Highly accurate spline collocation technique for the numerical solution of generalized Burgers-Fisher’s problem

Kukreja, Vijay Kumar; ., Shallu

doi:10.22034/cmde.2024.49824.2071

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	45
تعداد شماره‌ها	1,510
تعداد مقالات	18,422
تعداد مشاهده مقاله	59,910,334
تعداد دریافت فایل اصل مقاله	21,054,073

	Highly accurate spline collocation technique for the numerical solution of generalized Burgers-Fisher’s problem
Computational Methods for Differential Equations
مقاله 16، دوره 13، شماره 2، خرداد 2025، صفحه 578-591 اصل مقاله (2.46 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.49824.2071
نویسندگان
Vijay Kumar Kukreja^* ؛ Shallu .
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Punjab, India.
چکیده
This study employs the cubic B-spline collocation strategy to address the solution challenges posed by the nonlinear generalized Burgers-Fisher’s equation (gBFE), with some improvisation. This approach incorporates refinements within the spline interpolants, resulting in enhanced convergence rates along the spatial dimension. Temporal integration is achieved through the Crank-Nicolson methodology. The stability of the technique is assessed using the rigorous von Neumann method. Convergence analysis based on Green’s function reveals a fourth-order convergence along the space domain and a second-order convergence along the temporal domain. The results are validated by taking a number of examples. MATLAB 2017 is used for computational work.
کلیدواژه‌ها
Burgers-Fisher’s equation؛ B-splines collocation method؛ Crank Nicolson؛ Convergence analysis؛ Stability analysis

مراجع
[1] M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math, Biol., 41(6) (1979), 835-840. [2] G. Arora and V. Joshi, A computational approach for solution of one dimensional parabolic partial differential equation with application in biological processes, Ain Shams Eng. J., 9(4) (2018), 1141-1150. [3] G. Arora and G. S. Bhatia, A meshfree numerical technique based on radial basis function pseudospectral method for Fishers equation, Int. J. Nonlinear. Sci. Num. Simu., 21(1) (2020), 37-49. [4] S. Arora, R. Jain, and V. K. Kukreja, A robust Hermite spline collocation technique to study generalized BurgersHuxley equation, generalized Burgers-Fisher equation and Modified Burgers equation, J. Ocean Eng. Sci., (2022). [5] A. G. Bratsos, A fourth order improved numerical scheme for the generalized BurgersHuxley equation, Am. J. Comput. Math., 1(03) (2011), 152-158. [6] A. G. Bratsos and A. Q. M. Khaliq, An exponential time differencing method of lines for Burgers-Fisher and coupled Burgers equations, J. Comput. Appl. Math., 356 (2019), 182-197. [7] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mechn., 1 (1948), 171-199. [8] C. De Boor and B. Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10(4) (1973), 582-606. [9] E. Fan, Extended tanh-function method and its applications to nonlinear equations Phys. Lett. A, 277(4-5) (2000), 212-218. [10] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7(4) (1937), 355-369. [11] A. Golbabai and M. Javidi, A spectral domain decomposition approach for the generalized BurgersFisher equation, Chaos Solitons Fractals, 39(1) (2009), 385-392. [12] M. Hussain and S. Haq, Numerical solutions of strongly non-linear generalized BurgersFisher equation via meshfree spectral technique, Int. J. Comput. Math., 98(9) (2021), 1727-1748. [13] H. N. Ismail, K. Raslan, and A. A. Abd Rabboh, Adomian decomposition method for Burger’sHuxley and Burger’sFisher equations, Appl. Math. Comput., 159(1) (2004), 291-301. [14] M. Javidi, Spectral collocation method for the solution of the generalized BurgerFisher equation, Appl. Math. Comput., 174(1) (2006), 345-352. [15] R. Kaur, Shallu, S. Kumar, and V. K. Kukreja, Numerical approximation of generalized Burgers-Fisher and generalized Burgers-Huxley equation by compact finite difference method, Adv. Math. Phys., (2021), 1-17. [16] D. Kaya and S. M. El-Sayed, A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation, Appl. Math. Comput., 152(2) (2004), 403-413. [17] J. Mendoza and C. Muriel, New exact solutions for a generalised Burgers-Fisher equation, Chaos, Solitons & Fractals, 152 (2021), 111360. [18] R. C. Mittal and A. Tripathi, Numerical solutions of generalized BurgersFisher and generalized BurgersHuxley equations using collocation of cubic B-splines, Int. J. Comput. Math, 92(5) (2015), 1053-1077. [19] M. Namjoo, M. Zeinadini, and S. Zibaei, Nonstandard finitedifference scheme to approximate the generalized BurgersFisher equation, Math. Mthd. Appl. Sci, 41(17) (2018), 8212-8228. [20] P. M. Prenter, Splines and Variational Methods, New York: Wiley-interscience publication, (1975). [21] U. Saeed and K. Gilani, CAS wavelet quasi-linearization technique for the generalized BurgerFisher equation, Math. Sci., 12(1) (2018), 61-69. [22] V. Sangwan and B. Kaur, An exponentially fitted numerical technique for singularly perturbed Burgers-Fisher equation on a layer adapted mesh, Int. J. Comput. Math., 96(7) (2019), 1502-1513. [23] M. Sari, G. Grarslan, and . Da, A compact finite difference method for the solution of the generalized BurgersFisher equation, Numer. Methods Partial Differ. Equ., 26(1) (2010), 125-134. [24] Shallu, A. Kumari, and V. K. Kukreja, An improved extrapolated collocation technique for singularly perturbed problems using cubic B-spline functions, Mediterr. J. Math., 18(4) (2021), 1-29. [25] Shallu and V. K. Kukreja, An improvised collocation algorithm with specific end conditions for solving modified Burgers equation, Numer. Methods Partial Differ. Equ., 37(1) (2021), 874-896. [26] Shallu and V. K. Kukreja, Analysis of RLW and MRLW equation using an improvised collocation technique with SSP-RK43 scheme, Wave Motion (2021), 102761. [27] Y. Shang and Q. Chen, The generalized ColeHopf transformation for a generalized BurgersFisher equation with spatiotemporal variable coefficients, Appl. Math. Lett., 117 (2021), 107074. [28] M. Tamsir, N. Dhiman, and V. K. Srivastava, Cubic trigonometric B-spline differential quadrature method for numerical treatment of Fishers reaction-diffusion equations, Alex. Eng. J., 57(3) (2018), 2019-2026. [29] M. Tamsir, V. K. Srivastava, N. Dhiman, and A. Chauhan, Numerical computation of nonlinear Fishers reaction-diffusion equation with exponential modified cubic B-spline differential quadrature method, Int. J. Comput. Math., 4(1) (2018), 1-13. [30] M. Tatari, B. Sepehrian, and M. Alibakhshi, New implementation of radial basis functions for solving BurgersFisher equation, Numer. Methods Partial Differ. Equ., 28(1) (2012), 248-262. [31] A. K. Verma and S. Kayenat, On the stability of Micken’s type NSFD schemes for generalized Burgers Fisher equation, J. Differ. Equ. Appl., 25(12) (2019), 1706-1737. [32] A. M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and BurgersFisher equations, Appl. Math. Comput., 169(1) (2005), 321-338.
آمار تعداد مشاهده مقاله: 375 تعداد دریافت فایل اصل مقاله: 678

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

Highly accurate spline collocation technique for the numerical solution of generalized Burgers-Fisher’s problem