Showing papers in "Computers & Mathematics With Applications in 2007"
TL;DR: The basic conceptual framework of variational iteration technique with application to nonlinear problems is outlined and a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested.
Abstract: Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.
722 citations
TL;DR: This paper proposes two operators Up and Lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of TOPSIS and suggests that these two operations are employed to find ideal and negative ideal solutions under a fuzzy environment.
Abstract: In this paper, we generalize TOPSIS to fuzzy multiple-criteria group decision-making (FMCGDM) in a fuzzy environment. TOPSIS is one of the well-known methods for multiple-criteria decision-making (MCDM). Most of the steps of TOPSIS can be easily generalized to a fuzzy environment, except max and min operations in finding the ideal solution and negative ideal solution. Thus we propose two operators Up and Lo which satisfy the partial ordering relation on fuzzy numbers to the generalization of TOPSIS. In generalized TOPSIS, these two operations (Up and Lo) are employed to find ideal and negative ideal solutions under a fuzzy environment. Then the FMCGDM problem can be solved effectively and efficiently.
373 citations
TL;DR: With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed.
Abstract: Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF) When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in cases of high order polynomial interpolation) as a key error mechanism This observation suggests that it can be advantageous to let the shape parameter vary spatially, rather than assigning a single value to it Benefits typically include improvements in both accuracy and numerical conditioning Still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas This idea is routinely used when working with splines or finite element methods However, local refinement with RBFs may cause RP-type errors unless we use a spatially variable shape paremeter With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed
328 citations
TL;DR: A novel initialization approach which employs opposition-based learning to generate initial population is proposed which can accelerate convergence speed and also improve the quality of the final solution.
Abstract: Population initialization is a crucial task in evolutionary algorithms because it can affect the convergence speed and also the quality of the final solution. If no information about the solution is available, then random initialization is the most commonly used method to generate candidate solutions (initial population). This paper proposes a novel initialization approach which employs opposition-based learning to generate initial population. The conducted experiments over a comprehensive set of benchmark functions demonstrate that replacing the random initialization with the opposition-based population initialization can accelerate convergence speed.
311 citations
TL;DR: The combined KdV-MKdV equation and the Liouville equation are chosen to illustrate the effectiveness and convenience of the proposed Exp-function method for seeking solitary solutions, periodic solutions, and compacton-like solutions of nonlinear differential equations.
Abstract: In this paper, the Exp-function method is used for seeking solitary solutions, periodic solutions, and compacton-like solutions of nonlinear differential equations. The combined KdV-MKdV equation and the Liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method.
249 citations
TL;DR: Comparison of the results obtained by the homotopy perturbation method with those obtaining by the variational iteration method reveals that the present methods are very effective and convenient.
Abstract: In this article, the homotopy perturbation method proposed by J.- H. He is adopted for solving linear fractional partial differential equations. The fractional derivatives are described in the Caputo sense. Comparison of the results obtained by the homotopy perturbation method with those obtained by the variational iteration method reveals that the present methods are very effective and convenient.
184 citations
TL;DR: An approximate analytical solution is obtained of the steady, laminar three-dimensional flow for an incompressible, viscous fluid past a stretching sheet using the homotopy perturbation method (HPM) proposed by He.
Abstract: An approximate analytical solution is obtained of the steady, laminar three-dimensional flow for an incompressible, viscous fluid past a stretching sheet using the homotopy perturbation method (HPM) proposed by He. The flow is governed by a boundary value problem (BVP) consisting of a pair of non-linear differential equations. The solution is simple yet highly accurate and compares favorably with the exact solutions obtained early in the literature. The methodology presented in the paper is useful for solving the BVPs consisting of more than one differential equation.
169 citations
TL;DR: A numerical scheme to solve the two-dimensional (2D) time-dependent Schrodinger equation using collocation points and approximating the solution using multiquadrics and the Thin Plate Splines Radial Basis Function (RBF).
Abstract: In this paper, we propose a numerical scheme to solve the two-dimensional (2D) time-dependent Schrodinger equation using collocation points and approximating the solution using multiquadrics (MQ) and the Thin Plate Splines (TPS) Radial Basis Function (RBF). The scheme works in a similar fashion as finite-difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.
162 citations
TL;DR: This work constructs a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features.
Abstract: We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains.
152 citations
TL;DR: It is proved that the sequences of Mann and Ishikawa iterates converge to a fixed point of T and a convergence theorem of Mann iterates for a mapping defined on a noncompact domain is proved.
Abstract: Let K be a nonempty compact convex subset of a uniformly convex Banach space, and T:K->P(K) a multivalued nonexpansive mapping. We prove that the sequences of Mann and Ishikawa iterates converge to a fixed point of T. This generalizes former results proved by Sastry and Babu [K.P.R. Sastry, G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J. 55 (2005) 817-826]. We also introduce both of the iterative processes in a new sense, and prove a convergence theorem of Mann iterates for a mapping defined on a noncompact domain.
149 citations
TL;DR: The variational iteration method is used for analytic treatment of the linear and nonlinear systems of partial differential equations and shows improvements over existing numerical techniques.
Abstract: In this work, the variational iteration method (VIM) is used for analytic treatment of the linear and nonlinear systems of partial differential equations. The method reduces the calculation size and overcomes the difficulty of handling nonlinear terms. Numerical examples are examined to highlight the significant features of the VIM method. The method shows improvements over existing numerical techniques.
TL;DR: This paper proposes two kinds of reduction methods for the reduction of the concept lattices based on rough set theory and presents the sufficient and necessary conditions for justifying whether an attribute and an object are dispensable or indispensable in the above concept lattice.
Abstract: Rough set theory and formal concept analysis are two complementary mathematical tools for data analysis. In this paper, we study the reduction of the concept lattices based on rough set theory and propose two kinds of reduction methods for the above concept lattices. First, we present the sufficient and necessary conditions for justifying whether an attribute and an object are dispensable or indispensable in the above concept lattices. Based on the above justifying conditions, we propose a kind of multi-step attribute reduction method and object reduction method for the concept lattices, respectively. Then, on the basis of the defined discernibility functions of the concept lattices, we propose a kind of single-step reduction method for the concept lattices. Additionally, the relations between the attribute reduction of the concept lattices in FCA and the attribute reduction of the information system in rough set theory are discussed in detail. At last, we apply the above multi-step attribute reduction method for the concept lattices based on rough set theory to the reduction of the redundant premises of the multiple rules used in the job shop scheduling problem. The numerical computational results show that the reduction method for the concept lattices is effective in the reduction of the multiple rules.
TL;DR: It is shown that an analytical solution is possible by employing a homotopy analysis method (HAM) and the convergence of the obtained solution is also taken into account.
Abstract: The steady flow of a second grade fluid in a porous channel is considered. The constitutive equations are those used for a second grade fluid. The fluid is electrically conducting in the presence of a uniform magnetic field applied in the transverse direction to the flow. It is shown that an analytical solution is possible by employing a homotopy analysis method (HAM). The convergence of the obtained solution is also taken into account. Assessment for the influence of various parameters of interest on the velocity is undertaken.
TL;DR: The convergence rate of the natural frequencies is shown to be fast and the stability of the numerical methodology is very good, while the effect of different grid point distributions on the convergence, the stability and the accuracy of the GDQ procedure is investigated.
Abstract: This paper deals with the dynamical behaviour of hemispherical domes and spherical shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C), simply supported (S) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. These results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Engineer, which also elaborate a three-dimensional analysis. The effect of different grid point distributions on the convergence, the stability and the accuracy of the GDQ procedure is investigated. The convergence rate of the natural frequencies is shown to be fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions.
TL;DR: It is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation.
Abstract: In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the complete set in the Trefftz method for the Laplace and biharmonic problems are embedded in the degenerate kernels of the MFS. The degenerate scale appears using the MFS when the geometrical matrix is singular. The occurring mechanism of the degenerate scale in the MFS is also studied by using circulant. The comparison of accuracy and efficiency of the two methods was addressed.
TL;DR: By means of He's homotopy perturbation method (HPM), an approximate solution of a boundary layer equation in unbounded domain is obtained and the method is very effective and simple.
Abstract: By means of He's homotopy perturbation method (HPM) an approximate solution of a boundary layer equation in unbounded domain is obtained. Comparison is made between the obtained results and those in open literature. The results show that the method is very effective and simple.
TL;DR: This paper presents two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections, and shows that projection of a vector or quaternions can be expressed concisely using three mutually perpendicular anti-involutions.
Abstract: An involution or anti-involution is a self-inverse linear mapping. In this paper we study quaternion involutions and anti-involutions. We review formal axioms for such involutions and anti-involutions. We present two mappings, one a quaternion involution and one an anti-involution, and a geometric interpretation of each as reflections. We present results on the composition of these mappings and show that the quaternion conjugate may be expressed using three mutually perpendicular anti-involutions. Finally, we show that projection of a vector or quaternion can be expressed concisely using three mutually perpendicular anti-involutions.
TL;DR: The steady, laminar, axisymmetric flow of a Newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet is investigated and a solution based upon He's homotopy perturbation method has been developed.
Abstract: The steady, laminar, axisymmetric flow of a Newtonian fluid due to a stretching sheet when there is a partial slip of the fluid past the sheet has been investigated. The flow is governed by a third-order non-linear boundary value problem whose exact numerical solution has been obtained non-iteratively in terms of the non-dimensional slip parameter @l. A perturbation solution valid for small @l and an asymptotic solution valid for large @l have been derived. Finally a solution based upon He's homotopy perturbation method has been developed. The latter, being analytical, is elegant but is sufficiently accurate for all values of @l.
TL;DR: Variational iteration and homotopy perturbation methods are applied to various evolution equations and it is revealed that both methods are capable of solving effectively a large number of nonlinear differential equations with high accuracy.
Abstract: He's variational iteration and homotopy perturbation methods are applied to various evolution equations. To assess the accuracy of the solutions, we compare the results with the exact solutions, revealing that both methods are capable of solving effectively a large number of nonlinear differential equations with high accuracy.
TL;DR: The study highlights the power of the VIM method, which is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials.
Abstract: In this work, He's variational iteration method (VIM) is used for analytic treatment of the linear and the nonlinear wave equations in bounded and unbounded domains. Wave-like equations are also investigated. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. The study highlights the power of the VIM method.
TL;DR: Numerical results obtained for well-known test problems show the efficiency of the new explicit Numerov-type method, which is useful only when a good estimate of the frequency of the problem is known in advance.
Abstract: In this paper a new explicit Numerov-type method is introduced. The construction is based on a modification of a sixth-order explicit Numerov-type method recently developed by Tsitouras [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 45 (2003) 37-42]. Two free parameters are added in order to nullify the phase-lag and the amplification. The method is useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. Numerical results obtained for well-known test problems show the efficiency of the new method.
TL;DR: A simple algorithm based on space marching mollification techniques is introduced for the numerical solution of the discrete problem of attempting to recover the boundary temperature and the heat flux functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor.
Abstract: The ill-posed problem of attempting to recover the boundary temperature and the heat flux functions from one measured transient data temperature at some interior point of a one-dimensional semi-infinite conductor when the governing linear diffusion equation is of fractional type is discussed. A simple algorithm based on space marching mollification techniques is introduced for the numerical solution of the discrete problem. Stability bounds, error estimates and numerical examples of interest are also presented.
TL;DR: The results show that the variational iteration method is of high accuracy, more convenient and efficient for solving integro-differential equations.
Abstract: In this paper, the variational iteration method proposed by Ji-Huan He is applied to solve both linear and nonlinear boundary value problems for fourth order integro-differential equations. The numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method. The results show that the variational iteration method is of high accuracy, more convenient and efficient for solving integro-differential equations.
TL;DR: Experimental results demonstrate that the proposed deterministic forecasting model outperforms the existing models in terms of accuracy, robustness, and reliability and adheres to the consistency principle that a shorter interval length leads to more accurate results.
Abstract: The fuzzy time series has recently received increasing attention because of its capability of dealing with vague and incomplete data. There have been a variety of models developed to either improve forecasting accuracy or reduce computation overhead. However, the issues of controlling uncertainty in forecasting, effectively partitioning intervals, and consistently achieving forecasting accuracy with different interval lengths have been rarely investigated. This paper proposes a novel deterministic forecasting model to manage these crucial issues. In addition, an important parameter, the maximum length of subsequence in a fuzzy time series resulting in a certain state, is deterministically quantified. Experimental results using the University of Alabama's enrollment data demonstrate that the proposed forecasting model outperforms the existing models in terms of accuracy, robustness, and reliability. Moreover, the forecasting model adheres to the consistency principle that a shorter interval length leads to more accurate results.
TL;DR: This work introduces a framework for obtaining exact solutions to linear and nonlinear diffusion equations for some diffusion processes of power law diffusitivies.
Abstract: In this work, we introduce a framework for obtaining exact solutions to linear and nonlinear diffusion equations. Exact solutions are developed for some diffusion processes of power law diffusitivies. He's variational iteration method (VIM) is used for analytic treatment of these equations. The powerful VIM method is capable of handling both linear and nonlinear equations in a direct manner.
TL;DR: This paper applies He's energy balance method to determine the frequency-amplitude relation of the Duffing-harmonic oscillator, which gives a good estimate for the angular frequency.
Abstract: This paper applies He's energy balance method to determine the frequency-amplitude relation of the Duffing-harmonic oscillator, which gives a good estimate for the angular frequency.
TL;DR: Variational iteration method overcomes the difficulty arising in calculating theAdomian's polynomials which is an important advantage over the Adomian decomposition method.
Abstract: This paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations. The variational iteration method and Adomian decomposition method are used for solving this equation and then numerical results are compared with each other, showing that the variational iteration method leads to more accurate results. Furthermore, variational iteration method overcomes the difficulty arising in calculating the Adomian's polynomials which is an important advantage over the Adomian decomposition method.
TL;DR: In this article, the double integral calculus of variations on time scales is considered and necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler-Lagrange equation.
Abstract: We consider a version of the double integral calculus of variations on time scales, which includes as special cases the classical two-variable calculus of variations and the discrete two-variable calculus of variations. Necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler-Lagrange equation.
TL;DR: Variational iteration technique can be viewed as an efficient and reliable method for solving a wide class of linear and nonlinear boundary value problems.
Abstract: In this paper, we apply the variational iteration method for solving fourth order boundary value problems. The analytical results are in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the method. Comparison is made to confirm the reliability of this technique. Variational iteration technique can be viewed as an efficient and reliable method for solving a wide class of linear and nonlinear boundary value problems.
TL;DR: Comparison with exact solution shows that the He's variational iteration method is very effective and convenient for solving integral equations.
Abstract: In this paper, several integral equations are solved by He's variational iteration method. Comparison with exact solution shows that the method is very effective and convenient for solving integral equations.