Quasilinearization approach to computations with singular potentials

We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r²+λr−α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries.We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.     

Quasilinearization approach to quantum mechanics

The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.     

Quasilinearization method and WKB

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers of ?, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schrödinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.     

Systems Identification by Quasilinearization and by Evolutionary Programming

Abstract

Two new methods for obtaining the values of the coefficients in the differential equations describing the dynamics of a system are developed. The first method is based on a quasilinearization procedure and is applicable in parameter identification problems where the plant is modeled by a system of linear differential equations, and noisy measurements of state and control variables are available. Computationally, this method is equivalent to a modification of the classical Newton-Raphson method. The second, a “directed random search” method, is based on a concept called evolutionary programming, and is also applicable for nonlinear problems.

Using recorded flight test data of an experimental aircraft, the two methods are compared for accuracy and computational efficiency.     

Construction of optimal feedback control for nonlinear systems via Chebyshev polynomials

A method is proposed to determine the optimal feedback control law of a class of nonlinear optimal control problems. The method is based on two steps. The first step is to determine the open-hop optimal control and trajectories, by using the quasilinearization and the state variables parametrization via Chebyshev polynomials of the first type. Therefore the nonlinear optimal control problem is replaced by a sequence of small quadratic programming problems which can easily be solved. The second step is to use the results of the last quasilinearization iteration, when an acceptable convergence error is achieved, to obtain the optimal feedback control law. To this end, the matrix Riccati equation and another n linear differential equations are solved using the Chebyshev polynomials of the first type. Moreover, the differentiation operational matrix of Chebyshev polynomials is introduced. To show the effectiveness of the proposed method, the simulation results of a nonlinear optimal control problem are shown.     

A new numerical learning approach to solve general Falkner–Skan model

A new numerical learning approach namely Rational Gegenbauer Least Squares Support Vector Machines (RG_LS_SVM), is introduced in this paper. RG_LS_SVM method is a combination of collocation method based on rational Gegenbauer functions and LS_SVM method. This method converts a nonlinear high order model on a semi-infinite domain to a set of linear/nonlinear equations with equality constraints which decreases computational costs. Blasius, Falkner–Skan and MHD Falkner–Skan models and the effects of various parameters over them are investigated to satisfy accuracy, validity and efficiency of the proposed method. Both Primal and Dual forms of the problems are considered and the nonlinear models are converted to linear models by applying quasilinearization method to get the better results. Comparing the results of RG_LS_SVM method with available analytical and numerical solutions show that the present methods are efficient and have fast convergence rate and high accuracy.

     

Squeezing flow of nanofluids of Cu–water and kerosene between two parallel plates by Gegenbauer Wavelet Collocation method

Operational matrices of Gegenbauer wavelets have significant role for approximate solution of differential equations. In the present study, approximate solutions of the squeezing nanofluids of Cu–kerosene and Cu–water between parallel plates with magnetic field are obtained by GW Collocation Method. The governing nonlinear PDEs may be turned into the nonlinear ODEs by similarity transformation. These nonlinear equations are turned into the set of linear ODEs by quasilinearization technique. The effective thermal conductivity and the effective dynamic viscosity of nanofluids have been taken as models of Maxwell–Garnetts and Brinkman. The effects of physical parameters have been displayed by graphs and tables.

     

An adaptive FEM with ITP approach for steady Schrödinger equation

ABSTRACT

In this paper, an adaptive numerical method is proposed for solving a 2D Schrödinger equation with an imaginary time propagation approach. The differential equation is first transferred via a Wick rotation to a real time-dependent equation, whose solution corresponds to the ground state of a given system when time approaches infinity. The temporal equation is then discretized spatially via a finite element method, and temporally utilizing a Crank–Nicolson scheme. A moving mesh strategy based on harmonic maps is considered to eliminate possible singular behaviour of the solution. Several linear and nonlinear examples are tested by using our method. The experiments demonstrate clearly that our method provides an effective way to locate the ground state of the equations through underlying eigenvalue problems.     

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.     

Fast corrected Uzawa methods for solving symmetric saddle point problems

Abstract We consider the solution of linear systems of saddle point problems by two nonlinear iterative methods, which are similar to Uzawa-type methods and called corrected Uzawa methods. Their convergence rates are analyzed. The results of numerical experiments are presented when we apply them to solve the Stokes equations discretized by mixed finite elements. Keywords: Saddle point problem, Uzawa-type algorithm, Schur complement, Stokes equation     

Numerical solutions of the regularized long-wave equation

The combined approach of quasilinearization and invariant imbedding is used for computing solutions of the nonlinear regularized long-wave (RLW) equation. The accuracy and efficiency of the scheme is tested by obtaining a solitary wave solution of the equation. In another example the development of an undular bore is discussed. The results are in good agreement with the available results.     

State feedback H ∞ control for a class of nonlinear systems using partition of unity method

The system model is nonlinear with respect to all its variables while the output model is linear. The nonlinear system model is firstly converted into an equivalent linear model with error by using partition of unity method. The stability with decay rate and the disturbance attenuation for the nonlinear system are discussed based on the equivalent model. A state feedback H _∞ controller is then proposed in terms of linear matrix inequalities (LMIs). Recommended by Editorial Board member Bin Jiang under the direction of Editor Jae Weon Choi. The authors would like to thank the anonymous reviewers and the editor for their constructive comments based on which this paper has been improved. Dong-Fang Han received the Ph.D. degree in Pure Mathematics from Shantou University in 2008. His research interests include nonlinear control, robust control and time-delay system. Yin-He Wang received the Ph.D. degree in Control Theory and Engineering from Northeast University in 1999. From 2000 to 2002, he was a Post-doctor in the department of automatic control, Northwestern Polytechnic University, China. His research interests include nonlinear systems, adaptive and robust control.     

Upper and lower eigenvalue summation bounds of the Lyapunov matrix differential equation and the application in a class time-varying nonlinear system

ABSTRACT

In this paper, we first show a class relation between the eigenvalue of functional matrix derivative and the derivative of function matrix eigenvalue. Applying the relation, we transform the time-varying linear matrix differential equation into eigenvalue differential equation. Furthermore, by using singular value decomposition and majorisation inequalities, we derive upper and lower bounds on eigenvalue summation of the solution for the Lyapunov matrix differential equation, which improve the recent results. As an application in control and optimisation, we show that our bounds could be used to discuss the stability of a class time-varying nonlinear system. Finally, we give a corresponding numerical example to show the superiority and effectiveness of the derived bounds.     

A New Approach to the Gas Dynamics Equation: An Application of the Decomposition Method

The Adomian decomposition method is used to implement the homogeneous gas dynamics equations. The analytic solution of the equation is calculated in the form of a series with easily computable components. The homogeneous problem is quickly solved by observing the self-canceling "noise" terms whose sum vanishes in the limit. Comparing the methodology with some other known techniques shows that the present approach is effective and powerful. Many test modeling problems from mathematical physics, both linear and nonlinear are discussed to illustrate the effectiveness and the performance of the decomposition method.     

Jacobi wavelet collocation method for the modified Camassa–Holm and Degasperis–Procesi equations

Matrices representations of integrations of wavelets have a major role to obtain approximate solutions of integral, differential and integro-differential equations. In the present work, operational matrix representation of rth integration of Jacobi wavelets is introduced and to find these operational matrices, all details of the processes are demonstrated for the first time. Error analysis of offered method is also investigated in present study. In the planned method, approximate solutions are constructed with the truncated Jacobi wavelets series. Approximate solutions of the modified Camassa–Holm equation and Degasperis–Procesi equation linearized using quasilinearization technique are obtained by presented method. Applicability and accuracy of presented method is demonstrated by examples. The proposed method is also convergent even when a minor number of grid points. The numerical results obtained by offered technique are compatible with those in the literature.

     

Alternative Method of Solution of the Regulator Equation: L2‐Space Approach

An alternative method for the proof of solvability of the differential equation that is a part of the regulator equation which arises from the solution of the output regulation problem. The proof uses the L²‐space based theory of solutions of partial differential equations for the case of the linear output regulation problem. In the nonlinear case, a sequence of linear equations is defined so that their solutions converge to the solution of the nonlinear problem. This is proved using the Banach Contraction Theorem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust stabilization of nonlinear systems via stable kernel representations with L2-gain bounded uncertainty

The approach to robust stabilization of linear systems using normalized left coprime factorizations with _∞ bounded uncertainty is generalized to nonlinear systems. A nonlinear perturbation model is derived, based on the concept of a stable kernel representation of nonlinear systems. The robust stabilization problem is then translated into a nonlinear disturbance feedforward _∞ optimal control problem, whose solution depends on the solvability of a single Hamilton-Jacobi equation.     

Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics

In this paper we propose a two-grid quasilinearization method for solving high order nonlinear differential equations. In the first step, the nonlinear boundary value problem is discretized on a coarse grid of size H. In the second step, the nonlinear problem is linearized around an interpolant of the computed solution (which serves as an initial guess of the quasilinearization process) at the first step. Thus, the linear problem is solved on a fine mesh of size h, h?H. On this base we develop two-grid iteration algorithms, that achieve optimal accuracy as long as the mesh size satisfies h=O(Hr²), r=1,2,… , where r is the rth Newton's iteration for the linearized differential problem. Numerical experiments show that a large class of NODEs, including the Fisher-Kolmogorov, Blasius and Emden-Fowler equations solving with two-grid algorithm will not be much more difficult than solving the corresponding linearized equations and at the same time with significant economy of the computations.     

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.     

A sixth order fast direct helmholtz equation solver

An O(h⁶) accurate difference approximation to solutions of the Helmholtz equation is derived. The discrete equations are solved using a reduction procedure and Fourier analysis. Its computational performance is compared with a fourth order similar method over a set of linear and mildly nonlinear elliptic boundary value problems.