A New Family of the Local Fractional PDEs

• Autorzy: Yang X.-J. , Machado J. A. T. , Nieto J. J.

Identyfikatory

DOI

10.3233/FI-2017-1479

• Języki publikacji: EN

Abstrakty

EN

A new family of the local fractional PDEs is investigated in this article. The linear, quasilinear, semilinear and nonlinear local fractional PDEs are presented. Furthermore, three types of the local fractional PDEs are discussed, namely, parabolic, hyperbolic and elliptic. Several examples illustrate the corresponding models in nonlinear mathematical physics.

Słowa kluczowe

EN

local fractional PDE; parabolic PDE; hyperbolic PDE; elliptic PDE; local fractional derivative

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 151, nr 1/4
• Strony: 63--75
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Yang X.-J.

• School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China

autor

Machado J. A. T.

• Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. António Bernardino de Almeida, 4249-015 Porto, Portugal

autor

Nieto J. J.

• Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, 15782 Santiago, Spain
• Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Bibliografia

• [1] Finan MB. A first course of partial differential equations in physical sciences and engineering, Arkansas Tech University, 2009. URL http://faculty.atu.edu/mfinan/4343/PDEbook.pdf.
• [2] Dawson CN, Wheeler MF, Woodward CS. A two-grid finite difference scheme for nonlinear parabolic equations, SIAM journal on Numerical Analysis, 1998; 35 (2): 435-452. doi: 10.1137/S0036142995293493.
• [3] Berger MJ, Oliger J. Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics, 1984; 53 (3): 484-512. doi: 10.1016/0021-9991(84)90073-1.
• [4] Boccardo L, Gallouët T, Vazquez JL. Nonlinear elliptic equations in RN without growth restrictions on the data, Journal of Differential Equations, 1993; 105 (2): 334-363. doi: 10.1006/jdeq.1993.1092.
• [5] Zopf C, Hoque SE, Kaliske M. Comparison of approaches to model viscoelasticity based on fractional time derivatives, Computational Materials Science, 2015; 98: 287-296. URL http://dx.doi.org/10.1016/j.commatsci.2014.11.012.
• [6] Tarasov VE. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, 2011. doi: 10.1007/978-3-642-14003-7.
• [7] Martynyuk V, Ortigueira M. Fractional model of an electrochemical capacitor, Signal Processing, 2015; 107: 355-360. URL http://dx.doi.org/10.1016/j.sigpro.2014.02.021.
• [8] Atanackovi TM, Pilipovi S, Stankovi B, Zorica D. Front Matter, John Wiley & Sons, 2014. doi: 10.1111/jofi.12212.
• [9] Machado JT, Mainardi F, Kiryakova V. Fractional Calculus: Quo Vadimus? (Where are we Going?), Fractional Calculus and Applied Analysis, 2015; 18 (2): 495-526. URL https://doi.org/10.1515/fca-2015-0031.
• [10] Micu S, Zuazua E. On the controllability of a fractional order parabolic equation, SIAM journal on Control and Optimization, 2006; 44 (6): 1950-1972. doi: 10.1137/S036301290444263X.
• [11] Jin B, Lazarov R, Zhou Z. Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM Journal on Numerical Analysis, 2013; 51 (l): 445-466. doi: 10.1137/120873984.
• [12] Abbas S, Benchohra M. Darboux problem for implicit impulsive partial hyperbolic fractional order differential equations, Electronic Journal of Differential Equations, 2011; 150: 1-14. doi: 10.1504/IJDSDE.2009.031110.
• [13] Ashyralyev A, Dal F, Pinar Z. A note on the fractional hyperbolic differential and difference equations, Applied Mathematics and Computation, 2011; 217 (9): 4654-4664. URL http://dx.doi.org/10.1016/j.amc.2010.11.017.
• [14] Angulo JM, Ruiz-Medina MD, Anh VV, Grecksch W. Fractional diffusion and fractional heat equation, Advances in Applied Probability, 2000; 32 (4): 1077-1099. URL: http://www.jstor.org/stable/1428518.
• [15] Wang H, Yang D. Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 2013; 51 (2): 1088-1107. doi: 10.1137/120892295.
• [16] Leonenko NN, Ruiz-Medina MD, Taqqu MS. Fractional elliptic, hyperbolic and parabolic random fields, Electronic Journal of Probability, 2011; 16 (40): 1134-1172. URL http://www.math.washington.edu/ejpecp/.
• [17] Yang XJ. Advanced local fractional calculus and its applications, World Science, New York, 2012. ISBN: 1938576012, 9781938576010.
• [18] Yang XJ, Srivastava HM, He JH, Baleanu D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Physics Letters A, 2013; 377 (28): 1696-1700. URL http://dx.doi.org/10.1016/j.physleta.2013.04.012.
• [19] Yang XJ, Baleanu D, Srivastava HM. Local fractional similarity solution for the diffusion equation defined on Cantor sets, Applied Mathematics Letters, 2015; 47: 54-60. URL http://dx.doi.org/10.1016/j.aml.2015.02.024.
• [20] Yang XJ, Srivastava HM. An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Communications in Nonlinear Science and Numerical Simulation, 2015; 29 (1): 499-504. URL http://dx.doi.org/10.1016/j.cnsns.2015.06.006.
• [21] Yang XJ, Machado JT, Hristov J. Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow, Nonlinear Dynamics, 2015. doi: 10.1007/s11071-015-2085-2.
• [22] Yang XJ, Baleanu D, Srivastava HM. Local Fractional Integral Transforms And Their Applications, Academic Press, 2015. ISBN-10: 0128040025, 13: 9780128040027.
• [23] Kolwankar KM, Gangal AD. Local fractional Fokker-Planck equation, Physical Review Letters, 1998; 80 (2): 214. doi: 10.1103/PhysRevLett.80.214.
• [24] Carpinteri A, Cornetti P, Kolwankar KM. Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos, Solitons, Fractals, 2004; 21 (3): 623-632. URL http://dx.doi.org/10.1016/j.chaos.2003.12.081.
• [25] Carpinteri A, Cornetti P, Sapora A. Static-kinematic fractional operators for fractal and non-local solids, Journal of Applied Mathematics and Mechanics, 2009; 89 (3): 207-217. doi: 10.1002/zamm.200800115.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).