The Grunwald-Letnikov method for fractional differential equations

被引：306

作者：

Scherer, Rudolf ^{[1
]}

Kalla, Shyam L. ^{[2
]}

Tang, Yifa ^{[3
]}

Huang, Jianfei ^{[3
]}

机构：

[1] Karlsruhe Inst Technol, Inst Appl & Numer Math, D-76128 Karlsruhe, Germany

[2] Vyas Inst Higher Educ, Inst Math, Jodhpur, Rajasthan, India

[3] Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China

来源：

COMPUTERS & MATHEMATICS WITH APPLICATIONS | 2011年 / 62卷 / 03期

关键词：

Fractional derivatives; Fractional differential equations; Grunwald-Letnikov approximation; Difference methods; Binomial coefficients; Stability; DIFFUSION-WAVE EQUATION; NUMERICAL-SOLUTION; OIL STRATA; SCHEME;

D O I：

10.1016/j.camwa.2011.03.054

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given. (C) 2011 Elsevier Ltd. All rights reserved.

引用

页码：902 / 917

页数：16

共 50 条

[41] A New Stability Theory for Grunwald-Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework
Hunek, Wojciech Przemyslaw
Wach, Lukasz
SYMMETRY-BASEL, 2019, 11 (10):
[42] Complex Grunwald-Letnikov, Liouville, Riemann-Liouville, and Caputo derivatives for analytic functions
Ortigueira, Manuel D.
Rodriguez-Germa, Luis
Trujillo, Juan J.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, 2011, 16 (11) : 4174 - 4182
[43] ROBUST FRACTIONAL-ORDER PERFECT CONTROL FOR NON-FULL RANK PLANTS DESCRIBED IN THE GRUNWALD-LETNIKOV IMC FRAMEWORK
Hunek, Wojciech P.
Feliks, Tomasz
FRACTIONAL CALCULUS AND APPLIED ANALYSIS, 2021, 24 (04) : 1257 - 1274
[44] Time-domain approximations to the Grunwald-Letnikov difference with application to modeling of fractional-order state space systems
Stanislawski, Rafal
Latawiec, Krzysztof J.
Lukaniszyn, Marian
Galek, Marcin
2015 20TH INTERNATIONAL CONFERENCE ON METHODS AND MODELS IN AUTOMATION AND ROBOTICS (MMAR), 2015, : 579 - 584
[45] Enhancement of MRI images of brain tumor using Grunwald Letnikov fractional differential mask
Wadhwa, Anjali
Bhardwaj, Anuj
MULTIMEDIA TOOLS AND APPLICATIONS, 2020, 79 (35-36) : 25379 - 25402
[46] Three-dimensional lattice models with long-range interactions of Grunwald-Letnikov type for fractional generalization of gradient elasticity
Tarasov, Vasily E.
MECCANICA, 2016, 51 (01) : 125 - 138
[47] A Grunwald-Letnikov based Manta ray foraging optimizer for global optimization and image segmentation
Abd Elaziz, Mohamed
Yousri, Dalia
Al-qaness, Mohammed A. A.
AbdelAty, Amr M.
Radwan, Ahmed G.
Ewees, Ahmed A.
ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE, 2021, 98
[48] An analysis of the Grunwald-Letnikov scheme for initial-value problems with weakly singular solutions
Chen, Hu
Holland, Finbarr
Stynes, Martin
APPLIED NUMERICAL MATHEMATICS, 2019, 139 : 52 - 61
[49] The Grunwald-Letnikov Formula and Its Equivalent Horner's Form Accuracy Comparison and Evaluation for Application to Fractional Order PID Controllers
Brzezinski, Dariusz W.
Ostalczyk, Piotr
2012 17TH INTERNATIONAL CONFERENCE ON METHODS AND MODELS IN AUTOMATION AND ROBOTICS (MMAR), 2012, : 579 - 584
[50] Variation of Constant Formulas of Linear Autonomous Grunwald-Letnikov-type Fractional Difference Equations
Anh Pham The
Jurgas, Piotr
Niezabitowski, Michal
Siegmund, Stefan
INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM-2018), 2019, 2116

← 1 2 3 4 5 →