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Keywords:
fractional-order Bessel functions; fractional operational matrix; error estimation
Summary:
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.
References:
[1] Agarwal, R., O'Regan, D., Hristova, S.: Stability of Caputo fractional differential equations by Lyapunov functions. Appl. Math., Praha 60 (2015), 653-676. DOI 10.1007/s10492-015-0116-4 | MR 3436567 | Zbl 1374.34005
[2] Baillie, R. T.: Long memory processes and fractional integration in econometrics. J. Econom. 73 (1996), 5-59. DOI 10.1016/0304-4076(95)01732-1 | MR 1410000 | Zbl 0854.62099
[3] Bhrawy, A. H., Alhamed, Y., Baleanu, D., Al-Zahrani, A.: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17 (2014), 1138-1157. DOI 10.2478/s13540-014-0218-9 | MR 3254684 | Zbl 1312.65166
[4] Bohannan, G. W.: Analog fractional order controller in temperature and motor control applications. J. Vib. Control 14 (2008), 1487-1498. DOI 10.1177/1077546307087435 | MR 2463074
[5] Caputo, M.: Linear models of dissipation whose $Q$ is almost frequency independent. II. Geophys. J. R. Astron. Soc. 13 (1967), 529-539. DOI 10.1111/j.1365-246X.1967.tb02303.x | MR 2379269 | Zbl 1210.65130
[6] Chen, Y., Sun, Y., Liu, L.: Numerical solution of fractional partial differential equations with variable coefficients using generalized fractional-order Legendre functions. Appl. Math. Comput. 244 (2014), 847-858. DOI 10.1016/j.amc.2014.07.050 | MR 3250624 | Zbl 1336.65173
[7] Chen, X., Wang, L.: The variational iteration method for solving a neutral functional differential equation with proportional delays. Comput. Math. Appl. 59 (2010), 2696-2702. DOI 10.1016/j.camwa.2010.01.037 | MR 2607972 | Zbl 1193.65145
[8] Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Legendre-Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336 (2018), 433-453. DOI 10.1016/j.amc.2018.05.017 | MR 3812592 | Zbl 07130448
[9] Dehestani, H., Ordokhani, Y., Razzaghi, M.: On the applicability of Genocchi wavelet method for different kinds of fractional-order differential equations with delay. Numer. Linear Algebra Appl. 26 (2019), Article ID e2259, 29 pages. DOI 10.1002/nla.2259 | MR 4011892
[10] Dehestani, H., Ordokhani, Y., Razzaghi, M.: Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. (to appear) in Math. Methods Appl. Sci., 18 pages. DOI 10.1002/mma.5840
[11] Doha, E. H., Bhrawy, A. H., Baleanu, D., Hafez, R. M.: A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations. Appl. Numer. Math. 77 (2014), 43-54. DOI 10.1016/j.apnum.2013.11.003 | MR 3145364 | Zbl 1302.65175
[12] Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44 (1996), 554-566. DOI 10.1109/8.489308 | MR 1382017 | Zbl 0944.78506
[13] Grosswald, E.: Bessel Polynomials. Lecture Notes in Mathematics 698, Springer, Berlin (1978). DOI 10.1007/bfb0063135 | MR 0520397 | Zbl 0416.33008
[14] He, J.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167 (1998), 57-68. DOI 10.1016/S0045-7825(98)00108-X | MR 1665221 | Zbl 0942.76077
[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.: Modified Laguerre wavelets method for delay differential equations of fractional-order. Egyptian J. Basic Appl. Sci. 2 (2015), 50-54. DOI 10.1016/j.ejbas.2014.10.004
[17] Jafari, H., Yousefi, S. A., Firoozjaee, M. A., Momani, S., Khalique, C. M.: Application of Legendre wavelets for solving fractional differential equations. Comput. Math. Appl. 62 (2011), 1038-1045. DOI 10.1016/j.camwa.2011.04.024 | MR 2824691 | Zbl 1228.65253
[18] Kazem, S.: Exact solution of some linear fractional differential equations by Laplace transform. Int. J. Nonlinear Sci. 16 (2013), 3-11. MR 3100782 | Zbl 1394.34015
[19] Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Modelling 37 (2013), 5498-5510. DOI 10.1016/j.apm.2012.10.026 | MR 3020667 | Zbl 06929800
[20] Kreyszig, E.: Introductory Functional Analysis with Applications. John Wiley & Sons, New York (1978). MR 0467220 | Zbl 0368.46014
[21] Kumar, P., Agrawal, O. P.: An approximate method for numerical solution of fractional differential equations. Signal Process. 86 (2006), 2602-2610. DOI 10.1016/j.sigpro.2006.02.007 | Zbl 1172.94436
[22] Li, Y.: Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 2284-2292. DOI 10.1016/j.cnsns.2009.09.020 | MR 2602712 | Zbl 1222.65087
[23] Li, X. Y., Wu, B. Y.: A continuous method for nonlocal functional differential equations with delayed or advanced arguments. J. Math. Anal. Appl. 409 (2014), 485-493. DOI 10.1016/j.jmaa.2013.07.039 | MR 3095056 | Zbl 1306.65225
[24] Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166 (2004), 209-219. DOI 10.1016/j.cam.2003.09.028 | MR 2057973 | Zbl 1036.82019
[25] Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. Fractals and Fractional Calculus in Continuum Mechanics CISM Courses and Lectures 378, Springer, Vienna (1997), 291-348. DOI 10.1007/978-3-7091-2664-6_7 | MR 1611587 | Zbl 0917.73004
[26] Mandelbrot, B.: Some noises with $1/f$ spectrum, a bridge between direct current and white noise. IEEE Trans. Inf. Theory 13 (1967), 289-298. DOI 10.1109/TIT.1967.1053992 | MR 1713511 | Zbl 0148.40507
[27] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[28] Moaddy, K., Momani, S., Hashim, I.: The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics. Comput. Math. Appl. 61 (2011), 1209-1216. DOI 10.1016/j.camwa.2010.12.072 | MR 2770523 | Zbl 1217.65174
[29] Momani, S., Al-Khaled, K.: Numerical solutions for systems of fractional differential equations by the decomposition method. Appl. Math. Comput. 162 (2005), 1351-1365. DOI 10.1016/j.amc.2004.03.014 | MR 2113975 | Zbl 1063.65055
[30] Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207 (2007), 96-110. DOI 10.1016/j.cam.2006.07.015 | MR 2332951 | Zbl 1119.65127
[31] Oldham, K. B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering 111, Academic Press, New York (1974). DOI 10.1016/S0076-5392(09)60219-8 | MR 0361633 | Zbl 0292.26011
[32] Parand, K., Nikarya, M.: Application of Bessel functions for solving differential and integro-differential equations of the fractional order. Appl. Math. Modelling 38 (2014), 4137-4147. DOI 10.1016/j.apm.2014.02.001 | MR 3233834 | Zbl 06992772
[33] Parand, K., Nikarya, M., Rad, J. A.: Solving non-linear Lane-Emden type equations using Bessel orthogonal functions collocation method. Celest. Mech. Dyn. Astron. 116 (2013), 97-107. DOI 10.1007/s10569-013-9477-8 | MR 3061372
[34] Petráš, I.: Fractional-order feedback control of a DC motor. J. Electr. Eng. 60 (2009), 117-128.
[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). MR 1658022 | Zbl 0924.34008
[36] Rahimkhani, P., Ordokhani, Y., Babolian, E.: Fractional-order Bernoulli wavelets and their applications. Appl. Math. Modelling 40 (2016), 8087-8107. DOI 10.1016/j.apm.2016.04.026 | MR 3529681
[37] Rahimkhani, P., Ordokhani, Y., Babolian, E.: Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J. Comput. Appl. Math. 309 (2017), 493-510. DOI 10.1016/j.cam.2016.06.005 | MR 3539800 | Zbl 06626265
[38] Rivlin, T. J.: An Introduction to the Approximation of Functions. Dover Books on Advanced Mathematics, Dover Publications, New York (1981). MR 0634509 | Zbl 0489.41001
[39] Saeed, U., Rehman, M. ur, Iqbal, M. A.: Modified Chebyshev wavelet methods for fractional delay-type equations. Appl. Math. Comput. 264 (2015), 431-442. DOI 10.1016/j.amc.2015.04.113 | MR 3351623 | Zbl 1410.65286
[40] Saeedi, H., Moghadam, M. M., Mollahasani, N., Chuev, G. N.: A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order. Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1154-1163. DOI 10.1016/j.cnsns.2010.05.036 | MR 2736623 | Zbl 1221.65354
[41] Tohidi, E., Nik, H. Saberi: A Bessel collocation method for solving fractional optimal control problems. Appl. Math. Modelling 39 (2015), 455-465. DOI 10.1016/j.apm.2014.06.003 | MR 3282588
[42] Wang, W.-S., Li, S.-F.: On the one-leg $\theta$-methods for solving nonlinear neutral functional differential equations. Appl. Math. Comput. 193 (2007), 285-301. DOI 10.1016/j.amc.2007.03.064 | MR 2385784 | Zbl 1193.34156
[43] Yin, F., Song, J., Wu, Y., Zhang, L.: Numerical solution of the fractional partial differential equations by the two-dimensional fractional-order Legendre functions. Abstr. Appl. Anal. 2013 (2013), Article ID 562140, 13 pages. DOI 10.1155/2013/562140 | MR 3129359 | Zbl 1291.65310
[44] Yuanlu, L., Weiwei, Z.: Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Comput. 216 (2010), 2276-2285. DOI 10.1016/j.amc.2010.03.063 | MR 2647099 | Zbl 1193.65114
[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).
[46] Yüzbaşi, Ş.: Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Appl. Math. Comput. 219 (2013), 6328-6343. DOI 10.1016/j.amc.2012.12.006 | MR 3018474 | Zbl 1280.65075
[47] Yüzbaşi, Ş., Şahin, N., Sezer, M.: Numerical solutions of systems of linear Fredholm \hbox{integro}-differential equations with Bessel polynomial bases. Comput. Math. Appl. 61 (2011), 3079-3096. DOI 10.1016/j.camwa.2011.03.097 | MR 2799833 | Zbl 1222.65154
[48] Zhang, X., Tang, B., He, Y.: Homotopy analysis method for higher-order fractional \hbox{integro}--differential equations. Comput. Math. Appl. 62 (2011), 3194-3203. DOI 10.1016/j.camwa.2011.08.032 | MR 2837752 | Zbl 1232.65120
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