A scale-dependent finite difference approximation for time fractional differential equation

2018 ◽  
Vol 63 (3) ◽  
pp. 429-442 ◽  
Author(s):  
XiaoTing Liu ◽  
HongGuang Sun ◽  
Yong Zhang ◽  
Zhuojia Fu
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Fractional Differential Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
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 Research on the numerical solution and dynamic properties of nonlinear fractional differential equations

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>

scholarly journals Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction

2021 ◽  
Vol 5 (3) ◽  
pp. 83
Author(s):  
Bilgi Görkem Yazgaç ◽  
Mürvet Kırcı
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Frequency Estimation ◽  
Signal Reconstruction ◽  
Reconstruction Method ◽  
Reconstruction Algorithms ◽  
Instantaneous Frequency Estimation ◽  
Proposed Model ◽  
Fractional Differential ◽  
Instantaneous Frequencies

In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter.

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