Songxin Liang

An efficient analytical approach for solving fourth order boundary value problems

Based on the homotopy analysis method (HAM), an e‐cient approach is proposed for obtaining approximate series solutions to fourth order two-point boundary value problems. We apply the approach to a linear problem which involves a parameter c and cannot be solved by other analytical methods for large values of c, and obtain convergent series solutions which agree very well with the exact solution, no matter how large the value of c is. Consequently, we give an a‐rmative answer to the open problem proposed by Momani and Noor in 2007 [S. Momani, M.A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput. 191(2007) 218-224]. We also apply the approach to a nonlinear problem, and obtain convergent series solutions which agree very well with the numerical solution given by the Runge-Kutta-Fehlberg 4-5 technique.

Approximate solutions to a parameterized sixth order boundary value problem

In this paper, the homotopy analysis method (HAM) is applied to solve a parameterized sixth order boundary value problem which, for large parameter values, cannot be solved by other analytical methods for finding approximate series solutions. Convergent series solutions are obtained, no matter how large the value of the parameter is.

An analytical approach for solving nonlinear boundary value problems in finite domains

Based on the homotopy analysis method (HAM), a general analytical approach for obtaining approximate series solutions to nonlinear two-point boundary value problems in finite domains is proposed. To demonstrate its effectiveness, this approach is applied to solve three nonlinear problems, and the analytical solutions obtained are more accurate than the numerical solutions obtained via the shooting method and the sinc-Galerkin method.

Automatic computation of the travelling wave solutions to nonlinear PDEs

Abstract 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: TWS Catalogue identifier: AEAM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAM_v1_0.html Program obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 1250 No. of bytes in distributed program, including test data, etc.: 78 101 Distribution format: tar.gz Programming language: Maple 10 Computer: A laptop with 1.6 GHz Pentium CPU Operating system: Windows XP Professional RAM: 760 Mbytes Classification: 5 Nature 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.

New travelling wave solutions to modified CH and DP equations

Abstract A new procedure for finding exact travelling wave solutions to the modified Camassa–Holm and Degasperis–Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them.

An algorithm for computing the complete root classification of a parametric polynomial

The Complete Root Classification for a univariate polynomial with symbolic coefficients is the collection of all the possible cases of its root classification, together with the conditions its coefficients should satisfy for each case. Here an algorithm is given for the automatic computation of the complete root classification of a polynomial with complex symbolic coefficients. The application of complete root classifications to some real quantifier elimination problems is also described.

The complete root classification of a parametric polynomial on an interval

Given a real parametric polynomial p(x) and an interval (a,b) ⊂ R, the Complete Root Classification (CRC) of p(x) on (a,b) is a collection of all possible cases of its root classification on (a,b), together with the conditions its coefficients must satisfy for each case. In this paper, a new algorithm is proposed for the automatic computation of the complete root classification of a parametric polynomial on an interval. As a direct application, the new algorithm is applied to some real quantifier elimination problems.

Rule-Based Simplification in Vector-Product Spaces

A vector-product space is a component-free representation of the common three-dimensional Cartesian vector space. The components of the vectors are invisible and formally inaccessible, although expressions for the components could be constructed. Expressions that have been built from the scalar and vector products can be simplified using a rule-based system. In order to develop and specify the system, an axiomatic system for a vector-product space is given. In addition, a brief description is given of an implementation in Aldor. The present work provides simplification functionality which overcomes difficulties encountered in earlier packages.

Analytic Approximations to Nonlinear Boundary Value Problems Modeling Beam-Type Nano-Electromechanical Systems

Abstract Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.

Unconstrained Parametric Minimization of a Polynomial: Approximate and Exact

We consider a monic polynomial of even degree with symbolic coefficients. We give a method for obtaining an expression in the coefficients (regarded as parameters) that is a lower bound on the value of the polynomial, or in other words a lower bound on the minimum of the polynomial. The main advantage of accepting a bound on the minimum, in contrast to an expression for the exact minimum, is that the algebraic form of the result can be kept relatively simple. Any exact result for a minimum will necessarily require parametric representations of algebraic numbers, whereas the bounds given here are much simpler. In principle, the method given here could be used to find the exact minimum, but only for low degree polynomials is this feasible; we illustrate this for a quartic polynomial. As an application, we compute rectifying transformations for integrals of trigonometric functions. The transformations require the construction of polynomials that are positive definite.