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ACI Avances en Ciencias e Ingenierías

SECTION A: EXACT SCIENCES

Vol. 12 No. 2 (2020)

Fractional calculus for differential equations

DOI
https://doi.org/10.18272/aci.v12i2.1946
Submitted
August 20, 2020
Published
2021-05-11

Abstract

In the present work two methodologies to find solutions to fractional partial differential equations are presented. The first methodology consist in the use of the Laplace transform through the employment of a product type solution.  On the other hand, the second methodology allow us to integrate the partial fractional differential equation. Finally, a discussion about the freedoms and restrictions of two methods is made.

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References

  1. Ansorge, R. & Sonar, T. (2009). Mathematical models of fluid dynamics. Wiley Online Library.
  2. Arfken, G. & Weber, H. (1999). Mathematical methods for physicists. American Association of Physics Teachers.
  3. Spiegelhalter, D., Best, N. & Carlin, B. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Metho- dology), 64(4), 583-639.
  4. https://doi.org/10.1111/1467-9868.00353
  5. Edelman, M. (2013). Fractional dynamical systems. https://doi.org/10.1109/ICFDA.2014.6967376
  6. Sumelka, W., Zaera, R. & Fernández-Sáez, J. (2015). A theoretical analysis of the free axial vibration of non-local rods with fractional continuum mechanics. Meccanica, 50(9), 2309-2323. https://doi.org/10.1007/s11012-015-0157-5
  7. Carpinteri, A., Cornetti, P. & Sapora, A. (2014). Nonlocal elasticity: an approach based on fractional calculus. Meccanica, 49 (11), 2551-2569. https://doi.org/10.1007/s11012-014-0044-5
  8. Craiem, D., Rojo, F., Atienza, J., Guinea, G. & Armentano, R. (2008). Fractional calculus applied to model arterial viscoelasticity. Latin American applied research, 38(2), 141-145.
  9. Applebaum, D. (2004). Lévy processes from probability to finance and quantum groups. Notices of the AMS, 51 (11), 1336-1347.
  10. Tarasova, V. & Tarasov, V. (2016). Elasticity for economic processes with memory: fractional differential calculus approach. Fractional Differential Calculus, 6 (2), 219-232. http://dx.doi.org/10.7153/fdc-06-14
  11. El-Ajou, A., Odibat, Z., Momani, S. & Alawneh, A. (2010). Construction of analytical solutions to fractional differential equations using homotopy analysis method.
  12. International Journal of Applied Mathematics, 40(2).
  13. Calderón, M., Rosales, J., Guzmán, R., González, A. & Alvarez, J. (2015). El cálculo diferencial e integral fraccionario y sus aplicaciones. Acta universitaria, 25 (2), 20-27. http://dx.doi.org/10.15174/au.2014.688
  14. Almusharrf, A. (2011). Development of fractional trigonometry and an application of fractional calculus to pharmacokinetic model.
  15. Abel, N. (1881). Solution de quelques probl`emes `a l'aide d'integrales definies. Oeuvres, 1, 11-27. https://doi.org/10.1017/CBO9781139245807.003
  16. Liouville, J. (1834). Memoire sur le theor`eme des fonctions complementaires. Journal für die reine und angewandte Mathematik, 1834 (11), 1-19. https://doi.org/10.1515/crll.1834.11.1
  17. Muñoz, J. M. S. (2011). Génesis y desarrollo del cálculo fraccional. Pensamiento Matemático, (1), 4.
  18. Rudin, W. (1964). Principles of mathematical analysis. McGraw-Hill New York.
  19. Haubold, H., Mathai, A. & Saxena, R. (2011). Mittag-leffler functions and their applications. Journal of Applied Mathematics, 2011.
  20. Samko, S., Kilbas, A. & Marichev, O. (1993). Fractional integrals and derivatives. Gordon y Breach Science Publishers, Yverdon Yverdon-les-Bains, Switzerland.
  21. Almeida, R. (2017). A caputo fractional derivative of a function with respect to another function. Communications in Nonlinear Science and Numerical Simulation, 44, 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006
  22. Alsaedi, A., Nieto, J. J. & Venktesh, V. (2015). Fractional electrical circuits. Advances in Mechanical Engineering, 7 (12), http://dx.doi.org/10.1177/1687814015618127.
  23. Baleanu, D., Golmankhaneh, A. K. & Golmankhaneh, A. K. (2009). The dual action of fractional multi time hamilton equations. International Journal of Theoretical Physics, 48 (9), 2558-2569. http://dx.doi.org/10.1007/s10773-009-0042-x
  24. Baleanu, D., Golmankhaneh, A. K., Golmankhaneh, A. K. & Nigmatullin, R. R. (2010). Newtonian law with memory. Nonlinear Dynamics, 60 (1-2), 81-86. https://doi.org/10.1007/s11071-009-9581-1
  25. Baleanu, D., Guvenc ̧, Z. B. & Machado, J. T. (Eds.). (2010). New trends in nano-technology and fractional calculus applications. Springer. doi:https://doi.org/10.1007/978-90-481-3293-5
  26. Caputo, M. (2014). The role of memory in modeling social and economic cycles of extreme events. En F. Francesco, M. Ram & M. N. Pietro (Eds.), A handbook of alternative theories of public economics. Edward Elgar Publishing. doi:https://doi.org/10.4337/9781781004715
  27. Caputo, M. & Cametti, C. (2008). Diffusion with memory in two cases of biological interest. Journal of theoretical biology, 254 (3), 697-703. https://doi.org/10.1016/j.jtbi.2008.06.021
  28. Caputo, M., Cametti, C. & Ruggero, V. (2008). Time and spatial concentration profile inside a membrane by means of a memory formalism. Physica A: Statistical Mechanics and its Applications, 387 (8-9), 2010-2018. https://doi.org/10.1016/j.physa.2007.11.033
  29. Caputo, M. & Fabrizio, M. (2015). Damage and fatigue described by a fractional derivative model. Journal of Computational Physics, 293, 400-408. https://doi.org/10.1016/j.jcp.2014.11.012
  30. Cesarone, F., Caputo, M. & Cametti, C. (2004). Memory formalism in the passive diffusion across a biological membrane. J. Membrane Sci, 250, 79-84. https://doi.org/10.1016/j.memsci.2004.10.018
  31. Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier.
  32. Schiff, J. (2013). The laplace transform: theory and applications. Springer Science & Business Media.
  33. Ferreira, M. & Vieira, N. (2016). Eigenfunctions and fundamental solutions of the fractional laplace and dirac operators: the riemann-liouville case. Complex Analysis and Operator Theory, 10(5), 1081-1100. https://doi.org/10.1007/s11785-015-0529-9
  34. Caballero, M. (2019). Una introducción a las ecuaciones integrales lineales. Recuperado desde https://hdl.handle.net/11441/90001
  35. Molano Cabrera, S. A. & Lesmes Acosta, M. d. C. (2007). La alternativa de fredholm.
  36. Ballén López, J. & León, P. (2017). Método de descomposición de Adomian (Tesis de maestría Universidad EAFIT).