A Robust Quintic Hermite Collocation Method for One-Dimensional Heat Conduction Equation
Yıl 2024,
Cilt: 7 Sayı: 2, 82 - 89, 31.08.2024
Selçuk Kutluay
,
Murat Yağmurlu
,
Ali Sercan Karakaş
Öz
In this work, a new robust numerical solution scheme constructed on Quintic Hermite Collocation Method (QHCM) utilizing the traditional Crank-Nicolson type approximation technique is developed for solving 1D heat conduction equation with certain initial and boundary conditions which is mostly handled as a prototype equation to support the reliability of many proposed new numerical methods. All temporal and spatial quantities in the equation are fully discretized using a usual Crank-Nicolson type finite difference approximation and a QHCM, respectively. In obtaining the present scheme, all the roots of the fourth degree Legendre and Chebyshev polynomials shifted to the unit interval are used as suitable inner collocation points. The obtained results from the developed scheme are found to be good enough and better than those from other schemes encountered in the literature. The scheme is also shown to be unconditionally stable by Fourier stability test.
Etik Beyan
The authors of the present manuscript clearly declare that all of the methods and schemes used in the manuscript do not need any ethical committee and/or legal special requirement or permission.
Destekleyen Kurum
This work has been supported by İnönü University Scientific Research Projects Unit under Grant No: FDK-2023-3402.
Kaynakça
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Yıl 2024,
Cilt: 7 Sayı: 2, 82 - 89, 31.08.2024
Selçuk Kutluay
,
Murat Yağmurlu
,
Ali Sercan Karakaş
Kaynakça
- [1] A. D. Amin, E. Amin, S. Hasan, A. Meysam, L. Yueming, W. Lian-Ping, J. Dengwei, X. Gongnan, A comprehensive review on multi-dimensional heat conduction of multi-layer and composite structures: analytical solutions, J. Therm. Sci., 30(6) (2021), 1875-1907.
- [2] H. J. Xu, Z. B. Xing, F. Q. Wangb, Z. M. Cheng, Review on heat conduction, heat convection, thermal radiation and phase change heat transfer of nanofluids in porous media: fundamentals and applications, Chem. Eng. Sci., 195 (2019), 462–483,
- [3] F. Suarez-Carreno, L. Rosales-Romero, Convergence and stability of explicit and implicit schemes in the simulation of the heat equation, Appl.Sci. 11 (2021).
- [4] N. F. Kaskar, Modified implicit method for solving one-dimensional heat equation, Int. J. Eng. Res. Comp. Sci. Eng., 8(9) (2021), 1-6.
- [5] G. Lozande-Cruz, C.E. Rubio-Mercedes, J. Rodrigues-Riberio, Numerical solution of heat equation with singular robin boundary condition, Tendencias em Matematica Aplicada e Computacional, 19(2) (2018), 209-220.
- [6] H. Sun, J. Zhang, A high-order compact boundary value method for solving one-dimensional heat equations, Numer. Methods Partial Differential Equations, 19(6) (2003), 846-857.
- [7] A. Yosaf, S. U. Rehman, F. Ahmad, M. Z. Ullah, A. S. Alshomrani, Eight-order compact finite difference scheme for 1D heat conduction equation, Adv. Numer. Anal., (2016), Article ID 8376061, 12 pages.
- [8] F. Han, W. Dai, New higher-order compact finite difference schemes for 1D heat conduction equations, Appl. Math. Modell., 37 (2013), 7940-7952.
- [9] S. Dhawan, S. Kumar, A numerical solution of one-dimensional heat equation using cubic b-spline basis functions, Int. J. Res. Rev. App. Sci., 1 (2009), 71-77.
- [10] H. Çağlar, M. Özer, N. Çağlar, The numerical solution of the one-dimensional heat equation by using third-degree b-spline functions, Chaos Solitons Fractals, 38 (2008), 1197–1201.
- [11] J. Goh, A. A. Majid, A. I. Ismail, Cubic b-spline collocation method for one-dimensional heat and advection-diffusion equations, J. Appl. Math., (2012), Article ID 458701, 8 pages.
- [12] M. H. Khabir, R. A. Farah, Cubic b-spline collocation method for one-dimensional heat equation, Pure Appl. Math. J., 6(1) (2017), 51-58.
- [13] B. Mebrate, Numerical solution of a one-dimensional heat equation with Dirichlet boundary conditions, Amer. J. Appl. Math., 3(6) (2015), 305-311.
- [14] N. Patel, J. U. Pandya, One-dimensional heat equation subject to both Neumann and Dirichlet initial boundary conditions and used a spline collocation method, Kalpa Publ. Comput., 2 (2017), 107-112.
- [15] M. R. Hooshmandasl, M. H. Heydari, F. M. Maalek Ghaini, Numerical solution of the one-dimensional heat equation by using Chebyshev wavelets method, J. Appl. Comp. Math., 1(6) (2012), 1-7.
- [16] T. Tarmizi, E. Safitri, S. Munzir, M. Ramli, On the numerical solutions of a one-dimensional heat equation: spectral and crank Nicolson method, AIP Conference Proceedings, (2020).
- [17] S. Kutluay, N. M. Ya˘gmurlu, A. S. Karakaş, An effective numerical approach based on cubic Hermite b-spline collocation method for Solving the 1D heat conduction equation, New Trends in Math. Sci., 10(4) (2022), 20-31.
- [18] H. Zeybek, S. B. G. Karakoç, A collocation algorithm based on quintic b-splines for the solitary wave simulation of the GRLW equation, Sci. Iran., 26(6) (2019), 3356-3368.
- [19] W. Wu, J. Manafian, K. A. Khalid, S. B. G. Karakoç, A. H.Taqi, M. A. Mahmoud, Numerical and analytical results of the 1D BBM equation and 2D coupled BBM-system by finite element method, Int. J. Mod. Phys. B, 36(28) (2022).
- [20] S. B. G. Karakoç, Y. Uçar, N. M. Ya˘gmurlu, Different linearization techniques for the numerical solution of the MEW equation, Selcuk J. Appl. Math., 13(2) (2012), 43-62.
- [21] S. Arora, I. Kaur, W. Tilahun, An exploration of quintic hermite splines to solve burgers’ equation, Arab. J. Math., (2020), 19–36.
- [22] S. Arora, I. Kaur, Applications of quintic Hermite collocation with time discretization to singularly perturbed problems, Appl. Math. Comput., 316 (2018) 409–421.
- [23] S. P. Kaur, A. K. Mittal, V. K. Kukreja, A. Kaundal, N. Parumasur, P. Singh, Analysis of a linear and non-linear model for diffusion–dispersion phenomena of pulp washing by using quintic hermite interpolation polynomials, Afr. Mat., 32 (2021), 997–1019.