In the present work, a new stochastic computing technique based on evolutionary cubic spline method (CSM) is introduced for solving nonlinear singular Thomas–Fermi system arising in atomic physics. The concept of cubic splines interpolation is engaged with an evolutionary optimization technique based on genetic algorithms (GAs) hybrid with sequential quadratic programming (SQP) to develop a proposed methodology, CSM-GASQP, that can solve nonlinear differential equations, and GA produces the optimized value for the coefficients of cubic splines, while SQP is used for rapid local refinements. The developed method CSM-GASQP for different lengths of the splines is applied effectively to solve the Thomas–Fermi equation for number of scenarios. Results show that proposed evolutionary paradigm CSM-GASQP is an effective, alternate, accurate, and reliable stochastic numerical solver for stiff nonlinear singular Thomas–Fermi systems.

A new heuristic computational solver for nonlinear singular Thomas–Fermi system using evolutionary optimized cubic splines

: It is shown that under certain approximations the basic system of equations governing the propagation of ion acoustic waves can be reduced to the Kortweg-deVries equation. (Author)

On the propagation of ion acoustic solitary waves of small amplitude.

Numerical difficulties arise in the solution of Burgers' equation for the case of a large Reynolds number. The aim of this paper is to extend the finite element method proposed by Caldwell et al. to the general case of n elements. The method is tested out on Burgers' equation for two different initial conditions and proves to be much more accurate than finite difference methods for a large Reynolds number.

Solution of Burgers' equation with a large Reynolds number

In this paper, the definition of fractional calculus introduces a newly constructed discrete chaotic map. Interestingly, when the order is extended to the improper fractional order, the map still shows a chaotic state with appropriate parameters. The related dynamical behavior is analyzed by a phase diagram, bifurcation model, maximum Lyapunov exponent diagram, and approximate entropy algorithm method. The research results show that both the chaotic range of the fractional-order and the improper fractional-order are larger than the chaotic range of the integer-order, and they have richer dynamical behaviors. Besides, we found that multi-stability phenomena also exist in discrete chaotic systems, and the multiple stability of fractional order is more complicated than that of integer order, and the types of coexisting attractors are more abundant. Finally, the digital circuit and pseudo-random sequence generator of the discrete system are designed. This research guides the application and teaching of discrete fractional-order systems.

Dynamical analysis of the improper fractional-order 2D-SCLMM and its DSP implementation

In this paper, we study the Adomian decomposition method (ADM for short) including its iterative scheme and convergence analysis, which is a simple and effective technique in dealing with some nonlinear problems. We take algebraic equations and fractional differential equations as applications to illustrate ADM’s efficiency.

/pdf/application-of-adomian-decomposition-method-to-nonlinear-4r9n59foa0.pdf

Application of Adomian decomposition method to nonlinear systems

In this paper, the Korteweg-de Vries-Burgers’ (KdVB) equation is solved numerically by anew differential quadrature method based on quintic B-spline functions. The weightingcoefficients are obtained by semi-explicit algorithm including an algebraic system with fivebandcoefficient matrix. The L2 and L∞ error norms and lowest three invariants 1 2 I , I and3 I have computed to compare with some earlier studies. Stability analysis of the method isalso given. The obtained numerical results show that the present method performs better thanthe most of the methods available in the literature.

https://journalskuwait.org/kjs/index.php/KJS/article/download/198/75

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

In this study, an effective differential quadrature method (DQM) which is based on modified cubic B-spline (MCB) has been implemented to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. After separating the Schrodinger equation into coupled real value differential equations,we have discretized using DQM and then obtained ordinary differential equation systems. For time integration, low storage strong stability-preserving Runge–Kutta method has been used. Numerical solutions of five different test problems have been obtained. The efficiency and accuracy of the method have been measured by calculating error norms L2 and Linfinity and two lowest invariants I1 and I2. Also relative changes of invariants are given. The newly obtained numerical results have been compared with the published numerical results and a comparison has shown that the MCB-DQM is an effective numerical scheme to solve the nonlinear Schrodinger equation.

An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method

In the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrodinger (NLS) equation. For this purpose, first of all, the Schrodinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, $L_{2}$
 and $L_{\infty}$
 , as well as the two lowest invariants, $I_{1}$
 and $I_{2}$
 , have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.

A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

The main purpose of this work is to obtain the numerical solutions for the Kawahara equation via the Crank–Nicolson–Differential Quadrature Method based on modified cubic B-splines (MCBC-DQM). First, the Kawahara equation has been discretized using Crank–Nicolson scheme. Then, Rubin and Graves linearization technique has been utilized and differential quadrature method has been applied to obtain algebraic equation system. Four different test problems, namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, and wave generation, have been solved. Next, to be able to test the efficiency and accuracy of the newly applied method, the error norms $$ L _{2}$$
 and $$ L _{\infty }$$
 as well as the three lowest invariants $$ I _{1}$$
 , $$ I _{2}$$
 , and $$ I _{3}$$
 have been computed. Besides those, the relative changes of invariants have been reported. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. The comparison of present results with earlier works showed that the newly method may be provide significant benefit in case of numerical solutions of the other nonlinear differential equations.

An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method

A numerical solution of the modified Burgers' equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing L 2 and L ∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

/pdf/two-different-methods-for-numerical-solution-of-the-modified-3jmn3tugfb.pdf

Ali Başhan

Papers

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method

A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method

Two different methods for numerical solution of the modified Burgers' equation.