Spline collocation methods for solving some types of nonlinear parabolic partial differential equations
Authors
B. A. Mahmood
- Department of Mathematics, College of Science, University of Duhok, Duhok, Iraq.
S. A. Tahir
- Department of Mathematics, College of Science, University of Sulaiumai, Sulaiumai, Iraq.
K. H. F. Jwamer
- Department of Mathematics, College of Science, University of Sulaiumai, Sulaiumai, Iraq.
Abstract
In this work, some types of nonlinear parabolic partial differential equations have been studied by means of the collocation method with cubic B-splines, without transformation or linearization. Here, the convergence analysis of the current scheme is also theoretically investigated. A few numerical examples are given to illustrate the viability and effectiveness of the proposed technique. The error norms \(l_2\) and \(l_{\infty}\) are used to assess the accuracy of the current method. In this respect, the proposed method, keeping the real features of such problems, is able to save the behavior of nonlinear terms without facing any conventional drawbacks. Furthermore, it is mathematically shown and numerically seen that there is a good agreement between the approximation and the exact solutions. The current approach reduces the cost of calculation as well as the need for storage space at various parameters.
Share and Cite
ISRP Style
B. A. Mahmood, S. A. Tahir, K. H. F. Jwamer, Spline collocation methods for solving some types of nonlinear parabolic partial differential equations, Journal of Mathematics and Computer Science, 31 (2023), no. 3, 262--273
AMA Style
Mahmood B. A., Tahir S. A., Jwamer K. H. F., Spline collocation methods for solving some types of nonlinear parabolic partial differential equations. J Math Comput SCI-JM. (2023); 31(3):262--273
Chicago/Turabian Style
Mahmood, B. A., Tahir, S. A., Jwamer, K. H. F.. "Spline collocation methods for solving some types of nonlinear parabolic partial differential equations." Journal of Mathematics and Computer Science, 31, no. 3 (2023): 262--273
Keywords
- Cubic B-spline
- collocation method
- numerical solution
- SSP-RK54 scheme
- nonlinear partial differential equations
MSC
References
-
[1]
K. S. Aboodh, Solving Fourth Order Parabolic PDE with Variable Coefficients Using Aboodh Transform Homotopy Perturbation Method, Pure Appl. Math. J., 4 (2015), 219–224
-
[2]
A. R. Hadhoud, F. E. Abd Alaal, A. A. Abdelaziz, T. Radwan, Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination, Demonstr. Math., 54 (2021), 436–451
-
[3]
H. Ahmad, T. A. Khan, S.-W. Yao, Numerical solution of second order Painlev´e differential equation, J. Math. Compu. Sci., 21 (2020), 150–157
-
[4]
B. Ahmad, A. Perviz, M. O. Ahmad, F. Dayan, Numerical Solution of Fourth Order Homogeneous Parabolic Partial Differential Equations (PDEs) Using Non-Polynomial Cubic Spline Method (NPCSM), Sci. Inquiry Rev., 5 (2021), 19–37
-
[5]
B. Ahmad, A. Perviz, M. O. Ahmed, F. Dayan, Solution of Parabolic Partial Differential Equations Via Non-Polynomial Cubic Spline Technique, Sci. Inquiry Rev., 5 (2021), 60–76
-
[6]
T. Akram, M. Abbas, A. Ali, A numerical study on time fractional Fisher equation using an extended cubic B-spline approximation, J. Math. Comput. Sci., 22 (2021), 85–96
-
[7]
A. M. AL-Rozbayani, M. O. Al-Amr, Discrete Adomian decomposition method for solving Burger’s-Huxley equation, Int. J. Contemp. Math. Sci., 8 (2013), 623-–631
-
[8]
E. A. Al-Said, M. A. Noor, Quartic Spline Method for Solving Fourth Order Obstacle Boundary Value Problems, J. Comput. Appl. Math., 143 (2002), 107–116
-
[9]
A. R. Appadu, Y. O. Tijani, 1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods, Front. Appl. Math. Stat., 7 (2022), 1–14
-
[10]
H. Benharzallah, A. Mennouni, D. Barrera, C1-Cubic Quasi-Interpolation Splines over a CT Refinement of a Type-1 Triangulation, Mathematics, 11 (2023), 11–19
-
[11]
J. Biazar, F. Mahmoodi, Application of differential transform method to the generalized Burgers-Huxley equation, Appl. Appl. Math., 5 (2010), 1726–1740
-
[12]
A. G. Bratsos, A Fourth Order Improved Numerical Scheme for the Generalized Burger-Huxley equation, Am. J. Comput. Math., 1 (2011), 152–158
-
[13]
Y. C¸ icek, S. Korkut, Numerical Solution of Generalized Burgers-Huxley Equation by Lie-trotter Splitting Method, Nume. Anal. Appl., 14 (2021), 90–120
-
[14]
L. Debtnath, Nonlinear Partial Differential Equations for Scientist and Engineers, Birkh¨auser Boston, Boston (1997)
-
[15]
K. C. Durga, T. B. Bahadur, Textbook of Differential Equation (for BSc Second year), Editor 2, Editor 2, A., Eds., Publishers and Distributors Pvt. Ltd, Kathmandu, (2018)
-
[16]
H. Eltayeb, D. E. Elgezouli, A. Kilicman, I. Bachar, Three-dimensional Laplace Adomian decomposition method and singular pseudoparabolic equations, J. Funct. Spaces, 2021 (2021), 15 pages
-
[17]
A. A. Gubaidullin, O. Y. Boldyreva, D. N. Dudko, Approach to the Numerical Study of Wave Processes in a Layered and Fractured Porous Media in a Two-Dimensional Formulation, Mathematics, 11 (2023), 1–13
-
[18]
H. N. A. Ismail, K. Raslan, A. A. A. Rabboh, Adomian decomposition method for Burger’s-Huxley and Burger’s-Fisher equations, Appl. Math. Comput., 159 (2004), 291–301
-
[19]
T. Hong, Y.-Z. Wang, Y.-S. Huo, Bogoliubov Quasiparticles Carried by Dark Solitonic Excitations in Nonuniform Bose- Einstein Condensates, Chinese Phys. Lett., 15 (1998), 550–552
-
[20]
B. Λ™Inan, A finite difference method for solving generalized FitzHugh-Nagumo equation, AIP Conf. Proc., 1962 (2018), 1–8
-
[21]
K. H. F. Jwamer, N. Abdullah, Lacunary Spline Function for Solving Second Order Boundary Value Problems, Gen. Math. Notes, 34 (2016), 37–55
-
[22]
M. K. Kadalbajoo, P. Arora, B-spline Collocation Method for the Singular-Perturbation Problem Using Artificial Viscosity, Comput. Math. Appl., 57 (2009), 650–663
-
[23]
J. Kafle, L. P. Bagale, D. J. K. C, Numerical Solution of Parabolic Partial Differential Equation by Using Finite Difference Method, J. Nepal Phys. Soc., 6 (2020), 57–65
-
[24]
I. Karafyllis, T. Ahmed-Ali, F. Giri, Sampled-data observers for 1-D parabolic PDEs with non-local outputs, Systems Control Lett., 133 (2019), 12 pages
-
[25]
R. Katz, E. Fridman, Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs, Automatica J. IFAC, 122 (2020), 10 pages
-
[26]
S. Kazem, M. Dehghan, Application of Finite Difference Method of Lines on the Heat Equation, Numer. Methods Partial Differential Equations, 34 (2017), 626–660
-
[27]
M. I. Liaqat, A. Akg¨ ul, H. Abu-Zinadah, Analytical Investigation of Some Time-Fractional Black–Scholes Models by the Aboodh Residual Power Series Method, Mathematics, 11 (2023), 1–19
-
[28]
A. C. Loyinmi, T. K. Akinfe, An Algorithm for Solving the Burgers–Huxley Equation Using the Elzaki Transform, SN Appl. Sci., 2 (2020), 1–17
-
[29]
B. Mebrate, Numerical Solution of a One-Dimensional Heat Equation with Dirichlet Boundary Conditions, Am. J. Appl. Math., 3 (2015), 305–311
-
[30]
R. Rohila, R. C. Mittal, Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method, Math. Sci., 12 (2018), 79–89
-
[31]
M. Modanli, B. Bajjah, S. Kus¸ulay, Two Numerical Methods for Solving the Schr¨odinger Parabolic and Pseudoparabolic Partial Differential Equations, Adv. Math. Phys., 2022 (2022), 10 pages
-
[32]
M. Sari, S. Ali Tahir, A. Bouhamidi, Behaviour of advection-diffusion-reaction processes with forcing terms, Carpathian J. Math., 35 (2019), 233–252
-
[33]
M. Namjoo, S. Zibaei, Numerical solutions of FitzHugh-Nagumo equation by exact finite-difference and NSFD schemes, Comput. Appl. Math., 37 (2018), 1395–1411
-
[34]
B. Batiha, M. S. M. Noorani, I. Hashim, Application of Variational Iteration Method to the Generalized Burger-Huxley equation, Chaos, Solitons Fractals, 36 (2008), 660–663
-
[35]
A. Pervaiz, M. O. Ahmad, Polynomial Cubic Spline Method for Solving Fourth-Order Parabolic Two-point Boundary Value Problems, Pakistan J. Sci., 67 (2015), 64–67
-
[36]
J. Rashidinia, R. Mohammadi, Non-polynomial Cubic Spline Methods for the Solution of Parabolic Equations, Int. J. Comput. Math., 85 (2008), 843–850
-
[37]
R. J. Spiteri, S. J. Ruuth, A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods, SIAM J. Numer. Anal., 40 (2002), 469–491
-
[38]
L.-Y. Sun, C.-G. Zhu, Cubic B-spline quasi-interpolation and an application to numerical solution of generalized Burgers- Huxley equation, Adv. Mech. Eng., 12 (2020), 1–8
-
[39]
S. A. Tahir, M. Sari, Simulation of Nonlinear Parabolic PDEs with Forcing Function without Linearization, Math. Slovaca, 71 (2021), 1005–1018
-
[40]
S. A. Tahir, M. Sari, A New Approach for the Coupled Advection-Diffusion Processes Including Source Effects, Appl. Numer. Math., 184 (2023), 391–405
-
[41]
X. Y. Wang, Z. S. Zhu, Y. K. Lu, Solitary Wave Solutions of the Generalized Burgers-Huxley Equation, J. Phys. A: Math. Gen., 23 (1990), 271–274
-
[42]
M. A. Yousif, B. A. Mahmood, Construction of Analytical Solution for Hirota–Satsuma Coupled KdV Equation According to Time Via New Approach: Residual Power Series, AIP Adv., 11 (2021), 7 pages
-
[43]
M. A. Yousif, B. A. Mahmood, M. M. Rashidi, Using differential transform method and Pade approximation for solving MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, J. Math. Compu. Sci., 17 (2017), 169– 178