In this paper, we discussed some new properties on the newly defined family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the first time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is efficient in solving various problems.
Citation: Chang Phang, Abdulnasir Isah, Yoke Teng Toh. Poly-Genocchi polynomials and its applications[J]. AIMS Mathematics, 2021, 6(8): 8221-8238. doi: 10.3934/math.2021476
In this paper, we discussed some new properties on the newly defined family of Genocchi polynomials, called poly-Genocchi polynomials. These polynomials are extensions from the Genocchi polynomials via generating function involving polylogarithm function. We succeeded in deriving the analytical expression and obtained higher order and higher index of poly-Genocchi polynomials for the first time. We also showed that the orthogonal version of poly-Genocchi polynomials could be presented as multiple shifted Legendre polynomials and Catalan numbers. Furthermore, we extended the determinant form and recurrence relation of shifted Genocchi polynomials sequence to shifted poly-Genocchi polynomials sequence. Then, we apply the poly-Genocchi polynomials to solve the fractional differential equation, including the delay fractional differential equation via the operational matrix method with a collocation scheme. The error bound is presented, while the numerical examples show that this proposed method is efficient in solving various problems.
[1] | F. A. Costabile, M. I. Gualtieri, A. Napoli, Polynomial sequences: elementary basic methods and application hints. A survey, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113 (2019), 3829–3862. |
[2] | C. B. Corcino, R. B. Corcino, Asymptotics of Genocchi polynomials and higher order Genocchi polynomials using residues, Afrika Matematika, (2020), 1–12. |
[3] | T. Usman, M. Aman, O. Khan, K. S. Nisar, S. Araci, Construction of partially degenerate Laguerre-Genocchi polynomials with their applications, AIMS Math., 5 (2020), 4399–4411. doi: 10.3934/math.2020280 |
[4] | A. Isah, C. Phang, Genocchi wavelet-like operational matrix and its application for solving non-linear fractional differential equations, Open Phys., 14 (2016), 463–472. doi: 10.1515/phys-2016-0050 |
[5] | H. Dehestani, Y. Ordokhani, M. Razzaghi, On the applicability of Genocchi wavelet method for different kinds of fractional order differential equations with delay, Numer. Linear Algebr., 26 (2019), e2259. |
[6] | H. Dehestani, Y. Ordokhani, M. Razzaghi, A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113 (2019), 3297–3321. |
[7] | J. R. Loh, C. Phang, A new numerical scheme for solving system of Volterra integro-differential equation, Alex. Eng. J., 57 (2018), 1117–1124. doi: 10.1016/j.aej.2017.01.021 |
[8] | A. Kanwal, C. Phang, U. Iqbal, Numerical solution of fractional diffusion wave equation and fractional Klein–Gordon equation via two-dimensional Genocchi polynomials with a Ritz–Galerkin method, Computation, 6 (2018), 40. doi: 10.3390/computation6030040 |
[9] | M. M. Matar, Existence of solution involving Genocchi numbers for nonlocal anti-periodic boundary value problem of arbitrary fractional order, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 112 (2018), 945–956. |
[10] | F. Rigi, H. Tajadodi, Numerical approach of fractional Abel differential equation by genocchi polynomials, International Journal of Applied and Computational Mathematics, 5 (2019), 1–11. |
[11] | H. Tajadodi, Efficient technique for solving variable order fractional optimal control problems, Alex. Eng. J., 59 (2020), 5179–5185. doi: 10.1016/j.aej.2020.09.047 |
[12] | B. Kurt, Identities and relation on the poly-Genocchi polynomials with a q-parameter, J. Inequal. Spec. Funct., 9 (2018), 1–8. |
[13] | N. U. Khan, T. Usman, M. Aman, Certain generating funtion of generalized apostol type legendre-based polynomials, Note di Matematica, 37 (2018), 21–44. |
[14] | C. S. Ryoo, W. A. Khan, On two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials, Mathematics, 8 (2020), 417. doi: 10.3390/math8030417 |
[15] | N. Khan, T. Usman, K. S. Nisar, A study of generalized Laguerre poly-Genocchi polynomials, Mathematics, 7 (2019), 219. doi: 10.3390/math7030219 |
[16] | F. A. Costabile, M. I. Gualtieri, A. Napoli, Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials, Integr. Transf. Spec. F., 30 (2019), 112–127. doi: 10.1080/10652469.2018.1537272 |
[17] | A. Kanwal, C. Phang, J. R. Loh, New collocation scheme for solving fractional partial differential equations, Hacet. J. Math. Stat., 49 (2020), 1107–1125. |
[18] | Ş. Yüzbaşi, N. Ismailov, An operational matrix method for solving linear Fredholm–Volterra integro-differential equations, Turk. J. Math., 42 (2018), 243–256. doi: 10.3906/mat-1611-126 |
[19] | M. H. Heydari, A. Atangana, Z. Avazzadeh, M. R. Mahmoudi, An operational matrix method for nonlinear variable-order time fractional reaction–diffusion equation involving Mittag-Leffler kernel, The European Physical Journal Plus, 135 (2020), 1–19. doi: 10.1140/epjp/s13360-019-00059-2 |
[20] | P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, A. Abdolahzadeh Ziabari, Numerical solution of full fractional Duffing equations with cubic-quintic-heptic nonlinearities, AIMS Math., 5 (2020), 1621–1641. doi: 10.3934/math.2020110 |
[21] | A. Isah, C. Phang, P. Phang, Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations, International Journal of Differential Equations, 2017 (2017). |
[22] | T. Kim, D. San Kim, G.-W. Jang, J. Kwon, Poly-Genocchi polynomials with umbral calculus viewpoint, J. Comput. Anal. Appl., 26 (2019). |
[23] | D. V. Dolgy, L.-C. Jang, Some identities on the poly-Genocchi polynomials and numbers, Symmetry, 12 (2020), 1007. doi: 10.3390/sym12061007 |
[24] | T. Kim, A note on the-Genocchi numbers and polynomials, J. Inequal. Appl., 2007 (2007), 1–8. |
[25] | S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput., 233 (2014), 599–607. doi: 10.1016/j.amc.2014.01.013 |
[26] | H. M. Srivastava, B. Kurt, Y. Simsek, Some families of Genocchi type polynomials and their interpolation functions, Integr. Transf. Spec. F., 23 (2012), 919–938. doi: 10.1080/10652469.2011.643627 |
[27] | T. Kim, D. San Kim, J. Kwon, H. Y. Kim, A note on degenerate Genocchi and poly-Genocchi numbers and polynomials, J. Inequal. Appl., 2020 (2020), 1–13. doi: 10.1186/s13660-019-2265-6 |
[28] | U. Duran, M. Acikgoz, S. Araci, Construction of the type 2 poly-Frobenius–Genocchi polynomials with their certain applications, Adv. Differ. Equ., 2020 (2020), 1–14. doi: 10.1186/s13662-019-2438-0 |
[29] | A. Isah, C. Phang, New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials, Journal of King Saud University-Science, 31 (2019), 1–7. |
[30] | A. H. Bhrawy, A. A. Al-Zahrani, Y. A. Alhamed, D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations, Rom. J. Phys., 59 (2014), 646–657. |
[31] | R. C. Duggan, A. M. Goodman, Pointwise bounds for a nonlinear heat conduction model of the human head, B. Math. Biol., 48 (1986), 229–236. doi: 10.1016/S0092-8240(86)80009-X |
[32] | P. Roul, U. Warbhe, A novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems, J. Comput. Appl. Math., 296 (2016), 661–676. doi: 10.1016/j.cam.2015.10.020 |
[33] | P. Roul, A new mixed MADM-collocation approach for solving a class of Lane–Emden singular boundary value problems, Journal of Mathematical Chemistry, 57 (2019), 945–969. doi: 10.1007/s10910-018-00995-x |
[34] | P. Roul, H. Madduri, An optimal iterative algorithm for solving Bratu-type problems, Journal of Mathematical Chemistry, 57 (2019), 583–598. doi: 10.1007/s10910-018-0965-7 |
[35] | M. A. Z. Raja, Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP, Connection Science, 26 (2014), 195–214. doi: 10.1080/09540091.2014.907555 |
[36] | S. Chandrasekhar, S. Chandrasekhar, An introduction to the study of stellar structure, volume 2. Courier Corporation, 1957. |
[37] | V. P. Dubey, R. Kumar, D. Kumar, Analytical study of fractional Bratu-type equation arising in electro-spun organic nanofibers elaboration, Physica A: Statistical Mechanics and its Applications, 521 (2019), 762–772. doi: 10.1016/j.physa.2019.01.094 |
[38] | H. Dehestani, Y. Ordokhani, M. Razzaghi, Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations, Math. Method. Appl. Sci., 42 (2019), 7296–7313. doi: 10.1002/mma.5840 |
[39] | H. Dehestani, Y. Ordokhani, M. Razzaghi, Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays, Int. J. Numer. Model. El., 32 (2019), e2606. |