Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems
Main Article Content
Abstract
This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
licenseTerms
Publication Dates
References
Hermann, M.; Saravi, M. Nonlinear ordinary differential equations: Analytical approximation and numerical methods. Springer India, 2016. DOI: https://doi.org/10.1007/978-81-322-2812-7
Kounadis, A. N. An efficient and simple approximate technique for solving nonlinear initial and boundary-value problems. Computational Mechanics, 1992, 9(3), 221-231.
Talib, I.; Tunc, C.; Noor, Z. A. New operational matrices of orthogonal Legendre polynomials and their operational. Journal of Taibah University for Science, 2019, 13(1), 377-389.
AL-Jawary, M. A.; Salih, O. M. Reliable iterative methods for 1D Swift–Hohenberg equation. Arab Journal of Basic and Applied Sciences, 2020, 27(1), 56-66.
Umesh; Kumar, M. Numerical solution of singular boundary value problems using advanced Adomian decomposition method. Engineering with Computers, 2021, 37(4), 2853-2863. DOI: https://doi.org/10.1007/s00366-020-00972-6
Harir, A.; Melliani, S.; El Harfi, H.; Chadli, L. S. Variational iteration method and differential transformation method for solving the SEIR epidemic model. International Journal of Differential Equations, 2020, 2020, 1-7.
Gohar, M.; Li, C.; Li, Z. Finite difference methods for Caputo–Hadamard fractional differential equations. Mediterranean Journal of Mathematics, 2020, 17(6), 1-26.
Roul, P.; Thula, K. A new high-order numerical method for solving singular two-point boundary value problems. Journal of Computational and Applied Mathematics, 2018, 343, 556-574. DOI: https://doi.org/10.1016/j.cam.2018.04.056
Ibraheem, K. I.; Mahmmood, H. S. Algorithm for solving fractional partial differential equations using homotopy analysis method with Pade approximation. International Journal of Electrical and Computer Engineering, 2022, 12(3), 3335-3342.
Sharma, B.; Kumar, S.; Paswan, M. K.; Mahato, D. Chebyshev operational matrix method for Lane-Emden problem. Nonlinear Engineering, 2019, 8(1), 1-9. DOI: https://doi.org/10.1515/nleng-2017-0157
Singh, S.; Patel, V. K.; Singh, V. K.; Tohidi, E. Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions. Computers and Mathematics with Applications, 2018, 75(7), 2280-2294. DOI: https://doi.org/10.1016/j.camwa.2017.12.003
Bhrawy, A. H.; Taha, T. M. An operational matrix of fractional integration of the Laguerre polynomials and its application on a semi-infinite interval. Mathematical Sciences, 2012, 6(1), 1-7. DOI: https://doi.org/10.1186/2251-7456-6-41
AL-Jawary, M. A.; Abdul Nabi, A. J. Three iterative methods for solving Jeffery-Hamel flow problem. Kuwait Journal of Science, 2020, 47(1), 1-13.
Agom, E. U.; Ogunfiditimi, F. O.; Bassey, E. V. Homotopy perturbation and Adomian decomposition methods on 12th-order boundary value problems. Advances in Mathematics: Scientific Journal, 2020, 9(12), 10671-10683.
Singh, R. A modified homotopy perturbation method for nonlinear singular Lane–Emden equations arising in various physical models. International Journal of Applied and Computational Mathematics, 2019, 5(3), 1-15.
Ibraheem, G. H.; AL-Jawary, M. A. The operational matrix of Legendre polynomials for solving nonlinear thin film flow problems. Alexandria Engineering Journal, 2020, 59(5), 4027-4033.
Gürbüz, B.; Sezer, M. Modified operational matrix method for second-order nonlinear ordinary differential equations with quadratic and cubic terms. An International Journal of Optimization and Control: Theories & Applications, 2020, 10(2), 218-225.
Al-Humedi, H. O.; Al-Saadawi, F. A. The numerical technique based on shifted Jacobi-Gauss-Lobatto polynomials for solving two dimensional multi-space fractional Bioheat equations. Baghdad Science Journal, 2020, 17(4), 1271-1282.
Al-A’asam, J. A. Deriving the composite Simpson rule by using Bernstein polynomials for solving Volterra integral equations. Baghdad Science Journal, 2014, 11(3), 1274-1282.
AL-Jawary, M. A.; Ibraheem, G. H. Two meshless methods for solving nonlinear ordinary differential equations in engineering and applied sciences. Nonlinear Engineering, 2020, 9(1), 244-255.
Hasan, P. M.; Sulaiman, N. A. Convergence analysis for the homotopy perturbation method for a linear system of mixed Volterra-Fredholm integral equations. Baghdad Science Journal, 2020, 17(3 (Suppl.)), 1010-1018.
Ateeah, A. K. Approximate solution for fuzzy differential algebraic equations of fractional order using Adomian decomposition method. Ibn AL-Haitham Journal for Pure and Applied Science, 2017, 30(2), 202-213.
Abed, S. M.; AL-Jawary, M. A. Efficient iterative methods for solving the SIR epidemic model. Iraqi Journal of Science, 2021, 62(2), 613-622.
Sparis, P. D.; Mouroutsos, S. G. The operational matrix of differentiation for orthogonal polynomial series. International Journal of Control, 1986, 44(1), 1-15. DOI: https://doi.org/10.1080/00207178608933579
Yousefi, S. A.; Behroozifar, M. Operational matrices of Bernstein polynomials and their applications. International Journal of Systems Science, 2010, 41(6), 709-716. DOI: https://doi.org/10.1080/00207720903154783
Öztürk, Y. Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials. Journal of Taibah University for Science, 2018, 12(2), 155-162. DOI: https://doi.org/10.1080/16583655.2018.1451063
Bazm, S.; Hosseini, A. Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type. Computational and Applied Mathematics, 2020, 39(2), 1-20.
Abdelkawy, M. A.; Taha, T. M. An operational matrix of fractional derivatives of Laguerre polynomials. Walailak Journal of Science and Technology, 2014, 11(12), 1041-1055.
Turkyilmazoglu, M. Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane–Emden–Fowler type. Applied Mathematical Modelling, 2013, 37(14-15), 7539-7548. DOI: https://doi.org/10.1016/j.apm.2013.02.014
Turkyilmazoglu, M. An effective approach for numerical solutions of high-order Fredholm integro-differential equations. Applied Mathematics and Computation, 2014, 227(15), 384-398. DOI: https://doi.org/10.1016/j.amc.2013.10.079
Turkyilmazoglu, M. High-order nonlinear Volterra–Fredholm-Hammerstein integro-differential equations and their effective computation. Applied Mathematics and Computation, 2014, 247, 410-416. DOI: https://doi.org/10.1016/j.amc.2014.08.074
Turkyilmazoglu, M. Effective computation of solutions for nonlinear heat transfer problems in fins. Journal of Heat Transfer, 2014, 136(9), 091901. DOI: https://doi.org/10.1115/1.4027772
Turkyilmazoglu, M. Solution of initial and boundary value problems by an effective accurate method. International Journal of Computational Methods, 2017, 14(06), 1750069. DOI: https://doi.org/10.1142/S0219876217500694
Shaban, M.; Kazem, S.; Rad, J. A. A modification of the homotopy analysis method based on Chebyshev operational matrices. Mathematical and Computer Modelling, 2013, 57(5-6), 1227-1239. DOI: https://doi.org/10.1016/j.mcm.2012.09.024
Motsa, S. S.; Sibanda, P.; Shateyi, S. A new spectral-homotopy analysis method for solving a nonlinear second order BVP. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(9), 2293-2302. DOI: https://doi.org/10.1016/j.cnsns.2009.09.019
Manafian, J. An optimal Galerkin-homotopy asymptotic method applied to the nonlinear second-order BVPs. Proceedings of the Institute of Mathematics and Mechanics, 2021, 47(1), 156-182.
Waqas, M.; Gulzar, M. M.; Dogonchi, A. S.; Javed, M. A.; Khan, W. A. Darcy–Forchheimer stratified flow of viscoelastic nanofluid subjected to convective conditions. Applied Nanoscience, 2019, 9(8), 2031-2037.
Awartani, M. M.; Hamdan, M. H. Fully developed flow through a porous channel bounded by flat plates. Applied mathematics and computation, 2005, 169(2), 749-757. DOI: https://doi.org/10.1016/j.amc.2004.09.087
Adewumi, A. O.; Akindeinde, S. O.; Aderogba, A. A.; Ogundare, B. S. A hybrid collocation method for solving highly nonlinear boundary value problems. Heliyon, 2020, 6(3), 1-10.
Abbasbandy, S.; Shivanian, E.; Hashim, I. Exact analytical solution of forced convection in a porous-saturated duct. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(10), 3981-3989. DOI: https://doi.org/10.1016/j.cnsns.2011.01.009
El-Gamel, M.; El-Shenawy, A. A numerical solution of Blasius equation on a semi-infinity flat plate. SeMA Journal, 2018, 75(3), 475-484. DOI: https://doi.org/10.1007/s40324-017-0145-x
Khataybeh, S. N.; Hashim, I.; Alshbool, M. Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations. Journal of King Saud University-Science, 2019, 31(4), 822-826. DOI: https://doi.org/10.1016/j.jksus.2018.05.002
Liao, S. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(8), 2003-2016. DOI: https://doi.org/10.1016/j.cnsns.2009.09.002
Rani, D.; Mishra, V. Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations. Results in Physics, 2020, 16, 102836.
Yao, B.; Chen, J. A new analytical solution branch for the Blasius equation with a shrinking sheet. Applied Mathematics and Computation, 2009, 215(3), 1146-1153. DOI: https://doi.org/10.1016/j.amc.2009.06.057
Marinca, V.; Herişanu, N. The optimal homotopy asymptotic method for solving Blasius equation. Applied Mathematics and Computation, 2014, 231, 134-139. DOI: https://doi.org/10.1016/j.amc.2013.12.121
Wazwaz, A. M. The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Applied Mathematics and Computation, 2007, 188(1), 485-491. DOI: https://doi.org/10.1016/j.amc.2006.10.009
Abbasbandy, S. A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos, Solitons & Fractals, 2007, 31(1), 257-260. DOI: https://doi.org/10.1016/j.chaos.2005.10.071
Parand, K.; Taghavi, A. Rational scaled generalized Laguerre function collocation method for solving the Blasius equation. Journal of Computational and Applied Mathematics, 2009, 233(4), 980-989. DOI: https://doi.org/10.1016/j.cam.2009.08.106
Abbasbandy, S.; Hajishafieiha, J. Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials. Applied Mathematics and Mechanics, 2021, 42(10), 1449-1460.
Falkner, V. M.; Skan, S. W. LXXXV. Solutions of the boundary-layer equations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1931, 12(80), 865-896. DOI: https://doi.org/10.1080/14786443109461870
Bakodah, H. O.; Ebaid, A.; Wazwaz, A. M. Analytical and numerical treatment of Falkner-Skan equation via a transformation and Adomian’s method. Romanian Reports in Physics, 2018, 70(2), 1-17.
AL-Jawary, M. A.; Adwan, M. I. Reliable iterative methods for solving the Falkner-Skan equation. Gazi University Journal of Science, 2020, 33(1), 168-186.
Moallemi, N.; Shafieenejad, I.; Hashemi, S. F.; Fata, A. Approximate explicit solution of Falkner-Skan equation by homotopy perturbation method. Research Journal of Applied Sciences, Engineering and Technology, 2012, 4(17), 2893-2897.
Abbasbandy, S.; Hayat, T. Solution of the MHD Falkner-Skan flow by homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(9-10), 3591-3598. DOI: https://doi.org/10.1016/j.cnsns.2009.01.030
Elgazery, N. S. Numerical solution for the Falkner-Skan equation. Chaos, Solitons & Fractals, 2008, 35(4), 738-746. DOI: https://doi.org/10.1016/j.chaos.2006.05.040
Kuo, B. L. Application of the differential transformation method to the solutions of Falkner-Skan wedge flow. Acta Mechanica, 2003, 164(3), 161-174. DOI: https://doi.org/10.1007/s00707-003-0019-4
Temimi, H.; Ben-Romdhane, M. Numerical solution of Falkner-Skan equation by iterative transformation method. Mathematical Modelling and Analysis, 2018, 23(1), 139-151. DOI: https://doi.org/10.3846/mma.2018.009
Guo, B. Y.; Shen, J.; Wang, Z. Q. A rational approximation and its applications to differential equations on the half line. Journal of scientific computing, 2000, 15(2), 117-147. DOI: https://doi.org/10.1023/A:1007698525506
Kajani, M. T.; Maleki, M.; Allame, M. A numerical solution of Falkner-Skan equation via a shifted Chebyshev collocation method. AIP Conference Proceedings, 2014, 1629(1), 381-386. DOI: https://doi.org/10.1063/1.4902298
Calvert, V.; Razzaghi, M. Solutions of the Blasius and MHD Falkner-Skan boundary-layer equations by modified rational Bernoulli functions. International Journal of Numerical Methods for Heat & Fluid Flow, 2017, 27(8), 1687-1705. DOI: https://doi.org/10.1108/HFF-05-2016-0190
Azodi, H. D.; Yaghouti, M. R. Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order. Filomat, 2018, 32(10), 3623-3635.