchoids

In this paper we present two packages, implemented in the computer algebra system Maple, for dealing with offsets and conchoids to algebraic curves, respectively. Help pages and procedures are described. Also in an annex, we provide a brief atlas, created with these packages, and where the offset and the conchoid of several algebraic plane curves are obtained, their rationality is analyzed, and parametrizations are computed. Practical performance of the implemented algorithms shows that the packages execute in reasonable time; we include time cost tables of the computation of the offset and conchoid curves of two rational families of curves using the implemented packages.

Extended generalized $$(Zakh\frac{G^{\prime }}{G})$$ ( Z a k h G ′ G ) -expansion method for solving the nonlinear quantum Zakharov–Kuznetsov equation

In this article, we apply the extended generalized \((\frac{G^{\prime }}{G})\)-expansion method combined with the Jacobi elliptic equation to find new exact solutions of the nonlinear quantum Zakharov–Kuznetsov (QZK) equation with the aid of computer algebraic system Maple. Soliton solutions, periodic solutions, rational functions solutions and Jacobi elliptic functions solutions are obtained. Based on reductive perturbation technique and a series of transformation, the nonlinear QZK had been derived by many authors which can be reduced to a nonlinear ordinary differential equation (ODE) using the wave transformation. The extended generalized \((\frac{G^{\prime }}{G})\)-expansion method is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.

Algorithm for construction of volume forms on toric varieties starting from a convex integer polytope

This paper presents a method and a corresponding algorithm for constructing volume forms (and related forms that act as kernels of integral representations) on toric varieties from a convex integer polytope. The algorithm is implemented in the Maple computer algebra system. The constructed volume forms are similar to the volume forms of the Fubini---Study metric on a complex projective space and can be used for constructing integral representations of holomorphic functions in polycircular regions of a multidimensional complex space.

Number of Forts in Iterated Logistic Mapping

Using the theory of complete discrimination system and the computer algebra systemMAPLE V.17, we compute the number of forts for the logistic mappingfλ(x)=λx(1-x)on[0,1]parameterized byλ∈(0,4]. We prove that if0<λ≤2then the number of forts does not increase under iteration and that ifλ>2then the number of forts is not bounded under iteration. Furthermore, we focus on the case ofλ>2and give for eachk=1,…,7some critical values ofλfor the change of numbers of forts.

On solving the nonlinear Biswas-Milovic equation with dual-power law nonlinearity using the extended tanh-function method

In this article, we apply the extended tanh-function method to …nd the exact traveling wave solutions of the nonlinear Biswas-Milovic equation (BME), which describes the prop-agation of solitons through optical …bers for trans-continental and trans-oceanic distances. This equation is a generalized version of the nonlinear Schrödinger equation with dual-power law nonlinearity. With the aid of computer algebraic system Maple, both constant and time-dependent coe¢ cients of BME are discussed. Comparison between our new results and the well-known results is given. The given method in this article is straightforward, concise and can be applied to other nonlinear partial di¤erential equations (PDEs) in mathematicalphysics.

Probabilistic effective characteristics of polymers containing rubber particles of Gaussian random diameter

This study is focused on the problem of statistical distribution of the size of rubber particles as fillers in elastomeric composites. This distribution (average diameter of the injected particles) is assumed to be Gaussian and uniquely defined by its mean value as well as standard deviation. The basic probabilistic parameters of the effective elasticity tensor of the entire elastomer are under consideration by using of the homogenization method. The basic computational ideology is based on strain deformation of the Representative Volume Element under uniaxial and biaxial loads. This deterministic method is enriched with the generalized stochastic perturbation technique and also by semi-analytical strategy, which are used together with the system ABAQUS® as the Stochastic Finite Element Method (SFEM) serving for a solution of the homogenization problem for such a composite. The basic stochastic characteristics of the homogenized elasticity tensor and its deterministic sensitivity coefficients are verified with such coming from analytical deterministic homogenization method extended towards random case in the computer algebra system MAPLE®. The computational study contains additionally computational error analysis as the homogenization problem is solved here with tetrahedral and hexahedral 3D solid finite elements with linear as well as with parabolic shape functions and their meshes with different densities.

A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines

The first elliptic function equation is used in this article to find a new kind of solutions of nonlinear partial differential equations (PDEs) based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method. New exact solutions to the Jacobi elliptic functions of a nonlinear PDE describing the nonlinear low-pass electrical lines are obtained with the aid of computer algebraic system Maple. Based on Kirchhoff’s law, the given nonlinear PDE has been derived and can be reduced to a nonlinear ordinary differential equation (ODE) using a simple transformation. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics. Further results may be obtained.

Involutions of polynomially parametrized surfaces

We provide an algorithm for detecting the involutions leaving a surface defined by a polynomial parametrization invariant. As a consequence, the symmetry axes, symmetry planes and symmetry center of the surface, if any, can be determined directly from the parametrization, without computing or making use of the implicit representation. The algorithm is based on the fact, proven in the paper, that any involution of the surface comes from an involution of the parameter space R2; therefore, by determining the latter, the former can be found. The algorithm has been implemented in the computer algebra system Maple 18. Evidence of its efficiency for moderate degrees, examples and a complexity analysis are also given.

Taylor collocation approach for delayed Lotka–Volterra predator–prey system

In this study, a numerical approach is proposed to obtain approximate solutions of the system of nonlinear delay differential equations defining Lotka–Volterra prey–predator model. By using the Taylor polynomials and collocation points, this method transforms the population model into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Taylor coefficients. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results. All numerical computations have been performed on the computer algebraic system Maple 15.

Formal solutions of completely integrable Pfaffian systems with normal crossings

In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffi an systems with normal crossings in several variables. This algorithm is a generalization of a method developed for the bivariate case based on a combination of several reduction techniques and is implemented in the computer algebra system Maple.

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

Reliability analysis of a corrugated I-beam girder with ribs

The main aim of this paper is to provide a reliability analysis of a corrugated-web I girder, including the vertical ribs distributed periodically throughout its length. The generalised stochastic perturbation method and its computer implementation – the stochastic finite-element method – are both used to accomplish this goal. They are numerically implemented using the finite-element method system Abaqus with its rectangular, four-noded, structural, thin-shell finite elements and computer algebra system Maple, in which all the probabilistic procedures are programmed. The perturbation-based results are compared with those from traditional Monte Carlo simulation and, separately, an analytical solution is obtained by way of symbolic integration, carried out also in Maple. The plate girder thickness is introduced as the basic input Gaussian random variable in this study, as the most influential geometrical parameter in this structure. The reliability indices are determined according to both first- and second-order reliability methods, and correspond to the serviceability limit state of the structure. The first four basic probabilistic characteristics of the maximum deflection are also computed – expectations, coefficients of variation, skewness and kurtosis. The results obtained here and the entire methodology may be extended further towards computational modelling of the corrosion process, which is frequently mathematically modelled in steel structures using certain time series with random coefficients.

Equality-constrained minimization of polynomial functions

This paper investigates the equality-constrained minimization of polynomial functions. Let ℝ be the field of real numbers, and ℝ[x1, …, xn] the ring of polynomials over ℝ in variables x1, …, xn. For an f ∈ ℝ[x1, …, xn] and a finite subset H of ℝ[x1, …, xn], denote by \(\mathcal{V}(f:H)\) the set {\(\left. {f\left( {\bar \alpha } \right)} \right|\bar \alpha \in \mathbb{R}^n\), and \(h(\bar \alpha ) = 0,\forall h \in H\)}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum \(inf \mathcal{V}(f:H)\) of \(\mathcal{V}(f:H)\) is a root of some polynomial in U whenever \(inf \mathcal{V}(f:H) \ne \pm \infty\). The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.

Numerical solution of the Duffing equation with random coefficients

The main aim of this work is numerical solution of the nonlinear vibrations of micro-resonators exhibiting bounded and Gaussian uncertainty in their parameters. The mechanical response in deterministic situation is described by the Duffing equation, whose numerical solution is obtained with the Runge–Kutta–Fehlenberg algorithm, while probabilistic analysis is carried out using the generalized stochastic perturbation technique enriched with automatic optimization of the approximating polynomial. Basic solution to this nonlinear vibration in the deterministic context is obtained with the use of the computer algebra system MAPLE, where all additional probabilistic procedures are also implemented. We compare each time expectations, coefficients of variation, skewness and kurtosis for the structural response to show probabilistic sensitivity of the MEMS accelerometer with respect to its design parameter expectation and coefficient of variation. An additional comparison of the proposed technique with the traditional Monte-Carlo sampling for the first four probabilistic moments is also provided.

Formula omitted]-invariant algorithm for computing SAGBI-Gröbner bases

This paper introduces a new algorithm for computing SAGBI-Gröbner bases for ideals of invariant rings of permutation groups. This algorithm is based on F4 algorithm. A first implementation of this algorithm has been made in SAGE and MAPLE computer algebra systems and has been successfully tried on a number of examples.

Application of Numerical Laplace Inversion Methods in Chemical Engineering with Maple®

Abstract In this paper, the procedure of choice of the effective numerical inversion method of Laplace transform for solving of the mathematical model of gas flow was shown. Two inversion methods are evaluated in this work: the Gaver-Stehfest method, and the Dubner and Abate method. The numerical tests has been made using test functions. Based on the test results the fast and accurate method was selected and applied for modeling of the gas flow. The Gaver-Stehfest method has proven the most effective. The model was developed using the computer algebra system Maple®. Implementation of the methods were done by authors.

Uncertainty in effective elastic properties of particle filled polymers by the Monte-Carlo simulation

The main issue of this paper is to present statistical computational procedure for a determination of the effective elastic parameters for polymers filled with rubber particles. The homogenized elasticity tensor components are derived both as the variational upper and lower bounds as well as the solution for the cell problem on the composite Representative Volume Element. Probabilistic simulation is provided using both Monte-Carlo simulation technique and the generalized stochastic perturbation technique implemented in the symbolic computer algebra system MAPLE and, separately, together with the FEM-oriented code MCCEFF. Finally, up to fourth order probabilistic moments and coefficients of the homogenized tensor for the elastomers are computed as functions of the polymer’s and particles’ Young’s moduli coefficients of variation. This study gives the basis for further homogenization-based experiments, where the hyperelastic constitutive model will replace the classical Hookean one applied here.

Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

Solving Abel integral equations of first kind via fractional calculus

We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.