Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Engineering with Computers
Erschienen in: Engineering with Computers 4/2021

12.02.2020 | Original Article

Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation

verfasst von: Farhad Fakhar-Izadi

Erschienen in: Engineering with Computers | Ausgabe 4/2021

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Distributed fractional derivative operators can be used for modeling of complex multiscaling anomalous transport, where derivative orders are distributed over a range of values rather than being just a fixed integer number. In this paper, we consider the space-time Petrov–Galerkin spectral method for a two-dimensional distributed-order time-fractional fourth-order partial differential equation. By applying a proper Gauss-quadrature rule to discretize the distributed integral operator, the problem is converted to a multi-term time-fractional equation. Then, the proposed method for solving the obtained equation is based on using Jacobi polyfractonomial, which are eigenfunctions of the first kind fractional Sturm–Liouville problem (FSLP), as temporal basis and Legendre polynomials for the spatial discretization. The eigenfunctions of the second kind FSLP are used as temporal basis in test space. This approach leads to finding the numerical solution of the problem through solving a system of linear algebraic equations. Finally, we provide some examples with smooth solutions and finite regular solutions to numerically demonstrate the efficiency, accuracy, and exponential convergence of the proposed method.
Vorheriger Artikel Subset simulation method including fitness-based seed selection for reliability analysis
Nächster Artikel Design and implementation of a new tuned hybrid intelligent model to predict the uniaxial compressive strength of the rock using SFS-ANFIS

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Literatur
1.
Abbaszadeh M, Dehghan M (2017) An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer Algorithms 75(1):173–211MathSciNetMATH
2.
Ainsworth M, Glusa C (2017) Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput Methods Appl Mech Eng 327:4–35MathSciNetMATH
3.
Ammi MRS, Jamiai I (2018) Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete Contin Dyn Syst Ser S 11(1)
4.
Ardakani AG (2016) Investigation of Brewster anomalies in one-dimensional disordered media having Lévy-type distribution. Eur Phys J B 89(3):76
5.
Armour KC, Marshall J, Scott JR, Donohoe A, Newsom ER (2016) Southern ocean warming delayed by circumpolar upwelling and equatorward transport. Nat Geosci 9(7):549
6.
Atanackovic T M, Pilipovic S, Zorica D (2009) Time distributed-order diffusion-wave equation. I. Volterra-type equation. Proc R Soc A Math Phys Eng Sci 465(2106):1869–1891MathSciNetMATH
7.
Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection–dispersion equation. Water Resour Res 36(6):1403–1412
8.
Bu W, Xiao A, Zeng W (2017) Finite difference/finite element methods for distributed-order time fractional diffusion equations. J Sci Comput 72(1):422–441MathSciNetMATH
9.
Chechkin A, Gorenflo R, Sokolov I (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E 66(4):046129
10.
Chen H, Lü S, Chen W (2016) Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain. J Comput Phys 315:84–97MathSciNetMATH
11.
Chen S, Shen J, Wang L-L (2018) Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. J Sci Comput 74(3):1286–1313MathSciNetMATH
12.
Cheng A, Wang H, Wang K (2015) A Eulerian–Lagrangian control volume method for solute transport with anomalous diffusion. Numer Methods Part Differ Equ 31(1):253–267MathSciNetMATH
13.
Coronel-Escamilla A, Gómez-Aguilar J, Torres L, Escobar-Jiménez R (2018) A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel. Phys A 491:406–424MathSciNet
14.
Diethelm K, Ford NJ (2001) Numerical solution methods for distributed order differential equations
15.
Diethelm K, Ford NJ (2009) Numerical analysis for distributed-order differential equations. J Comput Appl Math 225(1):96–104MathSciNetMATH
16.
Duan J-S, Baleanu D (2018) Steady periodic response for a vibration system with distributed order derivatives to periodic excitation. J Vib Control 24(14):3124–3131MathSciNet
17.
Edery Y, Dror I, Scher H, Berkowitz B (2015) Anomalous reactive transport in porous media: experiments and modeling. Phys Rev E 91(5):052130MathSciNet
18.
Fei M, Huang C (2019) Galerkin–Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation. Int J Comput Math 1–14
19.
Gao G-H, Alikhanov AA, Sun Z-Z (2017) The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J Sci Comput 73(1):93–121MathSciNetMATH
20.
Gao G-H, Sun H-W, Sun Z-Z (2015) Some high-order difference schemes for the distributed-order differential equations. J Comput Phys 298:337–359MathSciNetMATH
21.
Gardiner JD, Laub AJ, Amato JJ, Moler CB (1992) Solution of the sylvester matrix equation \({AX}{B}^{T}+ {CX}{D}^{ T}= {E}\). ACM Trans Math Softw (TOMS) 18(2):223–231MATH
22.
Gorenflo R, Luchko Y, Yamamoto M (2015) Time-fractional diffusion equation in the fractional Sobolev spaces. Fract Calculus Appl Anal 18(3):799–820MathSciNetMATH
23.
Guo S, Mei L, Zhang Z, Jiang Y (2018) Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction–diffusion equation. Appl Math Lett 85:157–163MathSciNetMATH
24.
Iwayama T, Murakami S, Watanabe T (2015) Anomalous eddy viscosity for two-dimensional turbulence. Phys Fluids 27(4):045104
25.
Ji C-C, Sun Z-Z, Hao Z-P (2016) Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J Sci Comput 66(3):1148–1174MathSciNetMATH
26.
Jin B, Lazarov R, Thomée V, Zhou Z (2017) On nonnegativity preservation in finite element methods for subdiffusion equations. Math Comput 86(307):2239–2260MathSciNetMATH
27.
Kharazmi E, Zayernouri M (2018) Fractional pseudo-spectral methods for distributed-order fractional PDEs. Int J Comput Math 95(6–7):1340–1361MathSciNet
28.
Kharazmi E, Zayernouri M, Karniadakis GE (2017a) Petrov–Galerkin and spectral collocation methods for distributed order differential equations. SIAM J Sci Comput 39(3):A1003–A1037MathSciNetMATH
29.
Kharazmi E, Zayernouri M, Karniadakis GE (2017b) A Petrov–Galerkin spectral element method for fractional elliptic problems. Comput Methods Appl Mech Eng 324:512–536MathSciNetMATH
30.
Kilbas A, Marichev O, Samko S (1993) Fractional integral and derivatives: theory and applications. Gordon and Breach, SwitzerlandMATH
31.
Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, volume 204. Elsevier Science Limited
32.
Klages R, Radons G, Sokolov IM (2008) Anomalous transport: foundations and applications. Wiley, Hoboken
33.
Konjik S, Oparnica L, Zorica D (2019) Distributed-order fractional constitutive stress-strain relation in wave propagation modeling. Zeitschrift für angewandte Mathematik und Physik 70(2):51MathSciNetMATH
34.
Li X, Rui H, Liu Z (2018) Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation. Numer Algorithms 1–27
35.
Li X, Wu B (2016) A numerical method for solving distributed order diffusion equations. Appl Math Lett 53:92–99MathSciNetMATH
36.
Li X, Xu C (2009) A space-time spectral method for the time fractional diffusion equation. SIAM Journal on Numerical Analysis 47(3):2108–2131MathSciNetMATH
37.
Liao H-L, Lyu P, Vong S, Zhao Y (2017) Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations. Numer Algorithms 75(4):845–878MathSciNetMATH
38.
Liu Y, Du Y, Li H, He S, Gao W (2015) Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem. Comput Math Appl 70(4):573–591MathSciNetMATH
39.
Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703–717MathSciNetMATH
40.
Macías-Díaz J (2018) An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun Nonlinear Sci Numer Simul 59:67–87MathSciNetMATH
41.
Mainardi F, Mura A, Gorenflo R, Stojanović M (2007) The two forms of fractional relaxation of distributed order. J Vib Control 13(9–10):1249–1268MathSciNetMATH
42.
Mao Z, Shen J (2016) Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J Comput Phys 307:243–261MathSciNetMATH
43.
Mao Z, Shen J (2018) Spectral element method with geometric mesh for two-sided fractional differential equations. Adv Comput Math 44(3):745–771MathSciNetMATH
44.
Meerschaert MM (2012) Fractional calculus, anomalous diffusion, and probability. In: Fractional dynamics: recent advances. World Scientific, pp 265–284
45.
Meerschaert MM, Sikorskii A (2011) Stochastic models for fractional calculus, vol 43. Walter de Gruyter, BerlinMATH
46.
Metzler R, Jeon J-H, Cherstvy AG, Barkai E (2014) Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys Chem Chem Phys 16(44):24128–24164
47.
Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339(1):1–77MathSciNetMATH
48.
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, HobokenMATH
49.
Naghibolhosseini M, Long GR (2018) Fractional-order modelling and simulation of human ear. Int J Comput Math 95(6–7):1257–1273MathSciNet
50.
Perdikaris P, Karniadakis GE (2014) Fractional-order viscoelasticity in one-dimensional blood flow models. Ann Biomed Eng 42(5):1012–1023
51.
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier, New YorkMATH
52.
Ran M, Zhang C (2018) New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order. Appl Numer Math 129:58–70MathSciNetMATH
53.
Samiee M, Kharazmi E, Zayernouri M, Meerschaert MM (2018) Petrov–Galerkin method for fully distributed-order fractional partial differential equations. arXiv preprint. arXiv:​1805.​08242
54.
Samiee M, Zayernouri M, Meerschaert MM (2019) A unified spectral method for FPDEs with two-sided derivatives; part I: a fast solver. J Comput Phys 385:225–243MathSciNetMATH
55.
Shraiman BI, Siggia ED (2000) Scalar turbulence. Nature 405(6787):639
56.
Siddiqi SS, Arshed S (2015) Numerical solution of time-fractional fourth-order partial differential equations. Int J Comput Math 92(7):1496–1518MathSciNetMATH
57.
58.
Szegö G (1975) Orthogonal polynomials, vol. 23. In: American Mathematical Society Colloquium Publications
59.
Tomovski Ž, Sandev T (2018) Distributed-order wave equations with composite time fractional derivative. Int J Comput Math 95(6–7):1100–1113MathSciNet
60.
Wei L, He Y (2014) Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl Math Model 38(4):1511–1522MathSciNetMATH
61.
Zayernouri M, Karniadakis GE (2013) Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J Comput Phys 252:495–517MathSciNetMATH
62.
Zhang H, Yang X, Xu D (2019) A high-order numerical method for solving the 2D fourth-order reaction–diffusion equation. Numer Algorithms 80(3):849–877MathSciNetMATH
63.
Zhang P, Pu H (2017) A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation. Numer Algorithms 76(2):573–598MathSciNetMATH
64.
Zhang Y, Meerschaert MM, Baeumer B, LaBolle EM (2015) Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resour Res 51(8):6311–6337
65.
Zhang Y, Meerschaert MM, Neupauer RM (2016) Backward fractional advection dispersion model for contaminant source prediction. Water Resour Res 52(4):2462–2473
Metadaten
Titel
Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation
verfasst von
Farhad Fakhar-Izadi
Publikationsdatum
12.02.2020
Verlag
Springer London
Erschienen in
Engineering with Computers / Ausgabe 4/2021
Print ISSN: 0177-0667
Elektronische ISSN: 1435-5663
DOI
https://doi.org/10.1007/s00366-020-00968-2

Weitere Artikel der Ausgabe 4/2021

Engineering with Computers 4/2021 Zur Ausgabe

Original Article

Hybrid metaheuristic algorithm using butterfly and flower pollination base on mutualism mechanism for global optimization problems

Original Article

Dynamic stability of viscoelastic nanotubes conveying pulsating magnetic nanoflow under magnetic field

Original Article

Enhanced a hybrid moth-flame optimization algorithm using new selection schemes

Original Article

Toward a robust optimal point selection: a multiple-criteria decision-making process applied to multi-objective optimization using response surface methodology

Original Article

A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions

Original Article

A variational iteration method (VIM) for nonlinear dynamic response of a cracked plate interacting with a fluid media

Neuer Inhalt

    Bildnachweise