This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms.
[1] S. Abbasbandy, M. S. Hashemi, and C. S. Liu, The Lie-group shooting method for solving the Bratu equation, Commun. Nonlinear Sci. Numer. Simul., 16(11) (2011), 4238–4249.
[2] W. M. Abd-Elhameed, New spectral solutions for high odd-order boundary value problems via generalized Jacobi polynomials, Bull. Malays. Math. Sci. Soc., 40(4) (2017), 1393–1412.
[3] W. M. Abd-Elhameed and A. M. Alkenedri, Spectral solutions of linear and nonlinear BVPs using certain Jacobi polynomials generalizing third-and fourth-kinds of Chebyshev polynomials, CMES Comput. Model. Eng. Sci., 126(3) (2021), 955–989.
[4] W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, New spectral second kind Chebyshev wavelets algorithm for solving linear and nonlinear second-order differential equations involving singular and Bratu type equations, Abst. Appl. Anal. (2013), Article ID 715756.
[5] W. M. Abd-Elhameed and Y. H. Youssri, A novel operational matrix of Caputo fractional derivatives of Fibonacci polynomials: spectral solutions of fractional differential equations, Entropy, 18(10) (2016), 345.
[6] W. M. Abd-Elhameed and Y. H. Youssri, Generalized Lucas polynomial sequence approach for fractional differ- ential equations, Nonlinear Dyn., 89 (2017), 1341–1355.
[7] W. M. Abd-Elhameed and Y. H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comput. Appl. Math.,37 (2018), 2897–2921.
[8] W. M. Abd-Elhameed and Y. H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul., 20(2) (2019), 191–203.
[9] W. M. Abd-Elhameed, Y. H. Youssri, and E. H. Doha, A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations, Math. Sci., 9(2) (2015), 93–102.
[10] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, volume 55 (1964), US Government printing office.
[11] M. Abukhaled, S. Khuri, and A. Sayfy, Spline-based numerical treatments of Bratu-type equations, Palestine J. Math., 1 (2012), 63–70.
[12] E. A. Al-Said,The use of cubic splines in the numerical solution of a system of second-order boundary value problems, Comput. Math. Appl., 42(6-7) (2001), 861–869.
[13] I. Ali, S. Haq, K. S. Nisar, and D. Baleanu, An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations, Adv. Difference Equ. 1 (2021), 1–24.
[14] N. Alonso III and K. L. Bowers, An alternating-direction Sinc–Galerkin method for elliptic problems, J. Complex- ity, 25(3) (2009), 237–252.
[15] S. Balaji and G. Hariharan, An efficient operational matrix method for the numerical solutions of the fractional Bagley–Torvik equation using wavelets, J. Math. Chem., 57 (8) (2019),1885–1901.
[16] K. E. Bisshopp and D. C. Drucker, Large deflection of cantilever beams, Quart. Appl. Math., 3 (3) (1945), 272–275.
[17] J. P. Boyd, One-point pseudospectral collocation for the one-dimensional Bratu equation, Appl. Math. Comp., 217(12) (2011), 5553–5565.
[18] H. Caglar, N. Caglar, M. Ozer, A. Valarıstos, and A. N. Anagnostopoulos, B-spline method for solving Bratu’s problem, Int.J.Comput. Math., 87(8) (2010), 1885–1891.
[19] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics. Springer-Verlag, (1988).
[20] E. Deeba, S. A. Khuri, and S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys., 159(2) (2000), 125–138.
[21] E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials, Collect. Math., 64(3) (2013), 373–394.
[22] E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, Efficient spectral-Petrov–Galerkin methods for the in- tegrated forms of third-and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials, Appl. Math. Comp., 218(15) (2012), 7727–7740.
[23] E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, Fully Legendre spectral Galerkin algorithm for solving linear one-dimensional telegraph type equation, Int. J. Comput. Methods, 16 (2019), Article ID 1850118.
[24] E. H. Doha, A. H. Bhrawy, D. Baleanu, and R. M. Hafez,Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type, Comput. Math. Math. Phys., 53(9) (2013), 1292–1302.
[25] W. Glabisz, The use of Walsh-wavelet packets in linear boundary value problems, Comps. Strs.,82(2-3) (2004), 131–141.
[26] S. Haq and I. Ali, Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials, Eng. Comput. (2021), DOI:10.1007/s00366-021-01327-5.
[27] S. Hichar, A. Guerfi, S. Douis, and M. T. Meftah, Application of nonlinear Bratu’s equation in two and three dimensions to electrostatics, Rep. Math. Phys., 76(3) (2015), 283–290.
[28] R. Jiwari, S. Pandit, and R. C. Mittal, Numerical simulation of two-dimensional sine-Gordon solitons by differ- ential quadrature method, Comput. Phys. Commun.,183(3) (2012), 600–616.
[29] R. Jiwari, Barycentric rational interpolation and local radial basis functions based numerical algorithms for mul- tidimensional sine-Gordon equation, Numer. Methods Partial Differential Equations,37(3) (2021), 1965–1992.
[30] H. H. Keller and E. S. Holdredge, Radiation heat transfer for annular fins of trapezoidal profile, J. Heat Transf., 92(1) (1970), 113–116.
[31] A. M. M. Khodier and A. Y. Hassan, One-dimensional adaptive grid generation, Int. J. Math. Math. Sci, 20(3) (1997), 77–584.
[32] S. A. Khuri, A new approach to Bratu’s problem, Appl.Math. Comput., 147(1) (2004), 131–136.
[33] A. B. Koc, M. C¸ akmak, and A. Kurnaz, A matrix method based on the Fibonacci polynomials to the generalized pantograph equations with functional arguments, Adv. Math. Phys., 2014 (2014), 1–5.
[34] M. Lakestani and M. Dehghan, The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal B-spline wavelets, Int. J. Comput. Math.,83(8-9) (2006), 685–694.
[35] L. B. Liu, H. W. Liu, and Y. Chen, Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions, Appl. Math. Comput., 217(16) (2011), 6872–6882.
[36] Y. L. Luke, Inequalities for generalized hypergeometric functions, J.Approx. Theory, 5(1) (1972), 41–65.
[37] T. Y. Na, computational Methods in Engineering Boundary Value Problems, Academic Press, (1980).
[38] A. Napoli and W. M. Abd-Elhameed, An innovative harmonic numbers operational matrix method for solving initial value problems, Calcolo, 54(1) (2017), 57–76.
[39] A. Napoli and W. M. Abd-Elhameed, A new collocation algorithm for solving even-order boundary value problems via a novel natrix method, Mediterr. J. Math.,14(4) (2017), 1–20.
[40] A. K. Nasab, Z. P. Atabakan, and A. Kılı¸cman,An efficient approach for solving nonlinear Troesch’s and Bratu’s problems by wavelet analysis method, Math. Probl. Eng., 2013(1) (2013), Article ID 825817.
[41] M. A. Noor, I. A. Tirmizi, and M. A. Khan, Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems, Appl. Math. Comp., 179(1) (2006), 153–160.
[42] O¨ . Oruc, A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 14–25.
[43] S. Pandit, R. Jiwari, K. Bedi, and M. E. Koksal, Haar wavelets operational matrix based algorithm for computa- tional modelling of hyperbolic type wave equations, Eng. Computations (2017).
[44] M. A. Ramadan, I. F. Lashien, and W. K. Zahra, Polynomial and nonpolynomial spline approaches to the nu- merical solution of second order boundary value problems, Appl. Math. Comput., 184(2) (2007), 476–484.
[45] M. N. Sahlan and H. Afshari, Lucas polynomials based spectral methods for solving the fractional order electrohy- drodynamics flow model, Commun. Nonlinear Sci. Numer. Simul., 107 (2022), 106–108.
[46] S. C. Shiralashetti, A. B. Deshi, and P. B. Mutalik Desai, Haar wavelet collocation method for the numerical solution of singular initial value problems, Ain Shams Eng. J.,7(2) (2016), 663–670.
[47] S. C. Shiralashetti and S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J., 57(4) (2018), 2591–2600.
[48] S. C. Shiralashetti and K. Srinivasa, Hermite wavelets method for the numerical solution of linear and nonlinear singular initial and boundary value problems, Comput. Methods DEs, 7(2) (2019), 177–198.
[49] M. I. Syam and A. Hamdan, An efficient method for solving Bratu equations, Appl. Math. Comput.,176(2) (2006), 704–713.
[50] I. A. Tirmizi and E. H. Twizell, Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems, Appl. Math. Lett.,15(7)(2002), 897–902.
[51] C. Tun¸c and E. Tun¸c, On the asymptotic behavior of solutions of certain second-order differential equations, J. Franklin Inst.,344(5) (2007), 391–398.
[52] C. Tun¸c and O. Tun¸c, On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, J. Adv. Res., 7(1) (2016), 165–168.
[53] S. G. Venkatesh, S. K. Ayyaswamy, and S. R. Balachandar, The Legendre wavelet method for solving initial value problems of Bratu-type, Comput. Math. Appl., 63(8) (2012), 1287–1295.
[54] F. Wang, Q. Zhao, Z. Chen, and C. M. Fan, Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains, Appl. Math. Comput., 397 (2021), 125903.
[55] A. M. Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Appl. Math. Comput., 128(1) (2002), 45–57.
[56] C. Yang and J. Hou, Chebyshev wavelets method for solving Bratu’s problem, BVPs, 2013 (1) (2013), 1–9.
[57] A. Yıldırım and T. O¨ zi¸s, Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method, Phys. Lett. A, 369(1-2) (2007), 70–76.
[58] Y. H. Youssri, A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation, Adv. Difference Equ., 2017(1) (2017), 1–17.
[59] Y. H. Youssri, W. M. Abd-Elhameed, and A. G. Atta, Spectral Galerkin treatment of linear one-dimensional telegraph type problem via the generalized Lucas polynomials, Arab. J. Math., 11 (2022), 601–615.
Youssri, Y., Sayed, S., Mohamed, A., Aboeldahab, E., & Abd-Elhameed, W. (2023). Modified Lucas polynomials for the numerical treatment of second-order boundary value problems. Computational Methods for Differential Equations, 11(1), 12-31. doi: 10.22034/cmde.2022.50891.2115
MLA
Youssri Hassan Youssri; Shahenda Sayed; Amany Saad Mohamed; Emad Aboeldahab; Waleed Mohamed Abd-Elhameed. "Modified Lucas polynomials for the numerical treatment of second-order boundary value problems". Computational Methods for Differential Equations, 11, 1, 2023, 12-31. doi: 10.22034/cmde.2022.50891.2115
HARVARD
Youssri, Y., Sayed, S., Mohamed, A., Aboeldahab, E., Abd-Elhameed, W. (2023). 'Modified Lucas polynomials for the numerical treatment of second-order boundary value problems', Computational Methods for Differential Equations, 11(1), pp. 12-31. doi: 10.22034/cmde.2022.50891.2115
VANCOUVER
Youssri, Y., Sayed, S., Mohamed, A., Aboeldahab, E., Abd-Elhameed, W. Modified Lucas polynomials for the numerical treatment of second-order boundary value problems. Computational Methods for Differential Equations, 2023; 11(1): 12-31. doi: 10.22034/cmde.2022.50891.2115
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