×

A novel exact solution for the fractional Ambartsumian equation. (English) Zbl 1487.45005

Summary: Fractional calculus (FC) is useful in studying physical phenomena with memory effect. In this paper, a fractional form of Ambartsumian equation is considered utilizing the Caputo fractional derivative. The Heaviside expansion formula in classical calculus (CC) is extended/developed in view of FC. Then, the extended Heaviside expansion formula is applied to obtain the exact solution in a simplest form. Several theorems and lemmas are proved to facilitate the evaluation of the inverse Laplace transform of specific expressions in fractional forms. The exact solution is established in terms of a one-parameter Mittag-Leffler function which is provided for the first time for the Ambartsumian equation in FC. The present solution reduces to the corresponding one in the relevant literature as the fractional order tends to one. Moreover, the convergence of the obtained solution is theoretically proved. Comparisons with another approach in the literature are performed. The advantage of the present analysis over the existing one in the relevant literature is discussed and analyzed.

MSC:

45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
34K37 Functional-differential equations with fractional derivatives
65L99 Numerical methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ambartsumian, V. A., On the fluctuation of the brightness of the milky way, Dokl. Akad. Nauk SSSR, 44, 223-226 (1994)
[2] Kato, T.; McLeod, J. B., The functional-differential equation \(y'(x)=ay(\lambda x)+by(x)\), Bull. Am. Math. Soc., 77, 891-935 (1971) · Zbl 0236.34064 · doi:10.1090/S0002-9904-1971-12805-7
[3] Patade, J.; Bhalekar, S., On analytical solution of Ambartsumian equation, Nat. Acad. Sci. Lett., 40, 291-293 (2017) · doi:10.1007/s40009-017-0565-2
[4] Bakodah, H. O.; Ebaid, A., Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018) · Zbl 1427.65141 · doi:10.3390/math6120331
[5] Alatawi, A. A.; Aljoufi, M.; Alharbi, F. M.; Ebaid, A., Investigation of the surface brightness model in the milky way via homotopy perturbation method, J. Appl. Math. Phys., 8, 3, 434-442 (2020) · doi:10.4236/jamp.2020.83033
[6] Khaled, S. M.; El-Zahar, E. R.; Ebaid, A., Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics, 7 (2019) · doi:10.3390/math7050425
[7] Kumar, D.; Singh, J.; Baleanu, D., Analysis of a fractional model of the Ambartsumian equation, Eur. Phys. J. Plus, 133, 133-259 (2018) · doi:10.1140/epjp/i2018-11954-7
[8] Adomian, G.; Rach, R., On the solution of algebraic equations by the decomposition method, J. Math. Anal. Appl., 105, 141-166 (1985) · Zbl 0552.60060 · doi:10.1016/0022-247X(85)90102-7
[9] Adomian, G.; Rach, R., Algebraic equations with exponential terms, J. Math. Anal. Appl., 112, 1, 136-140 (1985) · Zbl 0579.60058 · doi:10.1016/0022-247X(85)90280-X
[10] Adomian, G.; Rach, R., Algebraic computation and the decomposition method, Kybernetes, 15, 1, 33-37 (1986) · Zbl 0604.60064 · doi:10.1108/eb005727
[11] Fatoorehchi, H.; Abolghasemi, H., Finding all real roots of a polynomial by matrix algebra and the Adomian decomposition method, J. Egypt. Math. Soc., 22, 524-528 (2014) · Zbl 1302.65128 · doi:10.1016/j.joems.2013.12.018
[12] Alshaery, A.; Ebaid, A., Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method, Acta Astronaut., 140, 27-33 (2017) · doi:10.1016/j.actaastro.2017.07.034
[13] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Boston: Kluwer Acad, Boston · Zbl 0802.65122 · doi:10.1007/978-94-015-8289-6
[14] Wazwaz, A. M., Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166, 652-663 (2005) · Zbl 1073.65068
[15] Wazwaz, A. M., The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. Math. Comput., 216, 1304-1309 (2010) · Zbl 1190.65199
[16] Ebaid, A., Approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, Z. Naturforsch. A, 66, 423-426 (2011) · doi:10.1515/zna-2011-6-707
[17] Duan, J. S.; Rach, R., A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput., 218, 4090-4118 (2011) · Zbl 1521.65071
[18] Ebaid, A., A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math., 235, 1914-1924 (2011) · Zbl 1209.65077 · doi:10.1016/j.cam.2010.09.007
[19] Wazwaz, A. M.; Rach, R.; Duan, J. S., Adomian decomposition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Appl. Math. Comput., 219, 5004-5019 (2013) · Zbl 1282.65082
[20] Ali, E. H.; Ebaid, A.; Rach, R., Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions, Comput. Math. Appl., 63, 1056-1065 (2012) · Zbl 1247.65097 · doi:10.1016/j.camwa.2011.12.010
[21] Sheikholeslami, M.; Ganji, D. D.; Ashorynejad, H. R., Investigation of squeezing unsteady nanofluid flow using ADM, Powder Technol., 239, 259-265 (2013) · doi:10.1016/j.powtec.2013.02.006
[22] Chun, C.; Ebaid, A.; Lee, M.; Aly, E. H., An approach for solving singular two point boundary value problems: analytical and numerical treatment, ANZIAM J., 53, 21-43 (2012) · Zbl 1333.65091 · doi:10.21914/anziamj.v53i0.4582
[23] Kashkari, B. S.; Bakodah, H. O., New modification of Laplace decomposition method for seventh order KdV equation, Appl. Math. Inf. Sci., 9, 5, 2507-2512 (2015)
[24] Ebaid, A.; Aljoufi, M. D.; Wazwaz, A. M., An advanced study on the solution of nanofluid flow problems via Adomian’s method, Appl. Math. Lett., 46, 117-122 (2015) · Zbl 1329.76255 · doi:10.1016/j.aml.2015.02.017
[25] Bhalekar, S.; Patade, J., An analytical solution of fishers equation using decomposition method, Am. J. Comput. Appl. Math., 6, 123-127 (2016) · Zbl 1362.34101
[26] Bakodah, H. O.; Al-Zaid, N. A.; Mirzazadeh, M.; Zhou, Q., Decomposition method for solving Burgers’ equation with Dirichlet and Neumann boundary conditions, Optik, 130, 1339-1346 (2017) · doi:10.1016/j.ijleo.2016.11.140
[27] Ebaid, A.; Al-Enazi, A.; Albalawi, B. Z.; Aljoufi, M. D., Accurate approximate solution of Ambartsumian delay differential equation via decomposition method, Math. Comput. Appl., 24, 1 (2019)
[28] Kaur, D.; Agarwal, P.; Rakshit, M.; Chand, M., Fractional calculus involving (p,q)-Mathieu type series, Appl. Math. Nonlinear Sci., 5, 2, 15-34 (2020) · Zbl 1524.26013 · doi:10.2478/amns.2020.2.00011
[29] Agarwal, P., Mondal, S.R., Nisar, K.S.: On fractional integration of generalized Struve functions of first kind. Thai J. Math. (2021, to appear)
[30] Agarwal, P.; Singh, R., Modelling of transmission dynamics of Nipah virus (Niv): a fractional order approach, Phys. A, Stat. Mech. Appl., 547, 1 (2020) · Zbl 07530164 · doi:10.1016/j.physa.2020.124243
[31] Alderremy, A. A.; Saad, K. M.; Agarwal, P.; Aly, S.; Jain, S., Certain new models of the multi space-fractional Gardner equation, Phys. A, Stat. Mech. Appl., 545, 1 (2020) · doi:10.1016/j.physa.2019.123806
[32] Feng, Y.-Y., Yang, X.-J., Liu, J.-G., Chen, Z.-Q.: New perspective aimed at local fractional order memristor model on Cantor sets, Fractals (2021, to appear). doi:10.1142/S0218348X21500110 · Zbl 1497.94192
[33] Feng, Y.-Y.; Yang, X.-J.; Liu, J.-G., On overall behavior of Maxwell mechanical model by the combined Caputo fractional derivative, Chin. J. Phys., 66, 269-276 (2020) · doi:10.1016/j.cjph.2020.05.006
[34] Sweilam, N. H.; Al-Mekhlafi, S. M.; Assiri, T.; Atangana, A., Optimal control for cancer treatment mathematical model using Atangana-Baleanu-Caputo fractional derivative, Adv. Differ. Equ., 2020 (2020) · Zbl 1485.92053 · doi:10.1186/s13662-020-02793-9
[35] Atangana, A.; Qureshi, S., Mathematical modeling of an autonomous nonlinear dynamical system for malaria transmission using Caputo derivative, Fractional Order Analysis: Theory, Methods and Applications (2020) · doi:10.1002/9781119654223.ch9
[36] Agarwal, P.; El-Sayed, A. A., Vieta-Lucas polynomials for solving a fractional-order mathematical physics model, Adv. Differ. Equ., 2020 (2020) · Zbl 1487.65161 · doi:10.1186/s13662-020-03085-y
[37] Yassen, M. F.; Attiya, A. A.; Agarwal, P., Subordination and superordination properties for certain family of analytic functions associated with Mittag-Leffler function, Symmetry, 12 (2020) · doi:10.3390/sym12101724
[38] Agarwal, P.; El-Sayed, A. A.; Tariboon, J., Vieta-Fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations, J. Comput. Appl. Math., 382 (2021) · Zbl 1471.65066 · doi:10.1016/j.cam.2020.113063
[39] Podlubny, I., Fractional Differential Equations (1999), San Diego: Academic Press, San Diego · Zbl 0924.34008
[40] Spiegel, M. R., Laplace Transforms (1965), New York: McGraw-Hill, New York
[41] Chung, W. S.; Kim, T.; Kwon, H., On the q-analog of the Laplace transform (English summary), Russ. J. Math. Phys., 21, 2, 156-168 (2014) · Zbl 1311.44002 · doi:10.1134/S1061920814020034
[42] Bhalekar, S.; Patade, J., Series solution of the pantograph equation and its properties, Fractal Fract., 1 (2017) · doi:10.3390/fractalfract1010016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.