scholarly journals Numerical solutions of electromagnetic wave model of fractional derivative using class of finite difference scheme

2021 ◽  
Author(s):  
Vijay Patel ◽  
Dhirendra Bahuguna
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Electromagnetic Waves ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Wave Model ◽  
Dielectric Media

In this article, a numerical scheme is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The numerical scheme is based on the Hermite formula. The Caputo’s fractional derivatives in time are discretized by a finite difference scheme of order O((k^(4-alpha)) & O(k^(4-beta)), 1

scholarly journals A Laplace transform finite difference scheme for the Fisher-KPP equation

2021 ◽  
Vol 15 ◽  
pp. 174830262199958
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

scholarly journals C-N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jinsong Hu ◽  
Bing Hu ◽  
Youcai Xu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Damping Term ◽  
Discrete Energy ◽  
Initial Boundary Problem ◽  
Long Wave

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank-Nicolson nonlinear-implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify the theoretical analysis.

scholarly journals A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiaojie Li ◽  
Zhoushun Zheng ◽  
Shuang Wang ◽  
Jiankang Liu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
One Dimensional ◽  
Explicit Finite Difference ◽  
Boltzmann Method

An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmann method. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have been compared with the exact solutions reported in previous studies. TheL2, L∞and Root-Mean-Square (RMS) errors in the solutions show that the scheme is accurate and effective.

On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Midpoint Method

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

COMPARISON OF FINITE‐DIFFERENCE SCHEMES FOR THE GROSS‐PITAEVSKII EQUATION

2009 ◽  
Vol 14 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Vyacheslav A. Trofimov ◽  
Nikolai Peskov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Pitaevskii Equation ◽  
Conservative Scheme ◽  
Conservative Finite Difference Scheme

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L 2 norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

A STUDY OF THE SHORT WAVE COMPONENTS IN COMPUTATIONAL ACOUSTICS

1993 ◽  
Vol 01 (01) ◽  
pp. 1-30 ◽  
Author(s):  
CHRISTOPHER K. W. TAM ◽  
JAY C. WEBB ◽  
ZHONG DONG
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Euler Equations ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Long Waves ◽  
Short Waves

It is shown by using a Dispersion-Relation-Preserving [Formula: see text] finite difference scheme that it is feasible to perform direct numerical simulation of acoustic wave propagation problems. The finite difference equations of the [Formula: see text] scheme have essentially the same Fourier-Laplace transforms and hence dispersion relations as the original linearized Euler equations over a broad range of wavenumbers (here referred to as long waves). Thus it is guaranteed that the acoustic waves, the entropy and the vorticity waves computed by the [Formula: see text] scheme are good approximations of those of the exact solutions of Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves of a computation scheme are, therefore, contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration and standard dispersive wave theory. Numerical results of direct simulations of these waves are reported. These waves can be generated by discontinuous initial conditions. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.

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