Stationarity-conservation laws for certain linear fractional differential equations

Małgorzata Klimek

Type: Article

Publication Date: 2001-07-27

Citations: 10

DOI: https://doi.org/10.1088/0305-4470/34/31/311

Abstract

The Leibniz rule for fractional Riemann-Liouville derivative is studied in algebra of functions defined by Laplace convolution. This algebra and the derived Leibniz rule are used in construction of explicit form of stationary-conserved currents for linear fractional differential equations. The examples of the fractional diffusion in 1+1 and the fractional diffusion in d+1 dimensions are discussed in detail. The results are generalized to the mixed fractional-differential and mixed sequential fractional-differential systems for which the stationarity-conservation laws are obtained. The derived currents are used in construction of stationary nonlocal charges.

Locations

Journal of Physics A Mathematical and General - View
arXiv (Cornell University) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+ PDF Chat	Fractional-Order Sequential Linear Differential Equations with Nabla Derivatives on Time Scales	2024	Cheng-Cheng Zhu Zhu Jiang
+	Laplace transform method for linear sequential Riemann Liouville and Caputo fractional differential equations	2017	A. S. Vatsala M. Sowmya
+	Overview to mathematical analysis for fractional diffusion equations - new mathematical aspects motivated by industrial collaboration	2010	Junichi Nakagawa Kenichi Sakamoto Masahiro Yamamoto
+	New Generalized Results for Modified Atangana-Baleanu Fractional Derivatives and Integral Operators	2025	Gauhar Rahman Muhammad Samraiz Çetin Yıldız Thabet Abdeljawad Manar A. Alqudah
+	Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann-Liouville Sense	2022	Yuri Luchko
+	The Cauchy problems for q-difference equations with the Caputo fractional derivatives	2022	Serikbol Shaimardan N.S. Tokmagambetov A. M. Temirkhanova
+ PDF Chat	Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense	2022	Yuri Luchko
+	Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives	2013	Marek Błasik Małgorzata Klimek
+	Some results on dynamics of fractional derivatives via difference sequences	2020	P. Baliarsingh L. Nayak Hemen Dutta
+	On a Generic Fractional Derivative Associated with the Riemann-Liouville Fractional Integral	2024	Yuri Luchko
+ PDF Chat	A New Approach for Solving Fractional Partial Differential Equations in the Sense of the Modified Riemann-Liouville Derivative	2014	Bin Zheng Qinghua Feng
+ PDF Chat	Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations	2023	A‎. ‎K‎. Alomari Mohammad Alaroud Nedal Tahat Adel Almalki
+ PDF Chat	Nonlinear Sequential Riemann-Liouville and Caputo Fractional Differential Equations with Nonlocal and Integral Boundary Conditions	2019	Suphawat Asawasamrit Nawapol Phuangthong Sotiris K. Ntouyas Jessada Tariboon
+ PDF Chat	Application of Symmetry Analysis and Conservation Laws to Fractional-Order Nonlinear Conduction-Diffusion Model	2024	Harish Dhull Amit Tomar Hemant Gandhi Dimple Singh
+ PDF Chat	On the Nonlinear (k, Ψ)-Hilfer Fractional Differential Equations	2021	Kishor D. Kucche Kishor D. Kucche Audumbar Digambar Mali
+ PDF Chat	New exact solutions of fractional Cahn–Allen equation and fractional DSW system	2018	Shumaila Javeed Summaya Saif Dumitru Băleanu
+ PDF Chat	Fractional vector calculus and fractional Maxwell’s equations	2008	Vasily E. Tarasov
+	A study of double general transform for solving fractional partial differential equations	2023	Rania Saadeh Bayan Ghazal Aliaa Burqan
+	Fractional Derivatives of Some Fractional Functions and Their Applications	2020	Chii-Huei Yu
+	Fractional Derivatives of Some Fractional Functions and Their Applications	2020	Chii-Huei Yu

Works That Cite This (10)

Action	Title	Year	Authors
+	Constructing conservation laws for fractional-order integro-differential equations	2015	Stanislav Yu. Lukashchuk
+ PDF Chat	Duality for the left and right fractional derivatives	2014	Michèle Caputo Delfim F. M. Torres
+ PDF Chat	О построении законов сохранения для интегро-дифференциальных уравнений дробного порядка	2015	Stanislav Yu. Lukashchuk
+ PDF Chat	Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives	2010	Ricardo Almeida Delfim F. M. Torres
+ PDF Chat	General Fractional Noether Theorem and Non-Holonomic Action Principle	2023	Vasily E. Tarasov
+	General fractional classical mechanics: Action principle, Euler–Lagrange equations and Noether theorem	2023	Vasily E. Tarasov
+ PDF Chat	Conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces	2002	Małgorzata Klimek
+	Stationarity$ndash$conservation laws for fractional differential equations with variable coefficients	2002	Małgorzata Klimek
+ PDF Chat	Fractional Hamilton’s equations of motion in fractional time	2007	Sami I. Muslih Dumitru Băleanu Eqab M. Rabei
+ PDF Chat	Phenomenology of local scale invariance: from conformal invariance to dynamical scaling	2002	Malte Henkel

Works Cited by This (25)

Action	Title	Year	Authors
+	Fractional Integrals and Derivatives: Theory and Applications	1993	Stefan Samko Anatoly A. Kilbas O. I. Marichev
+	An Introduction to the Fractional Calculus and Fractional Differential Equations	1993	Kenneth S. Miller Bertram Ross
+ PDF Chat	The integral analogue of the Leibniz rule	1972	Thomas J. Osler
+	A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval’s Formula	1972	Thomas J. Osler
+	Fractional diffusion and wave equations	1989	W. R. Schneider Walter Wyss
+	The realization of the generalized transfer equation in a medium with fractal geometry	1986	Raoul R. Nigmatullin
+ PDF Chat	Lévy flights in quenched random force fields	1998	Hans C. Fogedby
+	Fractional relaxation-oscillation and fractional diffusion-wave phenomena	1996	Francesco Mainardi
+ PDF Chat	Langevin equations for continuous time Lévy flights	1994	Hans C. Fogedby
+	Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series	1970	Thomas J. Osler