scholarly journals A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. Taghipour ◽  
H. Aminikhah
Keyword(s):  
Algebraic Equations ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Burgers Equations ◽  
Second Derivatives ◽  
Spatial Derivatives ◽  
Nicolson Scheme ◽  
Derivatives Of ◽  
Crank Nicolson

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

scholarly journals Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adel R. Hadhoud ◽  
H. M. Srivastava ◽  
Abdulqawi A. M. Rageh
Keyword(s):  
Operational Matrix ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Von Neumann ◽  
B Spline ◽  
Burgers Equations ◽  
Finite Difference Approximations ◽  
L1 Approximation ◽  
Spatial Derivatives ◽  
Numerical Approaches

AbstractThis paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

scholarly journals Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation

2019 ◽  
Vol 9 (5) ◽  
pp. 3720
Author(s):  
Kamel Al-Khaled ◽  
Issam Abu-Irwaq
Keyword(s):  
Analytical Solution ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Present Article ◽  
Point Theory ◽  
Algebraic Equations ◽  
Time Direction ◽  
Numerical Examples ◽  
Sivashinsky Equation ◽  
Crank Nicolson

The present article is designed to supply two different numerical<br />solutions for solving  Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for  Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

scholarly journals DEVELOPMENT AND ANALYSIS OF NEW APPROXIMATION OF EXTENDED CUBIC B-SPLINE TO THE NONLINEAR TIME FRACTIONAL KLEIN–GORDON EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040039 ◽  
Author(s):  
TAYYABA AKRAM ◽  
MUHAMMAD ABBAS ◽  
MUHAMMAD BILAL RIAZ ◽  
AHMAD IZANI ISMAIL ◽  
NORHASHIDAH MOHD. ALI
Keyword(s):  
Fractional Derivative ◽  
Numerical Solution ◽  
Gordon Equation ◽  
Basis Functions ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Caputo’S Fractional Derivative ◽  
Klein Gordon ◽  
Klein Gordon Equation

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

scholarly journals Fluctuation dynamos at finite correlation times using renewing flows

2015 ◽  
Vol 81 (5) ◽  
Author(s):  
Pallavi Bhat ◽  
Kandaswamy Subramanian
Keyword(s):  
Evolution Equation ◽  
Correlation Time ◽  
Leading Order ◽  
Longitudinal Correlation ◽  
Second Derivatives ◽  
Scaling Solution ◽  
Non Local ◽  
Spatial Derivatives ◽  
Derivatives Of ◽  
Finite Correlation

Fluctuation dynamos are generic to turbulent astrophysical systems. The only analytical model of the fluctuation dynamo, due to Kazantsev, assumes the velocity to be delta-correlated in time. This assumption breaks down for any realistic turbulent flow. We generalize the analytic model of fluctuation dynamos to include the effects of a finite correlation time, ${\it\tau}$, using renewing flows. The generalized evolution equation for the longitudinal correlation function $M_{L}$ leads to the standard Kazantsev equation in the ${\it\tau}\rightarrow 0$ limit, and extends it to the next order in ${\it\tau}$. We find that this evolution equation also involves third and fourth spatial derivatives of $M_{L}$, indicating that the evolution for finite-${\it\tau}$ will be non-local in general. In the perturbative case of small-${\it\tau}$ (or small Strouhal number), it can be recast using the Landau–Lifschitz approach, to one with at most second derivatives of $M_{L}$. Using both a scaling solution and the WKBJ approximation, we show that the dynamo growth rate is reduced when the correlation time is finite. Interestingly, to leading order in ${\it\tau}$, we show that the magnetic power spectrum preserves the Kazantsev form, $M(k)\propto k^{3/2}$, in the large-$k$ limit, independent of ${\it\tau}$.

Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
A. A. Alanazi ◽  
Sultan Z. Alamri ◽  
S. Shafie ◽  
Shazirawati Mohd Puzi
Keyword(s):  
Wave Packet ◽  
Kerr Nonlinearity ◽  
Momentum Conservation ◽  
Conserved Quantities ◽  
Unconditionally Stable ◽  
Content Type ◽  
First Order ◽  
Nicolson Scheme ◽  
Energy Momentum Conservation ◽  
Crank Nicolson

Purpose The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme. Design/methodology/approach The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities. Findings The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients. Originality/value The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

scholarly journals Numerical approximation of coupled 1D and 2D non-linear Burgers’ equations by employing Modified Quartic Hyperbolic B-spline Differential Quadrature Method

2021 ◽  
Vol 15 ◽  
pp. 37-55
Author(s):  
Mamta Kapoor ◽  
Varun Joshi
Keyword(s):  
Present Method ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Differential Quadrature Method ◽  
Quadrature Method ◽  
B Spline ◽  
Numerical Examples ◽  
Burgers Equations ◽  
Weighting Coefficients

In this paper, the numerical solution of coupled 1D and coupled 2D Burgers' equation is provided with the appropriate initial and boundary conditions, by implementing "modified quartic Hyperbolic B-spline DQM". In present method, the required weighting coefficients are computed using modified quartic Hyperbolic B-spline as a basis function. These coupled 1D and coupled 2D Burgers' equations got transformed into the set of ordinary differential equations, tackled by SSPRK43 scheme. Efficiency of the scheme and exactness of the obtained numerical solutions is declared with the aid of 8 numerical examples. Numerical results obtained by modified quartic Hyperbolic B-spline are efficient and it is easy to implement

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