[SADE] a Maple package for the symmetry analysis of differential equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie–Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented by the authors in the package QPSI) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given.

Program summary

Program title: SADECatalogue identifier: AEHL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 27 704No. of bytes in distributed program, including test data, etc.: 346 954Distribution format: tar.gzProgramming language: MAPLE 13 and MAPLE 14Computer: PCs and workstationsOperating system: UNIX/LINUX systems and WINDOWSClassification: 4.3Nature of problem: Determination of analytical properties of systems of differential equations, including symmetry transformations, analytical solutions and conservation laws.Solution method: The package implements in MAPLE some algorithms (discussed in the text) for the study of systems of differential equations.Restrictions: Depends strongly on the system and on the algorithm required. Typical restrictions are related to the solution of a large over-determined system of linear or non-linear differential equations.Running time: Depends strongly on the order, the complexity of the differential system and the object computed. Ranges from seconds to hours.     

PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x)=0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f(x)=0. The third module Verify checks whether all isolated solutions of f(x)=0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.AMS Subject Classification: Primary: 65H10 system of equations, secondary: 65H20 global methods, including homotopy approaches.     

Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems

Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.This revised version was published online in July 2005 with corrected volume and issue numbers.     

An extension of the Prelle-Singer method and a Maple implementation

The Prelle-Singer method is a semi-decision algorithm which can be used to solve analytically first order ordinary differential equations which have solutions in terms of elementary functions. In this paper we develop an extension to the Prelle-Singer method which deals with first order ordinary differential equations whose solutions lie outside the scope of the standard Prelle-Singer method. We present a software package in Maple V, Release 5 which implements both the Prelle-Singer method in its original form and our extension. Tests with ordinary differential equations taken from standard references show that our package is able to solve equations where Maple's standard solution routines fail.     

The Navier–Stokes-αβ equations as a platform for a spectral multigrid method to solve the Navier–Stokes equations

This paper describes a spectral multigrid method for spatially periodic homogeneous and isotropic turbulent flows. The method uses the Navier–Stokes-αβ equations to accelerate convergence toward solutions of the Navier–Stokes equations. The Navier–Stokes-αβ equations are solved on coarse grids at various levels and the Navier–Stokes equations are solved on the “nest grid”. The method uses Crank–Nicolson time-stepping for the viscous terms, explicit time-stepping for the remaining terms, and Richardson iteration to solve linear systems encountered at each time step and on each grid level. To explore the computational efficiency of the method, comparisons are made with results obtained from an analogous spectral multigrid method for the Navier–Stokes equations. These comparisons are based on computing work units and residuals for multigrid cycles. Most importantly, we examine how choosing different values of the length scales α and β entering the Navier–Stokes-αβ equations influence the efficiency and accuracy of these multigrid schemes.     

Construction of exact partial solutions of nonintegrable systems by means of formal Laurent and Puiseux series

A package facilitating search for analytic solutions of nonintegrable systems of autonomous ordinary differential equations is suggested. The algorithm discussed, which is a generalization of the Conte-Musette method, allows one to obtain single-valued and multivalued analytical solutions that were known to exist only in the form of formal Laurent and Puiseux series (obtained, for example, by means of the Painlevé analysis), respectively. Solutions in the form of the formal Laurent series and procedures from the package discussed are used not only for constructing elliptic solutions but also for proving impossibility of such construction. The suggested procedures have been implemented as packages in Maple and REDUCE.     

Generalized variational inclusions and generalized resolvent equations in banach spaces

In this paper, we introduce and study generalized variational inclusions and generalized resolvent equations in real Banach spaces. It is established that generalized variational inclusion problems in uniformly smooth Banach spaces are equivalent to fixed-point problems. We also establish a relationship between generalized variational inclusions and generalized resolvent equations. By using Nadler's fixed-point theorem and resolvent operator technique for m-accretive mappings in real Banach spaces, we propose an iterative algorithm for computing the approximate solutions of generalized variational inclusions. The iterative algorithms for finding the approximate solutions of generalized resolvent equations are also proposed. The convergence of approximate solutions of generalized variational inclusions and generalized resolvent equations obtained by the proposed iterative algorithms is also studied. Our results are new and represent a significant improvement of previously known results. Some special cases are also discussed.     

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.     

基于程序频谱的缺陷定位方法

软件测试是生产可靠软件的重要保障,对测试所发现缺陷的解决可以分为缺陷定位和缺陷修改两个步骤^[1],其中的缺陷定位是最耗时的.通常情况下,测试套件中成功执行的测试用例都占绝大多数,对基于程序频谱的缺陷定位方法,应该具备自主调节成功测试用例覆盖比重的能力,以提高方法的可用性.即,随着语句被成功测试用例覆盖的次数增多,该语句的覆盖次数对怀疑率的贡献度应逐渐减小,成功测试用例数的有效处理能提高缺陷定位方法的效果.基于此,本文提出EPStar（EP*）缺陷定位方法,该方法可以有效调整成功执行用例数的影响,以避免成功用例数量对缺陷定位效果的过度影响,从而提高缺陷定位的准确性,通过实验对比,说明了EP*方法比现有的几种缺陷定位方法具有更高的缺陷定位精度.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

On the solution of coupled Riccati equations used in mixed H2 and H∞ optimal control

This paper considers a set of three coupled Riccati equations. The solution of these equations constitutes necessary conditions for mixed H₂ and H_∞ control problems. The main contributions of the paper are related to the conditions under which the solution of these equations is unique as well as to characterize necessary and sufficient conditions for the existence of solutions.     

Some schwarz methods for integral equations on surfaces-h and p versions

Abstact We present new results from 11, 7, 12 on various Schwarz methods for the h and p versions of the boundary element methods applied to prototype first kind integral equations on surfaces. When those integral equations (weakly/hypersingular) are solved numerically by the Galerkin boundary element method, the resulting matrices become ill-conditioned. Hence, for an efficient solution procedure appropriate preconditioners are necessary to reduce the numbers of CG-iterations. In the p version where accuracy of the Galerkin solution is achieved by increasing the polynomial degree the use of suitable Schwarz preconditioners (presented in the paper) leads to only polylogarithmically growing condition numbers. For the h version where accuracy is achieved by reducing the mesh size we present a multi-level additive Schwarz method which is competitive with the multigrid method. Communicated by: U. Langer Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations

In this paper, we continue our investigation of the locally divergence-free discontinuous Galerkin method, originally developed for the linear Maxwell equations (J. Comput. Phys. 194 588–610 (2004)), to solve the nonlinear ideal magnetohydrodynamics (MHD) equations. The distinctive feature of such method is the use of approximate solutions that are exactly divergence-free inside each element for the magnetic field. As a consequence, this method has a smaller computational cost than the traditional discontinuous Galerkin method with standard piecewise polynomial spaces. We formulate the locally divergence-free discontinuous Galerkin method for the MHD equations and perform extensive one and two-dimensional numerical experiments for both smooth solutions and solutions with discontinuities. Our computational results demonstrate that the locally divergence-free discontinuous Galerkin method, with a reduced cost comparing to the traditional discontinuous Galerkin method, can maintain the same accuracy for smooth solutions and can enhance the numerical stability of the scheme and reduce certain nonphysical features in some of the test cases.This revised version was published online in July 2005 with corrected volume and issue numbers.     

PHoMpara – Parallel Implementation of the Polyhedral Homotopy Continuation Method for Polynomial Systems

The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral homotopy functions, tracing the solution curves of the homotopy equations, and verifying the obtained solutions. A software package PHoMpara parallelizes PHoM to solve a polynomial system of large size. Many characteristics of the polyhedral homotopy continuation method make parallel implementation efficient and provide excellent scalability. Numerical results include some large polynomial systems that had not been solved.     

An Efficient Dynamic Formulation for Multibody Systems

The objective of this article is to present an efficient extension ofRosenthal's order-n algorithm to multibody systems containing closedloops. The equations of motion are created by using relative coordinatesand partial velocity theory. Closed topological loops are handled by cutjoint technique. The set of constraint equations of cut joints isadjoined to the system's equation of motion by using Lagrangemultipliers. This results in the equation of motion as adifferential-algebraic equation (DAE) rather than an ordinarydifferential equation. This DAE is then solved by applying the extendedRosenthal's order-n algorithm proposed in this article. While solvingDAE, violation of the kinematic constraint equations of cut joints iscorrected by coordinate projection method. Some numerical simulationsare carried out to demonstrate efficiency of the proposed method.     

Boundary element method for membrane and torsion crack problems

Here we present as an application of [12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications.A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Gårding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed.The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.     

Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov–Kuznetsov and Kadomtsev–Petviashvili equations as examples.The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations.The code ConservationLawsMD.m has been applied to multi-dimensional versions of the Sawada–Kotera, Camassa–Holm, Gardner, and Khokhlov–Zabolotskaya equations.     

An efficient method for analyzing nonuniformly coupled microstrip lines

This article presents an efficient method for analyzing nonuniformly coupled microstrip lines. By choosing a modal‐transformation matrix, the coupled nonlinear differential equations describing the symmetric nonuniformly coupled microstrip lines are decoupled using even‐ and odd‐mode parameters; the original problem is thus transformed into two single nonuniform transmission lines. A power‐law function of arbitrary order and having two adjustable parameters is chosen to better approximate the equation coefficients. Closed‐form ABCD matrix solutions are obtained and used to calculate the S‐parameters of nonuniformly coupled microstrip lines. Numerical results for two examples are compared with those from a full‐wave commercial package and experimental ones in the literature in order to demonstrate the accuracy and efficiency of this method. This highly efficient method is employed to optimize a cosine‐shape 10‐dB codirectional coupler, which has good return loss and high directivity performance over a wide frequency range. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.     

Solving systems of algebraic equations by a general elimination method

A method of factorisation of a U-resultant into linear factors is given. Using this method we can obtain solutions and their multiplicities of a system of algebraic equations, provided the system of algebraic equations has finitely many solutions. We directly calculate a matrix à which gives all solutions of the system by using a Gröbner basis of the ideal generated by the polynomials of the system of algebraic equations.