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

In this paper, we consider the system of conformable time-fractional Robertson equations with one-dimensional diffusion having widely varying diffusion coefficients. Due to the mismatched nature of the initial and boundary conditions associated with Robertson equation, there are spurious oscillations appearing in many computational algorithms. Our goal is to obtain an approximate solutions of this system of equations using the q-homotopy analysis method (q-HAM) and examine the widely varying diffusion coefficients and the fractional order of the derivative.

On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion

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 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

The present article gives an analysis of the impact of Darcy–Forchheimer flow and partial slip along with heat transfer in single-wall carbon nanotube/multi-wall carbon nanotube (SWCNT/MWCNT)-water nanofluid flow over a stretching/shrinking rotating disk. The study considers the heat transfer in nanofluids, using both static and dynamic models, namely the Xue and Buongiorno models, respectively. The effects of thermal radiation and viscous dissipation are considered. Nonlinear transformed coupled equations are solved computationally using a spectral quasilinearisation method. The numerical results show the impact of several physical parameters on radial and tangential velocity, temperature and nanoparticle concentration fields. The slip parameter accounts for a significant enhancement in the radial velocity and a decline in the tangential/azimuthal velocity. The presence of a porous medium has the effect of reducing the amplitude of velocity and momentum layer thickness for both SWCNTs-water and MWCNTs-water nanofluids. The present study indicates that the tangential velocity of SWCNT/MWCNT-water nanofluids diminishes due to the increment in slip parameter, while the radial velocity exhibits the reverse trends. The results also show that the involvement of a porous matrix gives rise to reduced velocity of SWCNT/MWCNT-water nanofluids. It is also observed that MWCNTs make a greater contribution in augmenting the heat transfer rate compared to SWCNT nanoparticles.

https://link.springer.com/content/pdf/10.1007/s10665-019-10031-9.pdf

Thermodynamic effect in Darchy–Forchheimer nanofluid flow of a single-wall carbon nanotube/multi-wall carbon nanotube suspension due to a stretching/shrinking rotating disk: Buongiorno two-phase model

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

In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numeric...

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

In this paper, a method based on the differential quadrature method with quintic B- spline has been applied to simulate the solitary wave solution of the complex modified Korteweg- de Vries equation (CMKdV) Three test problems, namely single solitary wave, interaction of two solitary waves and interaction of three solitary waves have been investigated The efficiency and accuracy of the method have been measured by calculating maximum error norm L∞ for single solitary waves having analytical solutions Also, the three lowest conserved quantities and obtained numerical results have been compared with some of the published numerical results

https://iopscience.iop.org/article/10.1088/1742-6596/766/1/012028/pdf

Numerical solution of the complex modified Korteweg-de Vries equation by DQM

In this paper, a numerical solution of the modified Burgers equation is obtained by a cubic B-spline collocation method. In the solution process, a linearization technique based on quasi-linearization has been applied to deal with the non-linear term appearing in the equation. The computed results are compared with others selected from the available literature. The error norms $$L_{2}$$
 and $$L_{\infty }$$
 are computed and found to be sufficiently small. A Fourier stability analysis of the method is also investigated.

N. Murat Yağmurlu

Papers

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

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

Numerical solution of the complex modified Korteweg-de Vries equation by DQM

Numerical Solutions of the Modified Burgers Equation by a Cubic B-spline Collocation Method

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation