scholarly journals Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

2021 ◽  
Vol 6 (8) ◽  
pp. 8611-8624
Author(s):  
Yong Zhang ◽  
◽  
Xiaobing Bao ◽  
Li-Bin Liu ◽  
Zhifang Liang ◽  
...  
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Fractional Differential Equation ◽  
Finite Difference Scheme ◽  
Adaptive Grid ◽  
Fractional Differential

scholarly journals Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Andang Sunarto ◽  
Praveen Agarwal ◽  
Jumat Sulaiman ◽  
Jackel Vui Lung Chew ◽  
Elayaraja Aruchunan
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Iterative Method ◽  
Difference Scheme ◽  
Fractional Differential Equation ◽  
Relaxation Method ◽  
Mathematical Physics ◽  
One Dimensional ◽  
Fractional Differential ◽  
Preconditioned Iterative

AbstractThis paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.

scholarly journals Stability Analysis of the Laminar Boundary Layer Flow

1970 ◽  
Vol 29 ◽  
pp. 23-34
Author(s):  
Nazma Parveen ◽  
Md MK Chowdhury
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Stability Analysis ◽  
Finite Difference ◽  
Difference Scheme ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Finite Difference Scheme ◽  
Linear Set ◽  
Layer Flow

In this paper, stability analysis of incompressible laminar boundary layer flow is presented. For this approach, the partial differential equation is converted to ordinary differential equation by suitable approximation. The implicit finite difference scheme is used to find the point of separations of the boundary layer equations. The finite difference equations for the given flow at each longitudinal position form a linear set with a tridiagonal coefficient matrix. To ensure the correct results, the methods are checked with standard flows like flow past circular cylinder, Howarth’s linear decelerating flows. These methods are demonstrated to compute accurately the separation points of several flows for which comparisons are made with previously published results. Then various series are tested with computer codes. At last, the stability diagram for plane poiseuille flow is shown. Key words: Stability; finite difference scheme; point of separation GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 23-34  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8512

A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shufang Hu ◽  
Wenlin Qiu ◽  
Hongbin Chen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Integro Differential Equation ◽  
Compact Finite Difference Method ◽  
Backward Euler ◽  
Predictor Corrector

Abstract A predictor–corrector compact finite difference scheme is proposed for a nonlinear partial integro-differential equation. In our method, the time direction is approximated by backward Euler scheme and the Riemann–Liouville (R–L) fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, a two-step predictor–corrector (P–C) algorithm called MacCormack method is used. A fully discrete scheme is constructed with space discretization by compact finite difference method. Numerical experiment presents the scheme is in good agreement with the theoretical analysis.

Sign in / Sign up

Export Citation Format

Share Document