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Advanced Analysis of Local Fractional Calculus Applied to the Rice Theory in Fractal Fracture Mechanics

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Methods of Mathematical Modelling and Computation for Complex Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 373))

Abstract

In this chapter, the recent results for the analysis of local fractional calculus are considered for the first time. The local fractional derivative (LFD) and the local fractional integral (LFI) in the fractional (real and complex) sets, the series and transforms involving the Mittag-Leffler function defined on Cantor sets are introduced and reviewed. The uniqueness of the solutions of the local fractional differential and integral equations and the local fractional inequalities are considered in detail. The local fractional vector calculus is applied to describe the Rice theory in fractal fracture mechanics.

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References

  1. Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer Science & Business Media (2011)

    Google Scholar 

  2. Cattani, C., Srivastava, H.M., Yang, X.J.: Fractional Dynamics. De Gruyter Open, Berlin (2015)

    Book  MATH  Google Scholar 

  3. West, B., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer Science & Business Media (2012)

    Google Scholar 

  4. Baskin, E., Iomin, A.: Electrostatics in fractal geometry: fractional calculus approach. Chaos, Solitons Fractals 44(4–5), 335–341 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tarasov, V.E.: Fractional Fokker-Planck equation for fractal media. Chaos Interdiscip. J. Nonlinear Sci. 15(2), 023102 (2005)

    Google Scholar 

  6. Carpinteri, A., Cornetti, P., Kolwankar, K.M.: Calculation of the tensile and flexural strength of disordered materials using fractional calculus. Chaos, Solitons Fractals 21(3), 623–632 (2004)

    Article  MATH  Google Scholar 

  7. Carpinteri, A., Cornetti, P.: A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos, Solitons Fractals 13(1), 85–94 (2002)

    Article  MATH  Google Scholar 

  8. Carpinteri, A., Chiaia, B., Cornetti, P.: Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 191(1–2), 3–19 (2001)

    Article  MATH  Google Scholar 

  9. Yang, X.J.: Local fractional integral transforms. Progress Nonlinear Sci. 4(1), 1–225 (2011)

    Google Scholar 

  10. Yang, X.J.: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher Limited, Hong Kong (2011)

    Google Scholar 

  11. Yang, X.J.: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York (2012)

    Google Scholar 

  12. Yang, X.J., Baleanu, D., Srivastava, H.M.: Local Fractional Integral Transforms and Their Applications. Academic, New York (2015)

    MATH  Google Scholar 

  13. Yang, X.J., Baleanu, D., Srivastava, H.M.: Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 47, 54–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, H.Y., He, J.H., Li, Z.B.: Fractional calculus for nanoscale flow and heat transfer. Int. J. Numer. Methods Heat Fluid Flow 24(6), 1227–1250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jafari, H., Jassim, H.K., Tchier, F., Baleanu, D.: On the approximate solutions of local fractional differential equations with local fractional operators. Entropy 18(4), 150 (2016)

    Article  Google Scholar 

  16. Yang, X.J., Baleanu, D., Machado, J.A.T.: Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. Boundary Value Problems 2013(1), 131 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Debbouche, A., Antonov, V.: Finite-dimensional diffusion models of heat transfer in fractal mediums involving local fractional derivatives. Nonlinear Stud. 24(3), 527–535 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Yang, X.J., Machado, J.A.T., Baleanu, D.: Exact traveling-wave solution for local fractional Boussinesq equation in fractal domain. Fractals 25(04), 1740006 (2017)

    Article  MathSciNet  Google Scholar 

  19. Hemeda, A.A., Eladdad, E.E., Lairje, I.A.: Local fractional analytical methods for solving wave equations with local fractional derivative. Math. Methods Appl. Sci. 41(6), 2515–2529 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Yang, X.J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, X.J., Baleanu, D.: Fractal heat conduction problem solved by local fractional variation iteration method. Thermal Sci. 17(2), 625–628 (2013)

    Article  Google Scholar 

  22. Kumar, D., Singh, J., Baleanu, D.: A hybrid computational approach for Klein-Gordon equations on Cantor sets. Nonlinear Dyn. 87(1), 511–517 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, X.J., Srivastava, H.M., He, J.H., Baleanu, D.: Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A 377(28–30), 1696–1700 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, X.J., Machado, J.A.T., Baleanu, D., Cattani, C.: On exact traveling-wave solutions for local fractional Korteweg-de Vries equation. Chaos Interdiscip. J. Nonlinear Sci. 26(8), 084312 (2016)

    Google Scholar 

  25. Ye, S.S., Mohyud-Din, S.T., Belgacem, F.B.M.: The Laplace series solution for local fractional Korteweg-de Vries equation. Thermal Sci. 20(3), S867–S870 (2016)

    Article  Google Scholar 

  26. Yang, X.J., Machado, J.A.T., Cattani, C., Gao, F.: On a fractal LC-electric circuit modeled by local fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 47, 200–206 (2017)

    Article  MATH  Google Scholar 

  27. Zhao, X.H., Zhang, Y., Zhao, D., Yang, X.J.: The RC circuit described by local fractional differential equations. Fundamenta Informaticae 151(1–4), 419–429 (2017)

    Google Scholar 

  28. Yang, X.J., Machado, J.A.T., Gao, F., Carlo, C.: On linear and nonlinear electric circuits: A local fractional calculus approach, Chapter 11. In: Azar, A.T., Radwan, A., Vaidyanathan, S. (eds.) Fractional Order Systems: Optimization, Control, Circuit Realizations and Applications. Academic, NY, USA (2018)

    Google Scholar 

  29. Yang, X.J., Machado, J.A.T., Hristov, J.: Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow. Nonlinear Dyn. 84(1), 3–7 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Singh, J., Kumar, D., Nieto, J.J.: A reliable algorithm for a local fractional tricomi equation arising in fractal transonic flow. Entropy 18(6), 206 (2016)

    Article  MathSciNet  Google Scholar 

  31. Zhang, Y., Srivastava, H.M., Baleanu, M.C.: Local fractional variational iteration algorithm II for non-homogeneous model associated with the non-differentiable heat flow. Adv. Mech. Eng. 7(10), 1–7 (2015)

    Article  Google Scholar 

  32. Jafari, H., Tajadodi, H., Johnston, J.S.: A decomposition method for solving diffusion equations via local fractional time derivative. Thermal Sci. 19(suppl.1), 123–129 (2015)

    Google Scholar 

  33. Yang, X.J., Machado, J.A.T.: A new insight into complexity from the local fractional calculus view point: modelling growths of populations. Math. Methods Appl. Sci. 40(17), 6070–6075 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, X.J., Gao, F., Srivastava, H.M.: Non-differentiable exact solutions for the nonlinear ODEs defined on fractal sets. Fractals 25(04), 1740002 (2017)

    Article  MathSciNet  Google Scholar 

  35. Sarikaya, M.Z., Budak, H.: Generalized Ostrowski type inequalities for local fractional integrals. Proc. Am. Math. Soc. 145(4), 1527–1538 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tunç, T., Sarikaya, M.Z., Srivastava, H.M.: Some generalized Steffensen’s inequalities via a new identity for local fractional integrals. Int. J. Anal. Appl. 13(1), 98–107 (2017)

    MATH  Google Scholar 

  37. Wu, S.H., Srivastava, H.M.: Some improvements and generalizations of Steffensen’s integral inequality. Appl. Math. Comput. 192, 422–428 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Erden, S., Sarikaya, M.Z.: Generalized Pompeiu type inequalities for local fractional integrals and its applications. Appl. Math. Comput. 274, 282–291 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Chen, G.S., Srivastava, H.M., Wang, P., Wei, W.: Some further generalizations of Hölder’s inequality and related results on fractal space. Abstract Appl. Anal. Article ID 832802, 1–7 (2014)

    Google Scholar 

  40. Sarikaya, M.Z., Tunc, T., Budak, H.: On generalized some integral inequalities for local fractional integrals. Appl. Math. Comput. 276, 316–323 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Budak, H., Sarikaya, M.Z., Yildirim, H.: New inequalities for local fractional integrals. Iranian J. Sci. Technol. Trans. Sci. 41(4), 1039–1046 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mo, H., Sui, X.: Hermite-Hadamard-type inequalities for generalized \(s\)-convex functions on real linear fractal set \(\mathbb{R}^\alpha \) (\(0<\alpha <1\)). Math. Sci. 11(3), 241–246 (2017)

    Google Scholar 

  43. Srivastava, H.M., Zhang, Z.H., Wu, Y.D.: Some further refinements and extensions of the Hermite-Hadamard and Jensen inequalities in several variables. Math. Comput. Model. 54, 2709–2717 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. K\(\imath \)l\(\imath \)çman, A., Saleh, W.: Notions of generalized \(s\)-convex functions on fractal sets. J. Inequalities Appl. 2015(1), 312 (2015)

    Google Scholar 

  45. Choi, J., Set, E., Tomar, M.: Certain generalized Ostrowski type inequalities for local fractional integrals. Commun. Korean Math. Soc. 32(3), 601–617 (2017)

    MathSciNet  MATH  Google Scholar 

  46. Liu, Q., Sun, W.: A Hilbert-type fractal integral inequality and its applications. J. Inequalities Appl. 2017(1), 83 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Liu, Q., Chen, D.: A Hilbert-type integral inequality on the fractal space. Integral Transf. Special Funct. 28(10), 772–780 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kiliçman, A., Saleh, W.: Generalized Convex Functions and their Applications. Selected Topics, Mathematical Analysis and Applications, pp. 77–99 (2018)

    Google Scholar 

  49. Lara, T., Meretes, N., Rosales, E., Sanchez, R.: Convexity on fractal sets. UPI J. Math. Biostat. 1(1), 22–31 (2018)

    Google Scholar 

  50. Srivastava, H.M., Golmankhaneh, A.K., Baleanu, D., Yang, X.J.: Local fractional Sumudu transform with application to IVPs on Cantor sets. Abstract Appl. Anal. 2014, 1–7 (2014)

    MathSciNet  MATH  Google Scholar 

  51. He, J.H.: Asymptotic methods for solitary solutions and compactons. Abstract Appl. Anal. 2012, 1–130 (2012)

    MathSciNet  MATH  Google Scholar 

  52. He, J.H.: A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53(11), 3698–3718 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yang, X.J., Liao, M.K., Chen, J.W.: A novel approach to processing fractal signals using the Yang-Fourier transforms. Procedia Eng. 29, 2950–2954 (2012)

    Article  Google Scholar 

  54. Yang, A.M., Zhang, Y.Z., Long, Y.: The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar. Thermal Sci. 17(3), 707–713 (2013)

    Article  Google Scholar 

  55. Zhong, W.P., Gao, F.: Application of the yang laplace transforms to solution to nonlinear fractional wave equation with local fractional derivative. In: International Conference on Computer Technology and Development, 3rd (ICCTD), ASME Press (2011)

    Google Scholar 

  56. Yan, S.P.: Local fractional Laplace series expansion method for diffusion equation arising in fractal heat transfer. Thermal Sci. 19(suppl.1), 131–135 (2015)

    Google Scholar 

  57. Liu, C.F., Kong, S.S., Yuan, S.J.: Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem. Thermal Sci. 17(3), 715–721 (2013)

    Article  Google Scholar 

  58. Jassim, H.K.: The analytical solutions for volterra integro-differential equations within local fractional operators by yang-laplace transform. Sahand Commun. Math. Anal. 6(1), 69–76 (2017)

    MATH  Google Scholar 

  59. Zhang, Y.Z., Yang, A.M., Long, Y.: Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform. Thermal Sci. 18(2), 677–681 (2014)

    Article  Google Scholar 

  60. Zhao, C.G., Yang, A.M., Jafari, H., Haghbin, A.: The Yang-Laplace transform for solving the IVPs with local fractional derivative. Abstract Appl. Anal. 2014, 1–5 (2014)

    MathSciNet  MATH  Google Scholar 

  61. Jassim, H.K., Ünlü, C., Moshokoa, S.P., Khalique, C.M.: Local fractional Laplace variational iteration method for solving diffusion and wave equations on Cantor sets within local fractional operators. Math. Problems Eng. 2015, 1–9 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  62. Hassan, K.J.: Analytical Solutions of Partial Differential Equations on Cantor Sets Within Local Fractional Derivative Operators. University of Mazandaran, Babolsar, Ph.D. Thesis (2016)

    Google Scholar 

  63. Zhong, W.P., Yang, X.J., Gao, F.: A Cauchy problem for some local fractional abstract differential equation with fractal conditions. J. Appl. Funct. Anal. 8(1), 92–99 (2013)

    MathSciNet  MATH  Google Scholar 

  64. Jafari, H., Jassim, H.K., Qurashi, M.A., Baleanu, D.: On the existence and uniqueness of solutions for local fractional differential equations. Entropy 18(11), 420 (2016)

    Article  Google Scholar 

  65. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematical Studies. Elsevier (North-Holland), Amsterdam (2006)

    Google Scholar 

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Author information

Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    **ao-Jun Yang

  2. Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, Ankara, 06790, Turkey

    Dumitru Baleanu

  3. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

  4. Department of Mathematics and Statistics, University of Victoria, V8W 3R4, Victoria, BC, Canada

    H. M. Srivastava

  5. Department of Medical Research, China Medical University Hospital China Medical University, Taichung, 40402, Taiwan, People’s Republic of China

    H. M. Srivastava

Authors
  1. **ao-Jun Yang
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  2. Dumitru Baleanu
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  3. H. M. Srivastava
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Correspondence to **ao-Jun Yang .

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Editors and Affiliations

  1. Department of Mathematics, JECRC University, Jaipur, Rajasthan, India

    Jagdev Singh

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  3. Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, India

    Devendra Kumar

  4. Department of Mathematics and Computer Sciences, Çankaya University, Ankara, Turkey

    Dumitru Baleanu

  5. Department of Chemical Engineering, University of Chemical Technology and Metallurgy, Sofia, Bulgaria

    Jordan Hristov

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Yang, XJ., Baleanu, D., Srivastava, H.M. (2022). Advanced Analysis of Local Fractional Calculus Applied to the Rice Theory in Fractal Fracture Mechanics. In: Singh, J., Dutta, H., Kumar, D., Baleanu, D., Hristov, J. (eds) Methods of Mathematical Modelling and Computation for Complex Systems. Studies in Systems, Decision and Control, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-030-77169-0_5

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  • DOI: https://doi.org/10.1007/978-3-030-77169-0_5

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