On the solvability and approximate solution of a one-dimensional singular problem for a p-Laplacian fractional differential equation

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.

Download Full-text

Approximate solution of inhomogeneous fractional differential equation

Advances in Nonlinear Analysis ◽

10.1515/anona-2012-0011 ◽

2012 ◽

Vol 1 (4) ◽

Cited By ~ 1

Author(s):

Biju K. Dutta ◽

Laxmi K. Arora

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Fractional Differential Equation ◽

Fractional Differential

Download Full-text

Research on the numerical solution and dynamic properties of nonlinear fractional differential equations

Lifelong Education ◽

10.18282/le.v7i2.776 ◽

2018 ◽

Vol 7 (2) ◽

pp. 42

Author(s):

Na Wang ◽

Yanqin Yang

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Finite Difference ◽

Fractional Calculus ◽

Fractional Differential Equation ◽

Fractional Order ◽

Fractional Integral ◽

Dynamic Properties ◽

Estimation Error ◽

Fractional Differential

<p>Fractional calculus is an important branch of mathematical analysis, which is specialized in the study of the mathematical properties and applications of arbitrary order integral and differential, and is the extension of the traditional integral calculus. At present, fractional integral and derivative operators are mainly used to calculate fractional calculus, among which the most famous ones are Riemann-Liouville fractional integral and derivative, Caputo fractional derivative, Grümwald-Letnikov fractional integral and derivative, etc. At present, the numerical algorithm of finite difference scheme is mainly used to solve the approximate solution of the equation, to solve the fractional differential equation. Through the finite difference of time fractional order or space fractional order, the approximate solution of the equation is obtained, and the stability, convergence and compatibility of the scheme are checked, and the convergence order and estimation error are calculated. At present, the theory and method of nonlinear fractional differential equation are widely used in the study of various intermediate processes and critical phenomena in finance, physics and mechanics, which can better fit some natural physical processes and dynamic system processes.</p>

Download Full-text

Approximate solution for a 2-D fractional differential equation with discrete random noise

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2020.109650 ◽

2020 ◽

Vol 133 ◽

pp. 109650 ◽

Cited By ~ 1

Author(s):

Nguyen Huy Tuan ◽

Dumitru Baleanu ◽

Tran Ngoc Thach ◽

Donal O’Regan ◽

Nguyen Huu Can

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Fractional Differential Equation ◽

Random Noise ◽

Fractional Differential

Download Full-text

Solution of Multi-order Fractional Differential Equation Based on Conformable Derivative by Shifted Legendre Polynomial

Cihan University-Erbil Scientific Journal ◽

10.24086/cuesj.v5n2y2021.pp64-68 ◽

2021 ◽

Vol 5 (2) ◽

pp. 64-68

Author(s):

Salim S. Mahmood ◽

Kamaran J. Hamad ◽

‎Milad A. Kareem ◽

Asrin F. Shex

Keyword(s):

Differential Equation ◽

Approximate Solution ◽

Linear Equation ◽

Fractional Differential Equation ◽

Legendre Polynomial ◽

Conformable Derivative ◽

Approximation Solution ◽

Non Linear Equation ◽

Non Linear ◽

Fractional Differential

The aim of this article is the way for finding approximation solution of multi-order fractional differential equation with conformable sense with use approximated function by shifted Legendre polynomial, the method is easy and powerful for get our results of the linear and non-linear equation, the background idea behind this method is finding system of algebra after achieving messing variable is that mean obtain approximate solution, a few examples illustrates for presented how much our method is capable.

Download Full-text