Fractional-order Bessel functions with various applications

Haniye Dehestani; Yadollah Ordokhani; Mohsen Razzaghi

Applications of Mathematics (2019)

  • Volume: 64, Issue: 6, page 637-662
  • ISSN: 0862-7940

Abstract

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We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.

How to cite

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Dehestani, Haniye, Ordokhani, Yadollah, and Razzaghi, Mohsen. "Fractional-order Bessel functions with various applications." Applications of Mathematics 64.6 (2019): 637-662. <http://eudml.org/doc/294541>.

@article{Dehestani2019,
abstract = {We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.},
author = {Dehestani, Haniye, Ordokhani, Yadollah, Razzaghi, Mohsen},
journal = {Applications of Mathematics},
keywords = {fractional-order Bessel functions; fractional operational matrix; error estimation},
language = {eng},
number = {6},
pages = {637-662},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Fractional-order Bessel functions with various applications},
url = {http://eudml.org/doc/294541},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Dehestani, Haniye
AU - Ordokhani, Yadollah
AU - Razzaghi, Mohsen
TI - Fractional-order Bessel functions with various applications
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 6
SP - 637
EP - 662
AB - We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.
LA - eng
KW - fractional-order Bessel functions; fractional operational matrix; error estimation
UR - http://eudml.org/doc/294541
ER -

References

top
  1. Agarwal, R., O'Regan, D., Hristova, S., 10.1007/s10492-015-0116-4, Appl. Math., Praha 60 (2015), 653-676. (2015) Zbl1374.34005MR3436567DOI10.1007/s10492-015-0116-4
  2. Baillie, R. T., 10.1016/0304-4076(95)01732-1, J. Econom. 73 (1996), 5-59. (1996) Zbl0854.62099MR1410000DOI10.1016/0304-4076(95)01732-1
  3. Bhrawy, A. H., Alhamed, Y., Baleanu, D., Al-Zahrani, A., 10.2478/s13540-014-0218-9, Fract. Calc. Appl. Anal. 17 (2014), 1138-1157. (2014) Zbl1312.65166MR3254684DOI10.2478/s13540-014-0218-9
  4. Bohannan, G. W., 10.1177/1077546307087435, J. Vib. Control 14 (2008), 1487-1498. (2008) MR2463074DOI10.1177/1077546307087435
  5. Caputo, M., 10.1111/j.1365-246X.1967.tb02303.x, Geophys. J. R. Astron. Soc. 13 (1967), 529-539. (1967) Zbl1210.65130MR2379269DOI10.1111/j.1365-246X.1967.tb02303.x
  6. Chen, Y., Sun, Y., Liu, L., 10.1016/j.amc.2014.07.050, Appl. Math. Comput. 244 (2014), 847-858. (2014) Zbl1336.65173MR3250624DOI10.1016/j.amc.2014.07.050
  7. Chen, X., Wang, L., 10.1016/j.camwa.2010.01.037, Comput. Math. Appl. 59 (2010), 2696-2702. (2010) Zbl1193.65145MR2607972DOI10.1016/j.camwa.2010.01.037
  8. Dehestani, H., Ordokhani, Y., Razzaghi, M., 10.1016/j.amc.2018.05.017, Appl. Math. Comput. 336 (2018), 433-453. (2018) Zbl07130448MR3812592DOI10.1016/j.amc.2018.05.017
  9. Dehestani, H., Ordokhani, Y., Razzaghi, M., 10.1002/nla.2259, Numer. Linear Algebra Appl. 26 (2019), Article ID e2259, 29 pages. (2019) MR4011892DOI10.1002/nla.2259
  10. Dehestani, H., Ordokhani, Y., Razzaghi, M., 10.1002/mma.5840, (to appear) in Math. Methods Appl. Sci., 18 pages. DOI10.1002/mma.5840
  11. Doha, E. H., Bhrawy, A. H., Baleanu, D., Hafez, R. M., 10.1016/j.apnum.2013.11.003, Appl. Numer. Math. 77 (2014), 43-54. (2014) Zbl1302.65175MR3145364DOI10.1016/j.apnum.2013.11.003
  12. Engheta, N., 10.1109/8.489308, IEEE Trans. Antennas Propag. 44 (1996), 554-566. (1996) Zbl0944.78506MR1382017DOI10.1109/8.489308
  13. Grosswald, E., 10.1007/bfb0063135, Lecture Notes in Mathematics 698, Springer, Berlin (1978). (1978) Zbl0416.33008MR0520397DOI10.1007/bfb0063135
  14. He, J., 10.1016/S0045-7825(98)00108-X, Comput. Methods Appl. Mech. Eng. 167 (1998), 57-68. (1998) Zbl0942.76077MR1665221DOI10.1016/S0045-7825(98)00108-X
  15. He, J., Nonlinear oscillation with fractional derivative and its applications, Proceedings of the International Conference on Vibrating Engineering, Dalian, 1998, pp. 288-291. 
  16. Iqbal, M. A., Saeed, U., Mohyud-Din, S. T., 10.1016/j.ejbas.2014.10.004, Egyptian J. Basic Appl. Sci. 2 (2015), 50-54. (2015) DOI10.1016/j.ejbas.2014.10.004
  17. Jafari, H., Yousefi, S. A., Firoozjaee, M. A., Momani, S., Khalique, C. M., 10.1016/j.camwa.2011.04.024, Comput. Math. Appl. 62 (2011), 1038-1045. (2011) Zbl1228.65253MR2824691DOI10.1016/j.camwa.2011.04.024
  18. Kazem, S., Exact solution of some linear fractional differential equations by Laplace transform, Int. J. Nonlinear Sci. 16 (2013), 3-11. (2013) Zbl1394.34015MR3100782
  19. Kazem, S., Abbasbandy, S., Kumar, S., 10.1016/j.apm.2012.10.026, Appl. Math. Modelling 37 (2013), 5498-5510. (2013) Zbl06929800MR3020667DOI10.1016/j.apm.2012.10.026
  20. Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley & Sons, New York (1978). (1978) Zbl0368.46014MR0467220
  21. Kumar, P., Agrawal, O. P., 10.1016/j.sigpro.2006.02.007, Signal Process. 86 (2006), 2602-2610. (2006) Zbl1172.94436DOI10.1016/j.sigpro.2006.02.007
  22. Li, Y., 10.1016/j.cnsns.2009.09.020, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 2284-2292. (2010) Zbl1222.65087MR2602712DOI10.1016/j.cnsns.2009.09.020
  23. Li, X. Y., Wu, B. Y., 10.1016/j.jmaa.2013.07.039, J. Math. Anal. Appl. 409 (2014), 485-493. (2014) Zbl1306.65225MR3095056DOI10.1016/j.jmaa.2013.07.039
  24. Liu, F., Anh, V., Turner, I., 10.1016/j.cam.2003.09.028, J. Comput. Appl. Math. 166 (2004), 209-219. (2004) Zbl1036.82019MR2057973DOI10.1016/j.cam.2003.09.028
  25. Mainardi, F., 10.1007/978-3-7091-2664-6_7, Fractals and Fractional Calculus in Continuum Mechanics CISM Courses and Lectures 378, Springer, Vienna (1997), 291-348. (1997) Zbl0917.73004MR1611587DOI10.1007/978-3-7091-2664-6_7
  26. Mandelbrot, B., 10.1109/TIT.1967.1053992, IEEE Trans. Inf. Theory 13 (1967), 289-298. (1967) Zbl0148.40507MR1713511DOI10.1109/TIT.1967.1053992
  27. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York (1993). (1993) Zbl0789.26002MR1219954
  28. Moaddy, K., Momani, S., Hashim, I., 10.1016/j.camwa.2010.12.072, Comput. Math. Appl. 61 (2011), 1209-1216. (2011) Zbl1217.65174MR2770523DOI10.1016/j.camwa.2010.12.072
  29. Momani, S., Al-Khaled, K., 10.1016/j.amc.2004.03.014, Appl. Math. Comput. 162 (2005), 1351-1365. (2005) Zbl1063.65055MR2113975DOI10.1016/j.amc.2004.03.014
  30. Momani, S., Odibat, Z., 10.1016/j.cam.2006.07.015, J. Comput. Appl. Math. 207 (2007), 96-110. (2007) Zbl1119.65127MR2332951DOI10.1016/j.cam.2006.07.015
  31. Oldham, K. B., Spanier, J., 10.1016/S0076-5392(09)60219-8, Mathematics in Science and Engineering 111, Academic Press, New York (1974). (1974) Zbl0292.26011MR0361633DOI10.1016/S0076-5392(09)60219-8
  32. Parand, K., Nikarya, M., 10.1016/j.apm.2014.02.001, Appl. Math. Modelling 38 (2014), 4137-4147. (2014) Zbl06992772MR3233834DOI10.1016/j.apm.2014.02.001
  33. Parand, K., Nikarya, M., Rad, J. A., 10.1007/s10569-013-9477-8, Celest. Mech. Dyn. Astron. 116 (2013), 97-107. (2013) MR3061372DOI10.1007/s10569-013-9477-8
  34. Petráš, I., Fractional-order feedback control of a DC motor, J. Electr. Eng. 60 (2009), 117-128. (2009) 
  35. Podlubny, I., Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198, Academic Press, San Diego (1999). (1999) Zbl0924.34008MR1658022
  36. Rahimkhani, P., Ordokhani, Y., Babolian, E., 10.1016/j.apm.2016.04.026, Appl. Math. Modelling 40 (2016), 8087-8107. (2016) MR3529681DOI10.1016/j.apm.2016.04.026
  37. Rahimkhani, P., Ordokhani, Y., Babolian, E., 10.1016/j.cam.2016.06.005, J. Comput. Appl. Math. 309 (2017), 493-510. (2017) Zbl06626265MR3539800DOI10.1016/j.cam.2016.06.005
  38. Rivlin, T. J., An Introduction to the Approximation of Functions, Dover Books on Advanced Mathematics, Dover Publications, New York (1981). (1981) Zbl0489.41001MR0634509
  39. Saeed, U., Rehman, M. ur, Iqbal, M. A., 10.1016/j.amc.2015.04.113, Appl. Math. Comput. 264 (2015), 431-442. (2015) Zbl1410.65286MR3351623DOI10.1016/j.amc.2015.04.113
  40. Saeedi, H., Moghadam, M. M., Mollahasani, N., Chuev, G. N., 10.1016/j.cnsns.2010.05.036, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1154-1163. (2011) Zbl1221.65354MR2736623DOI10.1016/j.cnsns.2010.05.036
  41. Tohidi, E., Nik, H. Saberi, 10.1016/j.apm.2014.06.003, Appl. Math. Modelling 39 (2015), 455-465. (2015) MR3282588DOI10.1016/j.apm.2014.06.003
  42. Wang, W.-S., Li, S.-F., 10.1016/j.amc.2007.03.064, Appl. Math. Comput. 193 (2007), 285-301. (2007) Zbl1193.34156MR2385784DOI10.1016/j.amc.2007.03.064
  43. Yin, F., Song, J., Wu, Y., Zhang, L., 10.1155/2013/562140, Abstr. Appl. Anal. 2013 (2013), Article ID 562140, 13 pages. (2013) Zbl1291.65310MR3129359DOI10.1155/2013/562140
  44. Yuanlu, L., Weiwei, Z., 10.1016/j.amc.2010.03.063, Appl. Math. Comput. 216 (2010), 2276-2285. (2010) Zbl1193.65114MR2647099DOI10.1016/j.amc.2010.03.063
  45. Yüzbaşi, Ş., Bessel Polynomial Solutions of Linear Differential, Integral and Integro-Differential Equations, M.Sc. Thesis, Graduate School of Natural and Applied Sciences, Mugla University, Kötekli (2009). (2009) 
  46. Yüzbaşi, Ş., 10.1016/j.amc.2012.12.006, Appl. Math. Comput. 219 (2013), 6328-6343. (2013) Zbl1280.65075MR3018474DOI10.1016/j.amc.2012.12.006
  47. Yüzbaşi, Ş., Şahin, N., Sezer, M., 10.1016/j.camwa.2011.03.097, Comput. Math. Appl. 61 (2011), 3079-3096. (2011) Zbl1222.65154MR2799833DOI10.1016/j.camwa.2011.03.097
  48. Zhang, X., Tang, B., He, Y., 10.1016/j.camwa.2011.08.032, Comput. Math. Appl. 62 (2011), 3194-3203. (2011) Zbl1232.65120MR2837752DOI10.1016/j.camwa.2011.08.032

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