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Existence and Uniqueness of Time-Fractional Diffusion Equation on a Metric Star Graph

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Computational Sciences - Modelling, Computing and Soft Computing (CSMCS 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1345))

Abstract

In this paper, we study the time-fractional diffusion equation on a metric star graph. The existence and uniqueness of the weak solution are investigated and the proof is based on eigenfunction expansions. Some priori estimates and regularity results of the solution are proved.

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References

  1. Adami, R., Caaciapuoti, C., Finco, D., Noja, D.: Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differ. Equ. 257, 3738–3777 (2014)

    Article  MathSciNet  Google Scholar 

  2. Below, J.V.: A characteristic equation associated to an eigenvalue problem on \(c^{\infty }\)-net. Linear Algebra Appl. 71, 309–325 (1985)

    Article  MathSciNet  Google Scholar 

  3. Below, J.V.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72(2), 316–337 (1988)

    Article  MathSciNet  Google Scholar 

  4. Bohannan, G.W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14, 1487–1498 (2008)

    Article  MathSciNet  Google Scholar 

  5. Dáger, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-d Flexible Multi-structures. Mathématiques & Applications [Mathematics & Applications], vol. 50. Springer, Berlin (2006). https://doi.org/10.1007/3-540-37726-3

  6. Friedrich, C.: Rheological material functions for associating comb-shaped or H-shaped polymers a fractional calculus approach. Philos. Mag. Lett. 66, 287–292 (1992)

    Article  Google Scholar 

  7. Leugering, G.: On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition. J. Comput. Appl. Math. 120, 133–157 (2000)

    Article  MathSciNet  Google Scholar 

  8. Grigor’yan, A., Lin, Y., Yang, Y.: Yamabe type equations on graph. J. Differ. Equ. 261, 4924–4943 (2016)

    Article  MathSciNet  Google Scholar 

  9. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  10. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980). https://doi.org/10.1007/978-3-642-66282-9

    Book  MATH  Google Scholar 

  11. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  12. Kumar, N., Mehra, M.: Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates. Optim. Control Appl. Methods 42, 417–444 (2021)

    Article  MathSciNet  Google Scholar 

  13. Kumar, N., Mehra, M.: Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost. Numer. Methods Partial Differ. Equ. 37, 1693–1724 (2021)

    Google Scholar 

  14. Lagnese, J.E., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures. Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston (1994). https://doi.org/10.1007/978-1-4612-0273-8

  15. Li, L., Jin, L., Fang, S.: Existence and uniqueness of the solution to a coupled fractional diffusion system. Adv. Differ. Equ. 2015(1), 1–14 (2015). https://doi.org/10.1186/s13662-015-0707-0

    Article  MathSciNet  Google Scholar 

  16. Li, Y.S., Wei, T.: An inverse time-dependent source problem for a time-space fractional diffusion equation. Appl. Math. Comput. 336, 257–271 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, 218–223 (2009)

    Article  MathSciNet  Google Scholar 

  18. Luchko, Y.: Some uniqueness and existence results for the initial-boundary value problems for the generalized time-fractional diffusion equations. Comput. Math. Appl. 59, 1766–1772 (2010)

    Article  MathSciNet  Google Scholar 

  19. Luchko, Y.: Anomalous diffusion: models, their analysis, and interpretation. In: Rogosin, S., Koroleva, A. (eds.) Advances in Applied Analysis, pp. 115–145. Springer, Basel (2012). https://doi.org/10.1007/978-3-0348-0417-2_3

    Chapter  MATH  Google Scholar 

  20. Lumer, G.: Connecting of local operators and evolution equations on a network. Lect. Notes Math. 787, 219–234 (1980)

    Article  MathSciNet  Google Scholar 

  21. Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, 23–28 (1996)

    Article  MathSciNet  Google Scholar 

  22. Mehandiratta, V., Mehra, M., Leugering, G.: An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems. Math. Methods Appl. Sci. 44, 3195–3213 (2020)

    Google Scholar 

  23. Mehandiratta, V., Mehra, M., Leugering, G.: Fractional optimal control problems on a star graph: optimality system and numerical solution. Math. Control Related Fields 11, 189–209 (2021)

    Article  MathSciNet  Google Scholar 

  24. Mehandiratta, V., Mehra, M., Leugering, G.: Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph. J. Math. Anal. Appl. 477, 1243–1264 (2019)

    Article  MathSciNet  Google Scholar 

  25. Mehandiratta, V., Mehra, M., Leugering, G.: Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: A study of fractional calculus on metric graph. Netw. Heterog. Media. 16, 155–185 (2021)

    Google Scholar 

  26. Mehra, M., Mallik, R.K.: Solutions of differential-difference equations arising from mathematical models of granulocytopoiesis. Differ. Equ. Dyn. Syst. 22(1), 33–49 (2014)

    Article  MathSciNet  Google Scholar 

  27. Mophou, G., Leugering, G., Fotsing, P.S.: Optimal control of a fractional Sturm-Liouville problem on a star graph. Optimization 70, 659–687 (2020)

    Article  MathSciNet  Google Scholar 

  28. Mugnolo, D.: Gaussian estimates for a heat equation on a network. Netw. Heterogen. Media 2, 55–79 (2007)

    Article  MathSciNet  Google Scholar 

  29. Nicaise, S.: Some results on spectral theory over networks, applied to nerve impulse transmission. In: Brezinski, C., Draux, A., Magnus, A.P., Maroni, P., Ronveaux, A. (eds.) Polynômes Orthogonaux et Applications. LNM, vol. 1171, pp. 532–541. Springer, Heidelberg (1985). https://doi.org/10.1007/BFb0076584

    Chapter  Google Scholar 

  30. Nicaise, S., Zair, O.: Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees. Revista Matematica Complutense 16, 151–178 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Patel, K.S., Mehra, M.: Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients. J. Comput. Appl. Math. 380, 112963 (2020)

    Article  MathSciNet  Google Scholar 

  32. Penkin, O.M., Pokornyi, Y.V., Provotorova, E.N.: On one vector boundary-value problem. Bound. Value Probl. 171, 64–70 (1983)

    Google Scholar 

  33. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  34. Pokornyi, Y.V., Borovskikh, A.V.: Differential equations on networks (geometric graphs). J. Math. Sci. 119, 691–718 (2004)

    Article  MathSciNet  Google Scholar 

  35. Provotorov, V.V.: Eigenfunctions of the Sturm-Liouville problem on a star graph. Sbornik Math. 199, 1523–1545 (2008)

    Article  MathSciNet  Google Scholar 

  36. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  37. Shukla, A., Mehra, M., Leugering, G.: A fast adaptive spectral graph wavelet method for the viscous burgers’ equation on a star-shaped connected graph. Math. Methods Appl. Sci. 43(13), 7595–7614 (2020)

    Article  MathSciNet  Google Scholar 

  38. Singh, A.K., Mehra, M.: Uncertainty quantification in fractional stochastic integro-differential equations using Legendre wavelet collocation method. Lect. Notes Comput. Sci. 12138, 58–71 (2020)

    Article  Google Scholar 

  39. Singh, A.K., Mehra, M.: Wavelet collocation method based on Legendre polynomials and its application in solving the stochastic fractional-integro differential equations. J. Comput. Sci. 51, 101342 (2021)

    Google Scholar 

  40. Walther, M.: Simulation-based model reduction for partial differential equations on networks. Ph.D. thesis, FAU Studies Mathematics and Physics, Erlangen (2018)

    Google Scholar 

  41. Yoshioka, H., Unami, K., Fujihara., M.: Burgers type equation models on connected graphs and their application to open channel hydraulics (2014). http://hdl.handle.net/2433/195771

  42. Zhang, W., Liu, W.: Existence and Ulam’s type stability results for a class of fractional boundary value problems on a star graph. Math. Methods Appl. Sci. 43, 8568–8594 (2020)

    Google Scholar 

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Acknowledgements

The author would like to thank the Indo-German exchange program “Multiscale Modelling, Simulation and Optimization for Energy, Advanced Materials and Manufacturing”. The program (grant number 1-3/2016 (IC)) is funded by University Grants Commission (India) and DAAD (Germany). The coordination of the program through the “Central Institute for Scientific Computing” at FAU, Erlangen is acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Delhi, India

    Vaibhav Mehandiratta & Mani Mehra

  2. Lehrstuhl Angewandte Mathematik II, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058, Erlangen, Germany

    Günter Leugering

Authors
  1. Vaibhav Mehandiratta
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  2. Mani Mehra
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  3. Günter Leugering
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Correspondence to Mani Mehra .

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Editors and Affiliations

  1. National Institute of Technology Calicut, Kozhikode, Kerala, India

    Ashish Awasthi

  2. National Institute of Technology Calicut, Kozhikode, Kerala, India

    Sunil Jacob John

  3. National Institute of Technology Calicut, Kozhikode, Kerala, India

    Satyananda Panda

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Mehandiratta, V., Mehra, M., Leugering, G. (2021). Existence and Uniqueness of Time-Fractional Diffusion Equation on a Metric Star Graph. In: Awasthi, A., John, S.J., Panda, S. (eds) Computational Sciences - Modelling, Computing and Soft Computing. CSMCS 2020. Communications in Computer and Information Science, vol 1345. Springer, Singapore. https://doi.org/10.1007/978-981-16-4772-7_2

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  • DOI: https://doi.org/10.1007/978-981-16-4772-7_2

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  • Online ISBN: 978-981-16-4772-7

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