scholarly journals Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 98
Author(s):  
Andang Sunarto ◽  
Praveen Agarwal ◽  
Jumat Sulaiman ◽  
Jackel Vui Lung Chew ◽  
Shaher Momani
Keyword(s):  
Numerical Methods ◽  
Linear System ◽  
Finite Difference ◽  
Fractional Derivative ◽  
Difference Scheme ◽  
Relaxation Method ◽  
Efficiency Analysis ◽  
Mathematical Equation ◽  
Large Matrix ◽  
Mathematical Equations

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.

scholarly journals NUMERICAL METHODS AND ERROR ESTIMATES FOR A SINGULAR BOUNDARY‐VALUE PROBLEM

2002 ◽  
Vol 7 (2) ◽  
pp. 271-284 ◽  
Author(s):  
P. M. Lima ◽  
A. M. Oliveira
Keyword(s):  
Asymptotic Expansion ◽  
Numerical Methods ◽  
Continuous Function ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Singular Boundary ◽  
Singular Boundary Value Problem ◽  
Numerical Examples

In this paper we analyze a class of equations of the form y? (x) = —g(x) xp (y(x)) q where p and q are real parameters satisfying p > _1 , g < _1 and g is a positive and continuous function on [0,1]. We search for positive solutions which satisfy the boundary conditions y'(0)=y(l) = 0. Numerical approximations of the solution are obtained by means of a finite difference scheme and the asymptotic expansion of the discretization error is deduced. Some numerical examples are analyzed.

scholarly journals Comparative analysis of the numerical methods for 3D continuation problem for parabolic equation with data on the part of the boundary

2021 ◽  
Vol 2092 (1) ◽  
pp. 012010
Author(s):  
Aleksei Prikhodko ◽  
Maxim Shishlenin
Keyword(s):  
Inverse Problem ◽  
Comparative Analysis ◽  
Parabolic Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Gradient Method ◽  
Finite Difference Scheme ◽  
Three Dimensional ◽  
Continuation Problem

Abstract The problem of continuation of the solution of a three-dimensional parabolic equation with data given on a time-like surface is investigated. Two numerical methods for solving the continuation problem are compared: the finite-difference scheme inversion and the solution of inverse problem by gradient method. The functional gradient formula is obtained. The results of numerical calculations are presented.

scholarly journals A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Zongqi Liang ◽  
Yubin Yan ◽  
Guorong Cai
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
The Stability ◽  
Predictor Corrector ◽  
Theoretical Results ◽  
Methods Numerical

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

scholarly journals Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation

2021 ◽  
Vol 6 (1) ◽  
pp. 23
Author(s):  
Dmitriy Tverdyi ◽  
Roman Parovik
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Riccati Equation ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Computational Accuracy ◽  
Order Of Accuracy ◽  
First Order

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

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 Application of Compact Finite Difference Method for Solving Some Type of Fractional Derivative Equations

2021 ◽  
Vol 15 ◽  
pp. 1324-1335
Author(s):  
Mahboubeh Molavi-Arabshahi ◽  
Zahra Saeidi
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Equilibrium Points ◽  
Accurate Solution ◽  
Order System ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

In this paper, the compact finite difference scheme as unconditionally stable method is applied to some type of fractional derivative equation. We intend to solve with this scheme two kinds of a fractional derivative, first a fractional order system of Granwald-Letnikov type 1 for influenza and second fractional reaction sub diffusion equation. Also, we analyzed the stability of equilibrium points of this system. The convergence of the compact finite difference scheme in norm 2 are proved. Finally, various cases are used to test the numerical method. In comparison to other existing numerical methods, our results show that the scheme yields an accurate solution that is quick to compute.

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