Automatic computation of the travelling wave solutions to nonlinear PDEs

Various extensions of the tanh-function method and their implementations for finding explicit travelling wave solutions to nonlinear partial differential equations (PDEs) have been reported in the literature. However, some solutions are often missed by these packages. In this paper, a new algorithm and its implementation called TWS for solving single nonlinear PDEs are presented. TWS is implemented in Maple 10. It turns out that, for PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.

Program summary

Program title:TWSCatalogue identifier:AEAM_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAM_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:1250No. of bytes in distributed program, including test data, etc.:78 101Distribution format:tar.gzProgramming language:Maple 10Computer:A laptop with 1.6 GHz Pentium CPUOperating system:Windows XP ProfessionalRAM:760 MbytesClassification:5Nature of problem:Finding the travelling wave solutions to single nonlinear PDEs.Solution method:Based on tanh-function method.Restrictions:The current version of this package can only deal with single autonomous PDEs or ODEs, not systems of PDEs or ODEs. However, the PDEs can have any finite number of independent space variables in addition to time t.Unusual features:For PDEs whose balancing numbers are not positive integers, TWS works much better than existing packages. Furthermore, TWS obtains more solutions than existing packages for most cases.Additional comments:It is easy to use.Running time:Less than 20 seconds for most cases, between 20 to 100 seconds for some cases, over 100 seconds for few cases.References:[1] E.S. Cheb-Terrab, K. von Bulow, Comput. Phys. Comm. 90 (1995) 102.[2] S.A. Elwakil, S.K. El-Labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299 (2002) 179.[3] E. Fan, Phys. Lett. 277 (2000) 212.[4] W. Malfliet, Amer. J. Phys. 60 (1992) 650.[5] W. Malfliet, W. Hereman, Phys. Scripta 54 (1996) 563.[6] E.J. Parkes, B.R. Duffy, Comput. Phys. Comm. 98 (1996) 288.     

USFKAD: An expert system for partial differential equations

The computation of the solution, by the separation of variables process, of the Poisson, diffusion, and wave equations in rectangular, cylindrical, or spherical coordinate systems, with Dirichlet, Neumann, or Robin boundary conditions, can be carried out in the time, Laplace, or frequency domains by a decision-tree process, using a library of eigenfunctions. We describe an expert system, USFKAD, that has been constructed for this purpose.

Program summary

Title of program:USFKADCatalogue identifier:ADYN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYN_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions:noneOperating systems under which the program has been tested: Windows, UNIXProgramming language used:C++, LaTeXNo. of lines in distributed program, including test data, etc.: 11 699No. of bytes in distributed program, including test data, etc.: 537 744Memory required to execute with typical data: 1.3 MegabytesDistribution format: tar.gzNature of mathematical problem: Analytic solution of Poisson, diffusion, and wave equationsMethod of solution: Eigenfunction expansionsRestrictions concerning the complexity of the problem: A few rarely-occurring singular boundary conditions are unavailable, but they can be approximated by regular boundary value problems to arbitrary accuracy.Typical running time:1 secondUnusual features of the program: Solutions are obtained for Poisson, diffusion, or wave PDEs; homogeneous or nonhomogeneous equations and/or boundary conditions; rectangular, cylindrical, or spherical coordinates; time, Laplace, or frequency domains; Dirichlet, Neumann, Robin, singular, periodic, or incoming/outgoing boundary conditions. Output is suitable for pasting into LaTeX documents.     

Coupled lateral bending-torsional vibration sensitivity of atomic force microscope cantilever

We study the influence of the contact stiffness and the ration between cantilever and tip lengths on the resonance frequencies and sensitivities of lateral cantilever modes. We derive expressions to determine both the effective resonance frequency and the mode sensitivity of an atomic force microscope (AFM) rectangular cantilever. Once the contact stiffness is given, the resonance frequency and the sensitivity of the vibration modes can be obtained from the expression. The results show that each mode has a different resonant frequency to variations in contact stiffness and each frequency increased until it eventually reached a constant value at very high contact stiffness. The low-order vibration modes are more sensitive to vibration than the high-order mode when the contact stiffness is low. However, the situation is reversed when the lateral contact stiffness became higher. Furthermore, increasing the ratio of tip length to cantilever length increases the vibration frequency and the sensitivity of AFM cantilever.     

Syntax and semantics of universal programming languages

This paper deals with certain characterizations of the sets of positive integers which when represented as strings on a finite alphabet, form tree adjunct languages, As the context free languages constitute a subfamily of tree adjunct languages, the results carry over to the former as well.     

A note on the Painlevé test for nonlinear variable-coefficient PDEs

For a generalized nonlinear PDEs with variable coefficients, it is not Painlevé integrable unless the variable coefficients satisfy certain constraint conditions. In this note a generalized algorithm is proposed for the Painlevé test of nonlinear variable-coefficient PDEs. For the three steps of Painlevé test, i.e. leading order analysis, resonance determination and verification of resonant conditions, the analysis of parametric constraints is similar to those of nonlinear PDEs with constant coefficients given in my previous work. The main difference lies in the coefficients of Laurent series should have proper dependence according to the types of variable coefficients. By this generalized algorithm, several important nonlinear variable-coefficient PDEs, including KdV equation, mKdV equation, KP equation, NLS equation and higher-order NLS equation are studied and, in addition to rederiving all known P-integrable conditions, some new P-integrable models are obtained with the assistance of Maple.     

Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations

A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh. Examples illustrate the key steps of the algorithm. Through discussion and example, parallels are drawn to the tanh-method for partial differential equations. The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed.

Program summary

Title of program: DDESpecialSolutions.mCatalogue identifier:ADUJProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUJProgram obtainable from:CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: Created using a PC, but can be run on UNIX and Apple machinesOperating systems under which the program has been tested: Windows 2000 and Windows XPProgramming language used: Mathematica, version 3.0 or higherMemory required to execute with typical data: 9 MBNumber of processors used: 1Has the code been vectorised or parallelized?: NoNumber of lines in distributed program, including test data, etc.: 3221Number of bytes in distributed program, including test data, etc.: 23 745Nature of physical problem: The program computes exact solutions to differential-difference equations in terms of the tanh function. Such solutions describe particle vibrations in lattices, currents in electrical networks, pulses in biological chains, etc.Method of solution: After the differential-difference equation is put in a traveling frame of reference, the coefficients of a candidate polynomial solution in tanh are solved for. The resulting traveling wave solutions are tested by substitution into the original differential-difference equation.Restrictions on the complexity of the program: The system of differential-difference equations must be polynomial. Solutions are polynomial in tanh.Typical running time: The average run time of 16 cases (including the Toda, Volterra, and Ablowitz-Ladik lattices) is 0.228 seconds with a standard deviation of 0.165 seconds on a 2.4 GHz Pentium 4 with 512 MB RAM running Mathematica 4.1. The running time may vary considerably, depending on the complexity of the problem.     

PSEUDO: applications of streams and lazy evaluation to integrable models

Procedures to manipulate pseudo-differential operators in MAPLE are implemented in the program PSEUDO to perform calculations with integrable models. We use lazy evaluation and streams to represent and operate with pseudo-differential operators. No order of truncation is needed since terms are produced on demand. We give a series of concrete examples.

Program summary

Title of program: PSEUDOCatalogue identifier: ADUOProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADUOProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: NoneComputers: IBM PCOperating systems under which the program has been tested: Windows systemsProgramming language used: MAPLE V Release 8Memory required to execute with typical data: Depends strongly on the problemNo. of lines in distributed program, including test data, etc.: 737No. of bytes in distributed program, including test data, etc.: 8822Distribution format: tar.gzNature of mathematical problem: Determination of equations of motion and conserved charges in the theory of integrable modelsMethods of solution: Pseudo-differential Lax operatorsRestrictions on the complexity of the problem: Handles only one-dimensional pseudo-differential operators with scalar coefficientsTypical running time: This depends strongly on the problem to be solved, usually taking from a few seconds to a few minutesUnusual features of the program: Use of delayed evaluation and streams     

Distinguished teaching in psychology: Ludy T. Benjamin, Jr.

The Distinguished Teaching in Psychology award, which includes a check for $1,000, is given to the recipient for his or her contributions to the teaching of psychology. The following guidelines are used to determine the recipient: (a) demonstrated influence as a teacher of students who become outstanding psychologists, (b) development of effective teaching methods and/or teaching materials, (c) engagement in significant research on teaching, (d) development of innovative curricula and courses, (e) outstanding performance as a classroom teacher, (f) being an especially effective trainer of teachers of psychology, and (g) being responsible for administrative facilitation of outstanding teaching. This article provides a citation and a biography for Ludy T. Benjamin, Jr., one of the recipients of the award for Distinguished Teaching in Psychology. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations

A direct approach with computerized symbolic computation is applied to construct a series of traveling wave solutions for nonlinear equations. Compared with most existing symbolic computation methods such as tanh method and Jacobi function method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solution according to some parameters.     

Discrete tanh method for nonlinear difference-differential equations

By introducing a simple difference equation to deduce the difference terms and a simple differential equation to deduce the differential terms, we proposed an unified algebraic method for constructing exact solutions to difference-differential equations (DDEs). This method could give many kinds of exact solutions including soliton solutions expressed by hyperbolic functions, periodic solutions expressed by trigonometric functions and rational solutions in a uniform way if solutions of these kinds exist. In this paper, we also give a generalization of the method to determine the degree of DDEs, and compared with the creativity work of D. Baldwin et al. [D. Baldwin, Ü. Göktas, W. Hereman, Comput. Phys. Comm. 162 (2004) 203-217] through the discrete Hybrid equation.