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.     

一类强非线性二阶微分方程的多模态近似解析解研究

利用自治力学系统的哈密顿函数为守恒量的性质,提出一种求非线性二阶微分方程多模态近似解析解的方法,称为哈密顿函数法.首先,介绍哈密顿函数法求多模态近似解的基本理论.其次,以质点在旋转的抛物线上运动为模型建立强非线性二阶微分方程.最后,用哈密顿函数法求得在给定初始条件和参数下强非线性二阶微分方程的三模态近似解析解表达式,作出三模态近似解析解的解曲线,并与直接用Mathematica软件作出的解曲线进行比较,讨论三模态近似解析解的精确性.结果表明：用哈密顿函数法求得的三模态近似解析解的解曲线与直接用Mathematica软件作出的解曲线十分吻合.     

A New Functional Iterative Algorithm for the Regularized Long-Wave Equation Using an Integral Equation Formalism

A fundamental question in computational nonlinear partial differential equations is raised to discover if one could construct a functional iterative algorithm for the regularized long-wave (RLW) equation (or the Benjamin–Bona–Mahony equation) based on an integral equation formalism? Here, the RLW equation is a third-order nonlinear partial differential equation, describing physically nonlinear dispersive waves in shallow water. For the question, the concept of pseudo-parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118–138, 2017), is introduced and incorporated into the RLW equation. Thereby, dual nonlinear integral equations of second kind involving the parameter are formulated. The application of the fixed point theorem to the integral equations results in a new (semi-analytic and derivative-free) functional iteration algorithm (as required). The new algorithm allows the exploration of new regimes of pseudo-parameters, so that it can be valid for a much wider range (in the complex plane) of pseudo-parameter values than that of Jang (2017). Being fairly simple (or straightforward), the iteration algorithm is found to be not only stable but accurate. Specifically, a numerical experiment on a solitary wave is performed on the convergence and accuracy of the iteration for various complex values of the pseudo-parameters, further providing the regions of convergence subject to some constraints in the complex plane. Moreover, the algorithm yields a particularly relevant physical investigation of the nonlinear behavior near the front of a slowly varying wave train, in which, indeed, interesting nonlinear wave features are demonstrated. As a consequence, the preceding question may be answered.     

GPG-stability of Runge-Kutta methods for generalized delay differential systems

The GP_G-stability of Runge-Kutta methods for the numerical solutions of the systems of delay differential equations is considered. The stability behaviour of implicit Runge-Kutta methods (IRK) is analyzed for the solution of the system of linear test equations with multiple delay terms. After an establishment of a sufficient condition for asymptotic stability of the solutions of the system, a criterion of numerical stability of IRK with the Lagrange interpolation process is given for any stepsize of the method.     

Tau-lines: A new hybrid approach to the numerical treatment of crack problems based on the Tau method

A new hybrid computational technique based on Ortiz' recursive formulation of the Tau method is introduced in this paper and applied to some model singular boundary value problems which are relevant to fracture mechanics (modes I and III). This technique, which we call Tau-lines, combines the method of lines with the Tau method. The former is used in the construction of a system of coupled ordinary differential equations which is the discretized model of a given partial differential equation; the latter is used to find an accurate approximation of the solution of such a system which involves no further discretization.Recent theoretical results on the Tau method show that its error is optimal in the sense that, for a given degree n, it has the same order of error as the best uniform approximation of the exact solution by algebraic polynomials of degree n.The present work may be considered as an encouraging first step towards the development of the Tau-lines approach into a useful and efficient computational tool for the numerical treatment of problems in fracture mechanics.     

An Algorithm to Determine Potential Systems in Mathematica

A symmetry analysis of differential equations plays an important role in discovering new solutions. In this article potential symmetries characterized by nonlocal transformations are introduced and an algorithm implemented in the computer algebra system Mathematica is presented which determines automatically potential systems and the corresponding potential symmetries. The possibilities of this algorithm are discussed by the examples of a nonlinear telegraph equation and the axial symmetric wave equation.     

On some explicit non-standard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity

We introduce non-standard, finite-difference schemes to approximate nonnegative solutions of a weakly hyperbolic (that is, a hyperbolic partial differential equation in which the second-order time-derivative is multiplied by a relatively small positive constant), nonlinear partial differential equation that generalizes the well-known equation of Fisher-KPP from mathematical biology. The methods are consistent of order 𝒪(Δ t+(Δ x)²). As a means to verify the validity of the techniques, we compare our numerical simulations with known exact solutions of particular cases of our model. The results show that there is an excellent agreement between the theory and the computational outcomes.     

Hybrid dynamical systems: robust control synthesis problems

The paper presents a new approach to robust control synthesis problems for hybrid dynamical systems. The hybrid system under consideration is a composite of a continuous plant and a discrete event controller. State and output feedback problems are considered. The main results are given in terms of the existence of suitable solutions to a dynamic programming equation and a Riccati differential equation of the H^∞ filtering type. These results show a connection between the theories of hybrid dynamical systems and robust and nonlinear control.     

An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations

In this work, we consider numerical solutions of the FitzHugh–Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the nonlinear space independent system of equations. The subequation approach is applied to derive dynamically consistent schemes for the submodels. This is followed by a consistent and systematic merging of the subschemes to give three explicit nonstandard finite difference schemes in the limit of fast extinction and slow recovery. A qualitative study of the schemes together with the error analysis is presented. Numerical simulations are given to support the theoretical results and verify the efficiency of the proposed schemes.     

A two-grid expanded mixed element method for nonlinear non-Fickian flow model in porous media

The full discrete scheme of expanded mixed finite element approximation is introduced for nonlinear parabolic integro-differential equations modelling non-Fickian flow in porous media. To solve the nonlinear problem efficiently, a two-grid algorithm is considered and analysed. This approach allows us to perform all of the nonlinear iterations on a coarse grid space and just execute a linear system on a fine grid space. Based on RT_k mixed element space, error estimates and convergence results are presented for solutions of the two-grid method. Some numerical examples are given to verify the theoretical predictions and show the efficiency of the two-grid method.     

Large amplitude solutions of the nonlinear wave equation for an ideal,cold three-fluid plasma

A method for obtaining numerical solutions of the nonlinear eigenvalue problem ? (e) e ? λ ? × (? × e) = 0 is described. The nonlinearity is due to the ponderomotive force exerted by the wave on a plasma, which modifies the densities. A centred finite-difference scheme is used, which avoids spectra pollution. Starting from the linear, small amplitude solution, a predictor-corrector method allows one to reach solutions of increasing amplitude.     

Numerical computations for singular semilinear elliptic boundary value problems

The paper studies a class of Dirichlet problems with homogeneous boundary conditions for singular semilinear elliptic equations in a bounded smooth domain in

. A numerical method is devised to construct an approximate Green's function by using radial basis functions and the method of fundamental solutions. An estimate of the error involved is also given. A weak solution of the above given problem is a solution of its corresponding nonlinear integral equation. A computational method is given to find the minimal weak solution U, and the critical index λ* (such that a weak solution U exists for λ < λ*, and U does not exist for λ > λ*).     

A symbolic-numerical method for solving the differential equation describing the states of polarizable particle in Coulomb potential

Methods for solving the differential equation describing the wave functions of a polarizable particle in the Coulomb potential are discussed. Relations between the coefficients under which the general solution of this equation can be found in analytical form are obtained. For the case of zero polarizability, the general solution to this equation in terms of special functions is obtained; for the first values of the parameter j, plots of the corresponding solutions are presented. For nonzero polarizability and certain specially chosen values of the energy level parameter, solutions possessing the required physical properties for the varying parameter j are constructed on fairly large intervals of the argument values using numerical methods and functional objects of the type DifferentialRoot. Instructions in Mathematica are presented that allow computer-aided analysis using numerical and analytical methods and visualization of the resulting solutions.     

Two iteration processes for computing positive definite solutions of the equation X - A*X−nA = Q

In the present paper, we suggest two iteration methods for obtaining positive definite solutions of nonlinear matrix equation X - A^*X⁻ⁿA = Q, for the integer n ≥ 1. We obtain sufficient conditions for existence of the solutions for the matrix equation. Finally, some numerical examples to illustrate the effectiveness of the algorithms and some remarks.     

Metodi Multistep con accelerazione per la risoluzione numerica dei sistemi di equazioni differenziali ordinarie

In this paper a procedure for the acceleration of the convergence is given. It allows the doubling of the order of the multistep methods for the numerical solution of the systems of ordinary differential equations: $$Y' = F(x,Y); Y_0 = Y(x_0 ) \begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \in [a,b]$$ whereY andF(x,Y) aret-vectors.     

A semi-implicit Hall-MHD solver using whistler wave preconditioning

The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form Δt∝²(Δx) for explicit calculations. A new semi-implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It is based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

Numerical method of bicharacteristics for hyperbolic partial functional differential equations

Classical solutions of mixed problems for first-order partial functional differential equations in several independent variables are approximated in this paper by solutions of a difference problem of Euler type. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. Convergence of explicit difference schemes is proved by consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type. Differential equations with deviated variables and differential integral equations can be obtained from the general model by specifying the given operators.      

Accelerazione della convergenza per metodi di risoluzione di equazioni differenziali ordinarie

In this paper a procedure for the acceleration of the convergence is given. It allows the doubling of the order of the multistep methods for the numerical solution of the ordinary differential equation $$y' = f(x,y),y_0 = y(x_0 );{}_{x_0 }^x \in [a,b].$$ This acceleration is applicable to any method of orderp≥1 whatsoever, and it requires the evaluation of the globalp-th derivate of the functionf(x, y). Special attention is confined to the 2⁰ and 3⁰ order methods, and a numerical exemple is provided.     

Structured matrix algorithms for inverse scattering on the line

Abstract In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schr?dinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation