Linearization methods in classical and quantum mechanics

The applicability and accuracy of linearization methods for initial-value problems in ordinary differential equations are verified on examples that include the nonlinear Duffing equation, the Lane-Emden equation, and scattering length calculations. Linearization methods provide piecewise linear ordinary differential equations which can be easily integrated, and provide accurate answers even for hypersingular potentials, for which perturbation methods diverge. It is shown that the accuracy of linearization methods can be substantially improved by employing variable steps which adjust themselves to the solution.     

An efficient analytical approach for solving fourth order boundary value problems

Based on the homotopy analysis method (HAM), an efficient 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 affirmative 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.     

Differential transform method for solving the linear and nonlinear Klein-Gordon equation

In this paper, we implemented relatively new, exact series method of solution known as the differential transform method for solving linear and nonlinear Klein-Gordon equation. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Piecewise quasilinearization techniques for singular boundary-value problems

Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons with exact solutions of several Lane-Emden equations, a singular problem of non-Newtonian fluid dynamics and the Thomas-Fermi equation. It is shown that the smooth piecewise quasilinearization method provides accurate solutions even near the singularity and is more precise than (iterative) second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth piecewise quasilinear method depends on the kind of singularity, nonlinearity and inhomogeneities of singular ordinary differential equations. For the Thomas-Fermi equation, it is shown that the piecewise quasilinearization method that provides globally smooth solutions is more accurate than that which only insures global continuity, and more accurate than global quasilinearization techniques which do not employ local linearization.     

Fortran programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap

Here we develop simple numerical algorithms for both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation describing the properties of Bose-Einstein condensates at ultra low temperatures. In particular, we consider algorithms involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation we consider the one-dimensional, two-dimensional circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. Numerical results for the chemical potential and root-mean-square size of stationary states are reported using imaginary-time propagation programs for all the cases and compared with previously obtained results. Also presented are numerical results of non-stationary oscillation for different trap symmetries using real-time propagation programs. A set of convenient working codes developed in Fortran 77 are also provided for all these cases (twelve programs in all). In the case of two or three space variables, Fortran 90/95 versions provide some simplification over the Fortran 77 programs, and these programs are also included (six programs in all).

Program summary

Program title: (i) imagetime1d, (ii) imagetime2d, (iii) imagetime3d, (iv) imagetimecir, (v) imagetimesph, (vi) imagetimeaxial, (vii) realtime1d, (viii) realtime2d, (ix) realtime3d, (x) realtimecir, (xi) realtimesph, (xii) realtimeaxialCatalogue identifier: AEDU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDU_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.: 122 907No. of bytes in distributed program, including test data, etc.: 609 662Distribution format: tar.gzProgramming language: FORTRAN 77 and Fortran 90/95Computer: PCOperating system: Linux, UnixRAM: 1 GByte (i, iv, v), 2 GByte (ii, vi, vii, x, xi), 4 GByte (iii, viii, xii), 8 GByte (ix)Classification: 2.9, 4.3, 4.12Nature of problem: These programs are designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-, two- or three-space dimensions with a harmonic, circularly-symmetric, spherically-symmetric, axially-symmetric or anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Solution method: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation, in either imaginary or real time, over small time steps. The method yields the solution of stationary and/or non-stationary problems.Additional comments: This package consists of 12 programs, see “Program title”, above. FORTRAN77 versions are provided for each of the 12 and, in addition, Fortran 90/95 versions are included for ii, iii, vi, viii, ix, xii. For the particular purpose of each program please see the below.Running time: Minutes on a medium PC (i, iv, v, vii, x, xi), a few hours on a medium PC (ii, vi, viii, xii), days on a medium PC (iii, ix).

Program summary (1)

Title of program: imagtime1d.FTitle of electronic file: imagtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (2)

Title of program: imagtimecir.FTitle of electronic file: imagtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (3)

Title of program: imagtimesph.FTitle of electronic file: imagtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 1 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (4)

Title of program: realtime1d.FTitle of electronic file: realtime1d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in one-space dimension with a harmonic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (5)

Title of program: realtimecir.FTitle of electronic file: realtimecir.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with a circularly-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (6)

Title of program: realtimesph.FTitle of electronic file: realtimesph.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77Typical running time: Minutes on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with a spherically-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (7)

Title of programs: imagtimeaxial.F and imagtimeaxial.f90Title of electronic file: imagtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (8)

Title of program: imagtime2d.F and imagtime2d.f90Title of electronic file: imagtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 2 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (9)

Title of program: realtimeaxial.F and realtimeaxial.f90Title of electronic file: realtimeaxial.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an axially-symmetric trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (10)

Title of program: realtime2d.F and realtime2d.f90Title of electronic file: realtime2d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Hours on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in two-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.

Program summary (11)

Title of program: imagtime3d.F and imagtime3d.f90Title of electronic file: imagtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum RAM memory: 4 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Few days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in imaginary time over small time steps. The method yields the solution of stationary problems.

Program summary (12)

Title of program: realtime3d.F and realtime3d.f90Title of electronic file: realtime3d.tar.gzCatalogue identifier:Program summary URL:Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandDistribution format: tar.gzComputers: PC/Linux, workstation/UNIXMaximum Ram Memory: 8 GByteProgramming language used: Fortran 77 and Fortran 90Typical running time: Days on a medium PCUnusual features: NoneNature of physical problem: This program is designed to solve the time-dependent Gross-Pitaevskii nonlinear partial differential equation in three-space dimensions with an anisotropic trap. The Gross-Pitaevskii equation describes the properties of a dilute trapped Bose-Einstein condensate.Method of solution: The time-dependent Gross-Pitaevskii equation is solved by the split-step Crank-Nicolson method by discretizing in space and time. The discretized equation is then solved by propagation in real time over small time steps. The method yields the solution of stationary and non-stationary problems.     

Accurate basis set by the CIP method for the solutions of the Schrödinger equation

In this paper, we propose a basis set approach by the Constrained Interpolation Profile (CIP) method for the calculation of bound and continuum wave functions of the Schrödinger equation. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution by the constraints from the spatial derivative of the original equation. Thus the solution even on the subgrid scale becomes consistent with the master equation. By increasing the order of the polynomial, this solution quickly converges. The method is tested on the one-dimensional Schrödinger equation and is proven to give solutions a few orders of magnitude higher in accuracy than conventional methods for the lower-lying eigenstates. The method is straightforwardly applicable to various types of partial differential equations.     

Structural reinforcement in a spring-block model of stress-induced fracture propagation

We present a mechanism-based model of fracture propagation in a two-dimensional elastic sheet subjected to biaxial stretching. The time evolution of lattice stretching is formulated using a set of coupled nonlinear differential equations describing the network dynamics of masses connected by springs. We show that reinforcement based on a Gaussian spatial distribution of failure thresholds is effective in hindering tear propagation, that is, it delays the onset of breakage and reduces the total fractured sites at equilibrium time. The method presented here is general—it can incorporate any type of load distribution and test any reinforcement procedures.     

Limit cycles in nonlinear excitation of clusters of classical oscillators

In this paper we develop a numerical procedure for detecting the existence of limit cycles in nonlinear excitation of clusters of classical harmonic oscillators. Our technique is able to compute also the main parameters of a limit cycle, that is the amplitudes and the period. The numerical method, based on the propagation matrix formalism, is transparent and easy to apply. It may find application in various areas where nonlinear excitations are involved, e.g., sound and mechanic vibrations in musical instruments, ground vibrations in volcanic areas, and sea tides.     

A parallel monotone iterative method for the numerical solution of multi-dimensional semiconductor Poisson equation

Various self-consistent semiconductor device simulation approaches require the solution of Poisson equation that describes the potential distribution for a specified doping profile (or charge density). In this paper, we solve the multi-dimensional semiconductor nonlinear Poisson equation numerically with the finite volume method and the monotone iterative method on a Linux-cluster. Based on the nonlinear property of the Poisson equation, the proposed method converges monotonically for arbitrary initial guesses. Compared with the Newton's iterative method, it is easy implementing, relatively robust and fast with much less computation time, and its algorithm is inherently parallel in large-scale computing. The presented method has been successfully implemented; the developed parallel nonlinear Poisson solver tested on a variety of devices shows it has good efficiency and robustness. Benchmarks are also included to demonstrate the excellent parallel performance of the method.     

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

A FORTRAN program is presented which solves a system of nonlinear simultaneous equations using the continuous analog of Newton's method (CANM). The user has the option of either to provide a subroutine which calculates the Jacobian matrix or allow the program to calculate it by a forward-difference approximation. Five iterative schemes using different algorithms of determining adaptive step size of the CANM process are implemented in the program.

Program summary

Title of program: CANMCatalogue number: ADSNProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSNProgram available from: CPC Program Library, Queen's University of Belfast, Northern IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested:Computers: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PCInstallation: Department of Chemistry, University of Toronto, Toronto, CanadaOperating systems under which the program has been tested: IRIX64 6.1, 6.4 and 6.5, AIX 3.4, HP-UX 9.01, Linux 2.4.7Programming language used: FORTRAN 90Memory required to execute with typical data: depends on the number of nonlinear equations in a system. Test run requires 80 KBNo. of bits in distributed program including test data, etc.: 15283Distribution format: tar gz formatNo. of lines in distributed program, including test data, etc.: 1794Peripherals used: line printer, scratch disc storeExternal subprograms used: DGECO and DGESL [1]Keywords: nonlinear equations, Newton's method, continuous analog of Newton's method, continuous parameter, evolutionary differential equation, Euler's methodNature of physical problem: System of nonlinear simultaneous equations

     

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as

     

Numerical procedure for the determination of an unknown coefficients in parabolic equations

A numerical procedure for an inverse problem of determination of unknown coefficients in a class of parabolic differential equations is presented. The approach of the proposed method is to approximate unknown coefficients by a piecewise linear function whose coefficients are determined from the solution of minimization problem based on the overspecified data. Some numerical examples are presented.     

Iteratively regularized Gauss-Newton method for atmospheric remote sensing

In this paper we present an inversion algorithm for nonlinear ill-posed problems arising in atmospheric remote sensing. The proposed method is the iteratively regularized Gauss-Newton method. The dependence of the performance and behaviour of the algorithm on the choice of the regularization matrices and sequences of regularization parameters is studied by means of simulations. A method for improving the accuracy of the solution when the identity matrix is used as regularization matrix is also discussed. Results are presented for atmospheric temperature retrievals from a far infrared spectrum observed by an airborne uplooking heterodyne instrument.     

Solving a set of truncated Dyson-Schwinger equations with a globally converging method

A globally converging numerical method to solve coupled sets of non-linear integral equations is presented. Such systems occur, e.g., in the study of Dyson-Schwinger equations of Yang-Mills theory and QCD. The method is based on the knowledge of the qualitative properties of the solution functions in the far infrared and ultraviolet. Using this input, the full solutions are constructed using a globally convergent modified Newton iteration. Two different systems will be treated as examples: The Dyson-Schwinger equations of 3-dimensional Yang-Mills-Higgs theory provide a system of finite integrals, while those of 4-dimensional Yang-Mills theory at high temperatures are only finite after renormalization.     

Genetically controlled random search: a global optimization method for continuous multidimensional functions

A new stochastic method for locating the global minimum of a multidimensional function inside a rectangular hyperbox is presented. A sampling technique is employed that makes use of the procedure known as grammatical evolution. The method can be considered as a “genetic” modification of the Controlled Random Search procedure due to Price. The user may code the objective function either in C++ or in Fortran 77. We offer a comparison of the new method with others of similar structure, by presenting results of computational experiments on a set of test functions.

Program summary

Title of program: GenPriceCatalogue identifier:ADWPProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWPProgram available from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer for which the program is designed and others on which it has been tested: the tool is designed to be portable in all systems running the GNU C++ compilerInstallation: University of Ioannina, GreeceProgramming language used: GNU-C++, GNU-C, GNU Fortran-77Memory required to execute with typical data: 200 KBNo. of bits in a word: 32No. of processors used: 1Has the code been vectorized or parallelized?: noNo. of lines in distributed program, including test data, etc.:13 135No. of bytes in distributed program, including test data, etc.: 78 512Distribution format: tar. gzNature of physical problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques are frequently trapped in local minima. Global optimization is hence the appropriate tool. For example, solving a nonlinear system of equations via optimization, employing a “least squares” type of objective, one may encounter many local minima that do not correspond to solutions, i.e. minima with values far from zero.Method of solution: Grammatical Evolution is used to accelerate the process of finding the global minimum of a multidimensional, multimodal function, in the framework of the original “Controlled Random Search” algorithm.Typical running time: Depending on the objective function.     

Density-driven flow and solute transport problems. A 2-D numerical model based on the network simulation method

The network simulation method, based on the formal equivalence between physical systems and electrical networks, solves numerical problems of relatively mathematical complexity in a versatile, efficient and computationally fast way. In this paper, the method is applied for the first time to the design of a general purpose model for simulating two-dimensional transient density-driven flow and solute transport through porous media, a mathematical model made up by coupled, nonlinear differential equations. Using the Boussinesq approximation and the stream function formulation, the model is used to solve two typical problems related with groundwater flows. Isochlor concentration and stream function curves are presented and successfully compared with those of other authors. Simulation is carried out using the digital computer program Pspice with relatively low computing times.     

Solving polynomial nonlinear matrix equations by a linearization method

This paper deals with some modification of a matrix linearization method. The scheme proposed makes it possible to find tuples of solutions for systems of polynomial nonlinear equations defined on a commutative matrix ring. The matrix linearization method reduces an initial polynomial nonlinear problem to a linear one with respect to matrices of solutions. Then, the method of elimination of unknowns is used to obtain a generalized eigenvalue problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 60–69, May–June 2006.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

近似线性化方法综述

回顾了精确线性化方法及目前存在的主要问题,讨论了基于平衡流型和工作点邻域的状态方程近似线性化问题,研究了输入输出近似线性化在自理非最小相位或不可控非线性问题时的作用。最后对未来的研究方向进行了探讨。     

GenMin: An enhanced genetic algorithm for global optimization

A new method that employs grammatical evolution and a stopping rule for finding the global minimum of a continuous multidimensional, multimodal function is considered. The genetic algorithm used is a hybrid genetic algorithm in conjunction with a local search procedure. We list results from numerical experiments with a series of test functions and we compare with other established global optimization methods. The accompanying software accepts objective functions coded either in Fortran 77 or in C++.

Program summary

Program title: GenMinCatalogue identifier: AEAR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAR_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.: 35 810No. of bytes in distributed program, including test data, etc.: 436 613Distribution format: tar.gzProgramming language: GNU-C++, GNU-C, GNU Fortran 77Computer: The tool is designed to be portable in all systems running the GNU C++ compilerOperating system: The tool is designed to be portable in all systems running the GNU C++ compilerRAM: 200 KBWord size: 32 bitsClassification: 4.9Nature of problem: A multitude of problems in science and engineering are often reduced to minimizing a function of many variables. There are instances that a local optimum does not correspond to the desired physical solution and hence the search for a better solution is required. Local optimization techniques are frequently trapped in local minima. Global optimization is hence the appropriate tool. For example, solving a nonlinear system of equations via optimization, employing a least squares type of objective, one may encounter many local minima that do not correspond to solutions (i.e. they are far from zero).Solution method: Grammatical evolution and a stopping rule.Running time: Depending on the objective function. The test example given takes only a few seconds to run.