scholarly journals Soliton Solution of Schrödinger Equation Using Cubic B-Spline Galerkin Method

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 108 ◽  
Author(s):  
Azhar Iqbal ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Galerkin Method ◽  
Time Discretization ◽  
Nls Equation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Basis ◽  
Space Discretization ◽  
Second Order Convergence ◽  
Nicolson Scheme ◽  
Good Agreement

The non-linear Schrödinger (NLS) equation has often been used as a model equation in the study of quantum states of physical systems. Numerical solution of NLS equation is obtained using cubic B-spline Galerkin method. We have applied the Crank–Nicolson scheme for time discretization and the cubic B-spline basis function for space discretization. Three numerical problems, including single soliton, interaction of two solitons and birth of standing soliton, are demonstrated to evaluate to the performance and accuracy of the method. The error norms and conservation laws are determined and found to be in good agreement with the published results. The obtained results show that the approach is feasible and accurate. The proposed method has almost second order convergence. The linear stability of the method is performed using the Von Neumann method.

Solution of non-linear time fractional telegraph equation with source term using B-spline and Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdul Majeed ◽  
Mohsin Kamran ◽  
Noreen Asghar
Keyword(s):  
Caputo Derivative ◽  
Numerical Solutions ◽  
Linear Time ◽  
Initial Boundary ◽  
Telegraph Equation ◽  
Test Problems ◽  
Von Neumann ◽  
B Spline ◽  
Nonlinear Part ◽  
Spline Basis

Abstract This article focusses on the implementation of cubic B-spline approach to investigate numerical solutions of inhomogeneous time fractional nonlinear telegraph equation using Caputo derivative. L1 formula is used to discretize the Caputo derivative, while B-spline basis functions are used to interpolate the spatial derivative. For nonlinear part, the existing linearization formula is applied after generalizing it for all positive integers. The algorithm for the simulation is also presented. The efficiency of the proposed scheme is examined on three test problems with different initial boundary conditions. The influence of parameter α on the solution profile for different values is demonstrated graphically and numerically. Moreover, the convergence of the proposed scheme is analyzed and the scheme is proved to be unconditionally stable by von Neumann Fourier formula. To quantify the accuracy of the proposed scheme, error norms are computed and was found to be effective and efficient for nonlinear fractional partial differential equations (FPDEs).

An Adaptive Wavelet Finite Element Method with High-Order B-Spline Basis Functions

2007 ◽  
Vol 345-346 ◽  
pp. 877-880 ◽  
Author(s):  
Satoyuki Tanaka ◽  
Hiroshi Okada
Keyword(s):  
Galerkin Method ◽  
Scaling Function ◽  
Adaptive Strategy ◽  
Basis Functions ◽  
Adaptive Wavelet ◽  
B Spline ◽  
Wavelet Galerkin Method ◽  
Wavelet Finite Element Method ◽  
Wavelet Finite Element ◽  
Spline Basis

In this paper, an adaptive strategy based on a B-spline wavelet Galerkin method is discussed. The authors have developed the wavelet Galerkin Method which utilizes quadratic and cubic B-spline scaling function/wavelet as its basis functions. The developed B-spline Galerkin Method has been proven to be very accurate in the analyses of elastostatics. Then the authors added a capability to adaptively adjust the special resolution of the basis functions by adding the wavelet basis functions where the resolution needs to be enhanced.

scholarly journals Wave-function visualization of core-induced interaction of a non-hydrogenic Rydberg atom in an electric field

2016 ◽  
Vol 94 (7) ◽  
pp. 671-678 ◽  
Author(s):  
Wei Gao ◽  
Min Deng ◽  
Hong Cheng ◽  
ShanShan Zhang ◽  
John B. Delos ◽  
...  
Keyword(s):  
Wave Function ◽  
Electric Field ◽  
Energy Levels ◽  
Rydberg Atom ◽  
Oscillator Strengths ◽  
Radial Part ◽  
B Spline ◽  
Line Intensities ◽  
Spline Basis ◽  
Good Agreement

In this paper we report a wave-function visualization of the core-induced interaction of a non-hydrogenic Rydberg atom in an electric field and its resultant redistribution of the spectral line intensities. The energy levels and oscillator strengths are calculated based on an effective potential where the radial part of the wavefunction is expanded in a B-spline basis. The calculation is in good agreement with the experimental measurements of the absorption spectrum of sodium atoms in an electric field of F = 840 V/cm, both below and above the saddle point Esp. The visualization of the wavefunction for the eigenstates can help us to see how they stem from the interaction between two or more red and blue hydrogenic states. An approximate localized symmetry close to the atomic core is also observed in the wavefunctions, which accounts for the alternation of oscillator strengths in the experiment.

scholarly journals A fourth-order B-spline collocation method for nonlinear Burgers–Fisher equation

2020 ◽  
Vol 14 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Aditi Singh ◽  
Sumita Dahiya ◽  
S. P. Singh
Keyword(s):  
Collocation Method ◽  
Fourth Order ◽  
Numerical Study ◽  
Fisher Equation ◽  
Spline Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Spline Collocation Method ◽  
The Stability ◽  
Nicolson Scheme

AbstractA fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics, gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

scholarly journals Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd. Majid ◽  
Ahmad Izani Md. Ismail ◽  
Abdur Rashid
Keyword(s):  
Space Dimension ◽  
Time Derivative ◽  
Nonlocal Boundary ◽  
Diffusion Problem ◽  
Computational Time ◽  
Von Neumann ◽  
B Spline ◽  
Boundary Constraints ◽  
Good Agreement ◽  
Diffusion Problems

A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of a nonclassical diffusion problem with nonlocal boundary constraints. The standard finite difference approach is applied to discretize the time derivative while cubic trigonometric B-spline is utilized as an interpolating function in the space dimension. The technique is shown to be unconditionally stable using the von Neumann method. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. TheL2andL∞error norms are also computed at different times for different space size steps to assess the performance of the proposed technique. The technique requires smaller computational time than several other methods and the numerical results are found to be in good agreement with known solutions and with existing schemes in the literature.

scholarly journals Modeling cesium migration through Opalinus clay: a benchmark for single- and multi-species sorption-diffusion models

2021 ◽  
Author(s):  
Jesús F. Águila ◽  
Vanessa Montoya ◽  
Javier Samper ◽  
Luis Montenegro ◽  
Georg Kosakowski ◽  
...  
Keyword(s):  
Time Discretization ◽  
Single Species ◽  
Opalinus Clay ◽  
Sorption Behavior ◽  
Diffusion Experiment ◽  
Modeling Tools ◽  
Cesium Sorption ◽  
Code Performance ◽  
Good Agreement

AbstractSophisticated modeling of the migration of sorbing radionuclides in compacted claystones is needed for supporting the safety analysis of deep geological repositories for radioactive waste, which requires robust modeling tools/codes. Here, a benchmark related to a long term laboratory scale diffusion experiment of cesium, a moderately sorbing radionuclide, through Opalinus clay is presented. The benchmark was performed with the following codes: CORE2DV5, Flotran, COMSOL Multiphysics, OpenGeoSys-GEM, MCOTAC and PHREEQC v.3. The migration setup was solved with two different conceptual models, i) a single-species model by using a look-up table for a cesium sorption isotherm and ii) a multi-species diffusion model including a complex mechanistic cesium sorption model. The calculations were performed for three different cesium boundary concentrations (10−3, 10−5, 10−7 mol / L) to investigate the models/codes capabilities taking into account the nonlinear sorption behavior of cesium. Generally, good agreement for both single- and multi-species benchmark concepts could be achieved, however, some discrepancies have been identified, especially near the boundaries, where code specific spatial (and time) discretization had to be improved to achieve better agreement at the expense of longer computation times. In addition, the benchmark exercise yielded useful information on code performance, setup options, input and output data management, and post processing options. Finally, the comparison of single-species and multi-species model concepts showed that the single-species approach yielded generally earlier breakthrough, because this approach accounts neither for cation exchange of Cs+ with K+ and Na+, nor K+ and Na+ diffusion in the pore water.

scholarly journals Efficient Numerical Evaluation of Thermodynamic Quantities on Infinite (Semi-)classical Chains

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Christian B. Mendl ◽  
Folkmar Bornemann
Keyword(s):  
Transfer Operator ◽  
Classical System ◽  
Free Energy Density ◽  
Classical Particle ◽  
Thermodynamic Quantities ◽  
Nls Equation ◽  
Classical Systems ◽  
Particle Chain ◽  
Dynamics Simulations ◽  
Good Agreement

AbstractThis work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger (NLS) equation and to a classical system on a cylindrical lattice. A comparison with molecular dynamics simulations performed for the NLS model shows very good agreement.

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