Pell-Lucas Collocation Method to Solve Second-Order Nonlinear Lane-Emden Type Pantograph Differential Equations

Abstract

In this article, we present a collocation method for second-order nonlinear Lane-Emden type pantograph differential equations under intial conditions. According to the method, the solution of the problem is sought depending on the Pell-Lucas polynomials. The Pell-Lucas polynomials are written in matrix form based on the standard bases. Then, the solution form and its the derivatives are also written in matrix forms. Next, a transformation matrix is constituted for the proportion delay of the solution form. By using the matrix form of the solution, the nonlinear term in the equation is also expressed in matrix form. By using the obtained matrix forms and equally spaced collocation points, the problem is turned into an algebraic system of equations. The solution of this system gives the coefficient matrix in the solution form. In addition, the error estimation and the residual improvement technique are also presented. All presented methods are applied to three examples. The results of applications are presented in tables and graphs. In addition, the results are compared with the results of other methods in the literature.

Keywords

• Collocation method
• Lane-Emden dierential equation
• nonlinear dierential equations
• pantograph dierential equations
• Pell-Lucas polynomials

References

1. Abbasbandy S., A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, Journal of Computational and Applied Mathematics, 207, 59-63, 2007.
2. Abbasbandy S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Applied Mathematics and Computation, 172, 485-490, 2006.
3. Adel W., Sabir Z., Solving a new design of nonlinear second-order Lane-Emden pantograph delay differential model via Bernoulli collocation method, The European Physical Journal Plus, 135(5), 427, 2020.
4. Akyüz-Daşcıoğlu A., Çerdik-Yaslan H., The solution of high-order nonlinear ordinary differential equations by Chebyshev series, Applied Mathematics and Computation, 217, 5658-5666, 2011.
5. Akyüz-Daşcıoğlu A., Yaslan H.Ç., An approximation method for solution of nonlinear integral equations, Applied Mathematics and Computation, 174, 619-629, 2006.
6. Alavizadeh S.R., Maalek Ghaini F.M., Numerical solution of higher-order linear and nonlinear ordinary differential equations with orthogonal rational Legendre functions, Journal of Mathematical Extension, 8(4), 109-130, 2014.
7. Bahgat M.S.M., Approximate analytical solution of the linear and nonlinear multi-pantograph delay differential equations, Physica Scripta, 95(5), 055219, 2020.
8. Bahşi M.M., Çevik M., Numerical solution of pantograph-type delay differential equations using perturbation-iteration algorithms, Journal of Applied Mathematics, 2015, Article ID 139821, 2015.

9. Başhan A., Karakoç S.B.G., Geyikli T., Approximation of the KdVB equation by the quintic B-spline differential quadrature method, Kuwait Journal of Science, 42, 67-92, 2015.
10. Bayin S.S., Solutions of Einstein’s field equations for static fluid spheres, Physical Review D, 18, 2745-2751, 1978.
11. Borghero F., Melis A., On Szebehely’s problem for holonomic systems involving generalized potential functions, Celestial Mechanics and Dynamical Astronomy, 49, 273-284, 1990.
12. Chen B., García-Bolós R., Jódar L., Roselló M.D., Chebyshev polynomial approximations for nonlinear differential initial value problems, Nonlinear Analysis, 63, e629-e637, 2005.
13. Dehghan M., Abbaszadeh M., Mohebbi A., The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions, Computers & Mathematics with Applications, 68, 212-237, 2014.
14. Dehghan M., Manafian J., Saadatmandi A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numerical Methods for Partial Differential Equations: An International Journal, 26, 448-479, 2010.
15. Dehghan M., Saadatmandi A., The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method, Mathematical and Computer Modelling, 46, 1434- 1441, 2007.
16. Dehghan M., Salehi R., The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem, Communications in Numerical Methods in Engineering, 27, 524-540, 2011.
17. Dehghan M., Shakeri F., Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, 13, 53-59, 2008.
18. Dönmez Demir D., Lukonde A.P., Kürkçü Ö.K., Sezer M., Pell-Lucas series approach for a class of Fredholm-type delay integro-differential equations with variable delays, Mathematical Sciences, 15, 55-64, 2021.
19. Eftekhari A., Saadatmandi A., DE sinc-collocation method for solving a class of second-order nonlinear BVPs, Mathematics Interdisciplinary Research, 6, 11-22, 2021.
20. El-Tawil M.A., Bahnasawi A.A., Abdel-Naby A., Solving Riccati differential equation using Adomian’s decomposition method, Applied Mathematics and Computation, 157, 503-514, 2004.
21. Eslahchi M.R., Dehghan M., Ahmadi Asl S., The general Jacobi matrix method for solving some nonlinear ordinary differential equations, Applied Mathematical Modelling, 36, 3387-3398, 2012.
22. Genga F., Lin Y., Cui M., A piecewise variational iteration method for Riccati differential equations, Computers and Mathematics with Applications, 58, 2518-2522, 2009.
23. Geyikli T., Karakoç S.B.G., Subdomain finite element method with quartic B-splines for the modified equal width wave equation, Computational Mathematics and Mathematical Physics, 3, 410-421, 2015.
24. Guirao J.L.G., Sabir Z., Saeed T., Design and Numerical Solutions of a Novel Third-Order Nonlinear Emden-Fowler Delay Differential Model, Mathematical Problems in Engineering, 2020, Article ID 7359242, 2020.
25. Gümgüm S., Baykuş Savaşaneril N., Kürkçü Ö.K., Sezer M., Lucas polynomial solution of nonlinear differential equations with variable delays, Hacettepe Journal of Mathematics and Statistics, 49, 553- 564, 2020.
26. He J.H., Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, 26, 695-700, 2005.
27. Horadam A.F., Mahon Bro J.M., Pell and Pell-Lucas polynomials, The Fibonacci Quarterly, 23, 7-20, 1985.
28. Horadam A.F., Swita B., Filipponi P., Integration and derivative sequences for Pell and Pell-Lucas polynomials, The Fibonacci Quarterly, 32(2), 130-135, 1994.
29. Imani A., Aminataei A., Imani A., Collocation method via Jacobi polynomials for solving nonlinear ordinary differential equations, International Journal of Mathematics and Mathematical Sciences, 2011, Article ID 673085, 2011.
30. Izadi M., Srivastava H.M., An efficient approximation technique applied to a non-linear Lane-Emden pantograph delay differential model, Applied Mathematics and Computation, 401, 126123, 2021.
31. Izadi M., Yüzbaşı Ş., Baleanu D., A Taylor-Chebyshev approximation technique to solve the 1D and 2D nonlinear Burgers equations, Mathematical Sciences, 2021.
32. Izadi M., Yüzbaşı Ş., Cattani C., Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains, Ricerche di Matematica, 2021.
33. Izadi M., Yüzbaşı Ş., Noeiaghdam S., Approximating solutions of non-linear Troesch’s problem via an efficient quasi-linearization Bessel approach, Mathematics, 9(16), 1841, 2021.
34. Katani R., Multistep block method for linear and nonlinear pantograph type delay differential equations with neutral term, International Journal of Applied and Computational Mathematics, 3, 1347-1359, 2017.
35. Kharrat B.N., Toma G., Differential transform method for solving initial value problems represented by strongly nonlinear ordinary differential equations, Middle-East Journal of Scientific Research, 27, 576-579, 2019.
36. Kumar A., Methi G., An efficient numerical algorithm for solution of nonlinear delay differential equations, Journal of Physics: Conference Series, 1849, 012014, 2021.
37. Lakestani M., Dehghan M., Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinalfunctions, Computer Physics Communications, 181, 957-966, 2010.
38. Maleknejad K., Mahmoudi Y., Taylor polynomial solutions of high-order nonlinear Volterra-Fredholm integro-differential equation, Applied Mathematics and Computation, 145, 641-653, 2003.
39. Markakis M.P., Closed-form solutions of certain Abel equations of the first kind, Applied Mathematics Letters, 22, 1401-1405, 2009.
40. Merdan M., On the solutions of nonlinear fractional Klein-Gordon equation with modified Riemann- Liouville derivative, Applied Mathematics and Computation, 242, 877-888, 2014.
41. Mittal R.C., Jiwari R., A higher order numerical scheme for some nonlinear differential equations models in biology, International Journal of Computational Methods in Engineering Science and Mechanics, 12(3), 134-140, 2011.
42. Noor M.A., Mohyud-Din S.T., Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method, Mathematical Problems in Engineering, 2008, Article ID 917407, 2008.
43. Noor M.A., Mohyud-Din S.T., Waheed A., Variation of parameters method for solving fifth-order boundary value problems, Applied Mathematics and Information Sciences, 2(2), 135-141, 2008.
44. Öztürk Y., Gülsu M., The approximate solution of high-order nonlinear ordinary differential equations by improved collocation method with terms of shifted chebyshev polynomials, International Journal of Applied and Computational Mathematics, 2, 519-531, 2016.
45. Rawashdeh M.S., Maitama S., Solving nonlinear ordinary differential equations using the NDM, Journal of Applied Analysis and Computation, 5, 77-88, 2015.
46. Razzaghi M., Yousefi S., Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Mathematics and Computers in Simulation, 70, 1-8, 2005.
47. Sabir Z., Raja M.A.Z., Le D.N., Aly A.A., A neuro-swarming intelligent heuristic for second-order nonlinear Lane-Emden multi-pantograph delay differential system, Complex & Intelligent Systems, 2021.
48. Sadollah A., Eskandar H., GuenYoo D., Hoon Kim J., Approximate solving of nonlinear ordinary differential equations using least square weight function and metaheuristic algorithms, Engineering Applications of Artificial Intelligence, 40, 117-132, 2015.
49. Şahin M., Sezer M., Pell-Lucas collocation method for solving high-order functional differential equations with hybrid delays, Celal Bayar University Journal of Science, 14, 141-149, 2018.
50. Vanani S.K., Aminataei A., On the numerical solution of nonlinear delay differential equations, Journal of Concrete and Applicable Mathematics, 8(4), 568-576, 2010.
51. Wazwaz A.M., El-Sayed S.M., A new modification of the Adomian decomposition method for linear and nonlinear operators, Applied Mathematics and Computation, 122, 393-404, 2001.
52. Yüzbaşı Ş., A collocation approach to solve the Riccati-type differential equation systems, International Journal of Computer Mathematics, 89, 2180-2197, 2012.
53. Yüzbaşı Ş., A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations, Applied Mathematics and Computation, 273, 142-154, 2016.
54. Yüzbaşı Ş., An operational method for solutions of Riccati type differential equations with functional arguments, Journal of Taibah University for Science, 14, 661-669, 2020.
55. Yüzbaşı Ş., A numerical approach for solving a class of the nonlinear Lane-Emden type equations arising in astrophysics, Mathematical Methods in the Applied Sciences, 34, 2218-2230, 2011.
56. Yüzbaşı Ş., A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential-difference equations, Computers & Mathematics with Applications, 64, 1691- 1705, 2012.
57. Yüzbaşı Ş., A numerical scheme for solutions of a class of nonlinear differential equations, Journal of Taibah University for Science, 11, 1165-1181, 2017.
58. Yüzbaşı Ş., Karaçayır M., A Galerkin-like scheme to solve Riccati equations encountered in quantum physics, Journal of Physics: Conference Series, 766, 012036, 2016.
59. Yüzbaşı Ş., Sezer M., An exponential approach for the system of nonlinear delay integro-differential equations describing biological species living together, Neural Computing and Applications, 27, 769- 779, 2016.
60. Yüzbaşı Ş., Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials, Applied Mathematics and Computation, 219(11), 6328-6343, 2013.
61. Yüzbaşı Ş., Şahin N., On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials, Journal of Numerical Mathematics, 20, 55-79, 2012.
62. Yüzbaşı Ş., Yıldırım G., Legendre collocation method to solve the riccati equations with functional arguments, International Journal of Computational Methods, 17(10), 2050011, 2020.
63. Yüzbaşı Ş., Yıldırım G., Pell-Lucas collocation method for numerical solutions of two population models and residual correction, Journal of Taibah University for Science, 14, 1262-1278, 2020.
64. Yüzbaşı Ş., Yıldırım G., Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction, Turkish Journal of Mathematics, 44(4), 1065- 1091, 2020.