A comparative study of numerical methods for singularly perturbed boundary value problems

Volume 37, Issue 2, pp 132--153 https://dx.doi.org/10.22436/jmcs.037.02.01

Publication Date: September 20, 2024 Submission Date: May 28, 2024 Revision Date: June 19, 2024 Accteptance Date: July 08, 2024

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Authors

Kamran - Department of Mathematics, ‎Islamia College Peshawar, Peshawar 25120, Khyber Pakhtoonkhwa, Pakistan. S. Aljawi - Department of Mathematical Sciences‎, Princess Nourah Bint Abdulrahman University, Riyadh, P.O. Box 84428, Saudi Arabia. M. Irfan - Department of Mathematics, Islamia College Peshawar‎, Peshawar 25120‎, Khyber Pakhtoonkhwa, Pakistan. D. Santina - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎P.O‎. ‎Box 66833‎, ‎Riyadh 11586, ‎Saudi Arabia. N. Mlaiki - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎P.O‎. ‎Box 66833‎, ‎Riyadh 11586, ‎Saudi Arabia.

Abstract

‎Singularly perturbed boundary value problems (SPBVPs) plays an important role in modeling various phenomena in engineering and other science fields‎. ‎The aim of this article is to compare the performance of numerical methods applied to solve the SPBVPs‎, ‎which are challenging due to the presence of small‎ ‎perturbation parameters leading to boundary or interior layers‎. ‎Four numerical methods are compared‎, ‎namely‎, ‎the local radial basis function method‎, ‎the‎ ‎improved Talbot's method‎, ‎the Euler's method‎, ‎and the Weeks method‎. ‎The main objective of this work is to improve the stability and accuracy of solutions for‎ ‎SPBVPs‎, ‎which are important for many applications in science and engineering‎. ‎We consider two-point SPBVPs with a boundary layer‎ ‎at one endpoint‎. ‎To evaluate the performance and effectiveness of the presented numerical techniques‎, ‎numerical approximations of four SPBVPs are derived and‎ ‎compared with the analytical solutions‎. ‎The results presented in the tables and figures validate that the proposed numerical schemes are highly accurate and‎ ‎clearly outperform the outcomes of other methods‎. ‎Furthermore‎, ‎results from the functional analysis were used to study the existence of the solution to the considered‎ ‎model and generate sufficient requirements for Ulam-Hyers stability‎.

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Keywords

• Singularly perturbed BVPs
• local radial basis functions
• Euler's method
• improved Talbot's method
• Weeks method
• Laplace transform

MSC

•  34Bxx
•  34Cxx
•  34D15
•  44A10

References

• [1] J. Abate, W. Whitt, A unified framework for numerically inverting Laplace transforms, INFORMS J. Comput., 18 (2006), 408–421
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Abdeljawad, A. Younus, M. A. Alqudah, U. Atta, On fuzzy conformable double Laplace transform with applications to partial differential equations, Comput. Model. Eng. Sci., 134 (2023), 2163–2191
  • View Article
  • Google Scholar


• [3] R. P. Agarwal, M. Meehan, D. O’regan, Fixed point theory and applications, Cambridge University Press, Cambridge (2001)
  • View Article
  • MathSciNet
  • Google Scholar


• [4] S. Ahmed, S. Jahan, K. J. Ansari, K. Shah, T. Abdeljawad, Wavelets collocation method for singularly perturbed differential difference equations arising in control system, Results Appl. Math., 21 (2024), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] M. Ahsan, M. Bohner, A. Ullah, A. A. Khan, S. Ahmad, A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions, Math. Comput. Simul., 204 (2023), 166–180
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. Ali, J. F. Gómez-Aguilar, A transform based local RBF method for 2D linear PDE with Caputo-Fabrizio derivative, C. R. Math. Acad. Sci. Paris, 358 (2020), 831–842
  • View Article
  • Google Scholar
  • MATH


• [7] I. Ali, S. Haq, R. Ullah, S. U. Arifeen, Approximate Solution of Second Order Singular Perturbed and Obstacle Boundary Value Problems Using Meshless Method Based on Radial Basis Functions, J. Nonlinear Math. Phys., 30 (2023), 215–234
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] Z. Avazzadeh, O. Nikan, J. Tenreiro Machado, M. N. Rasoulizadeh, Numerical analysis of time-fractional Sobolev equation for fluid-driven processes in impermeable rocks, Adv. Contin. Discrete Models, 2022 (2022), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] I. A. Bhat, L. N. Mishra, V. N. Mishra, C. Tunç, O. Tunç, Precision and efficiency of an interpolation approach to weakly singular integral equations, Int. J. Numer. Methods Heat Fluid Flow, 34 (2024), 1479–1499
  • View Article
  • Google Scholar


• [10] H. G. Debela, S. B. Kejela, A. D. Negassa, Exponentially Fitted Numerical Method for Singularly Perturbed Differential- Difference Equations, Int. J. Differ. Equ., 20 (2020), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] H. G. Debela, G. F. Duressa, Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition, J. Egypt. Math. Soc., 28 (2020), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] B. Dingfelder, J. A. C. Weideman, An improved Talbot method for numerical Laplace transform inversion, Numer. Algorithms, 68 (2015), 167–183
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] I. P. Gavrilyuk, V. L. Makarov, Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces, SIAM J. Numer. Anal., 43 (2005), 2144–2171
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] F. Z. Geng, S. P. Qian, S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, J. Comput. Appl. Math., 255 (2014), 97–105
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] M. B. Ghaemi, M. E. Gordji, B. Alizadeh, C. Park, Hyers-Ulam stability of exact second-order linear differential equations, Adv. Differ. Equ., 2012 (2012), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] V. Y. Glizer, Asymptotic analysis and solution of a finite-horizon H1 control problem for singularly-perturbed linear systems with small state delay, J. Optim. Theory. Appl., 117 (2003), 295–325
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] D. J. Halsted, D. E. Brown, Zakian’s technique for inverting Laplace transforms, Chem. Eng. J., 3 (1972), 312–313
  • View Article
  • Google Scholar


• [18] P. Hammachukiattikul, E. Sekar, A. Tamilselvan, R. Vadivel, N. Gunasekaran, P. Agarwal, Comparative study on numerical methods for singularly perturbed advanced-delay differential equations, J. Math., 2021 (2021), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] H. Hassanzadeh, M. Pooladi-Darvish, Comparison of different numerical Laplace inversion methods for engineering applications, Appl. Math. Comput., 189 (2007), 1966–1981
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] G. Honig, U. Hirdes, A method for the numerical inversion of Laplace transforms, J. Comput. Appl. Math., 10 (1984), 113–132
  • View Article
  • MathSciNet
  • Google Scholar


• [21] F. O. Ilicasu, D. H. Schultz, High-order finite-difference techniques for linear singular perturbation boundary value problems, Comput. Math. Appl., 47 (2004), 391–417
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. Jelbart, N. Pages, V. Kirk, J. Sneyd, M. Wechselberger, Process-oriented geometric singular perturbation theory and calcium dynamics, SIAM J. Appl. Dyna. Syst., 21 (2022), 982–1029
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] M. K. Kadalbajoo, P. Arora, V. Gupta, Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers, Comput. Math. Appl., 61 (2011), 1595–1607
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] Kamran, M. Irfan, F. M. Alotaibi, S. Haque, N. Mlaiki, K. Shah, RBF-based local meshless method for fractional diffusion equations, Fractal Fract., 7 (2023), 21 pages
  • View Article
  • Google Scholar


• [25] Kamran, S. U. Khan, S. Haque, N. Mlaiki, On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method, Symmetry, 15 (2023), 16 pages
  • View Article
  • Google Scholar


• [26] M. Kumar, P. Singh, H. K. Mishra, An initial-value technique for singularly perturbed boundary value problems via cubic spline, Int. J. Comput. Methods Eng. Sci. Mech., 8 (2007), 419–427
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] C. K. Lee, X. Liu, S. C. Fan, Local multiquadric approximation for solving boundary value problems, Comput. Mech., 30 (2003), 396–409
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] E. Liz, C. Lois-Prados, A note on the Lasota discrete model for blood cell production, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 701–713
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] A. Luongo, S. Casciati, D. Zulli, Perturbation method for the dynamic analysis of a bistable oscillator under slow harmonic excitation, Smart Struct. Syst., 18 (2016), 183–196
  • View Article
  • Google Scholar


• [30] M. Maruši´c, On , Math. Commun., 19 (2014), 545–559
  • View Article
  • Google Scholar
  • MATH


• [31] S. Natesan, J. Jayakumar, J. Vigo-Aguiar, Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers, J. Comput. Appl. Math., 158 (2003), 121–134
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York-London-Sydney (1973)
  • View Article
  • Google Scholar


• [33] N. N. Nefedov, L. Recke, K. R. Schneider, Existence and asymptotic stability of periodic solutions with an interior layer of reaction advection diffusion equations, J. Math. Anal. Appl., 405 (2013), 90–103
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] O. Nikan, Z. Avazzadeh, J. A. T. Machado, A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer, J. Adv. Res., 32 (2021), 45–60
  • View Article
  • Google Scholar


• [35] A. Noorizadegan, D. L. Young, C.-S. Chen, A novel local radial basis function collocation method for multi-dimensional piezoelectric problems, J. Intell. Mater. Syst. Struct., 33 (2022), 1574–1587
  • View Article
  • Google Scholar


• [36] C. E. Pearson, On a differential equation of boundary layer type, J. Math. Phys., 47 (1968), 134–154
  • View Article
  • Google Scholar


• [37] K. Phaneendra, Y. N. Reddy, G. B. S. L. Soujanya, Non-iterative numerical integration method for singular perturbation problems exhibiting internal and twin boundary layers, Int. J. Appl. Math. Comput., 3 (2011), 9–20
  • View Article
  • Google Scholar


• [38] J. Quinn, A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location, J. Comput. Appl. Math., 290 (2015), 500–515
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] P. Rai, K. K. Sharma, Numerical study of singularly perturbed differential difference equation arising in the modeling of neuronal variability, Comput. Math. Appl., 63 (2012), 118–132
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [40] H. Ramos, J. Vigo-Aguiar, S. Natesan, R. García-Rubio, M. A. Queiruga, Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm, J. Math. Chem., 48 (2010), 38–54
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] M. Safinejad, M. M. Moghaddam, A local meshless RBF method for solving fractional integro-differential equations with optimal shape parameters, Ital. J. Pure Appl. Math., 41 (2019), 382–398
  • View Article
  • Google Scholar


• [42] B. Šarler, R. Vertnik, Meshfree explicit local radial basis function collocation method for diffusion problems, Comput. Math. Appl., 51 (2006), 1269–1282
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [43] R. Schaback, Error estimates and condition numbers for radial basis function interpolation, Adv. Comput. Math., 3 (1995), 251–264
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] F. A. Shah, R. Abass, J. Iqbal, Numerical solution of singularly perturbed problems using Haar wavelet collocation method, Cogent Math., 3 (2016), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [45] F. A. Shah, Kamran, W. Boulila, A. Koubaa, N. Mlaiki, Numerical Solution of Advection-Diffusion Equation of Fractional Order Using Chebyshev Collocation Method, Fractal Fract., 7 (2023), 20 pages
  • View Article
  • Google Scholar


• [46] F. A. Shah, Kamran, D. Santina, N. Mlaiki, S. Aljawi, Application of a hybrid pseudospectral method to a new twodimensional multi-term mixed sub-diffusion and wave-diffusion equation of fractional order, Netw. Heterog. Media, 19 (2024), 44–85
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [47] F. A. Shah, Kamran, K. Shah, T. Abdeljawad, Numerical modelling of advection diffusion equation using Chebyshev spectral collocation method and Laplace transform, Results Appl. Math., 21 (2024), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [48] D. Sheen, I. H. Sloan, V. Thomée, A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer. Anal., 23 (2003), 269–299
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [49] J. L. Schiff, The Laplace transform: theory and applications, Springer-Verlag, New York (1999)
  • View Article
  • Google Scholar


• [50] Siraj-ul-Islam, R. Vertnik, B. Šarler, Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl. Numer. Math., 67 (2013), 136–151
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [51] H. Stehfest, Algorithm 368: Numerical inversion of Laplace transforms [D5], Commun. ACM., 13 (1970), 47–49
  • View Article
  • Google Scholar


• [52] A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl., 23 (1979), 97–120
  • View Article
  • Google Scholar
  • MATH


• [53] L. N. Trefethen, D. Bau III, Numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1997)
  • View Article
  • MathSciNet
  • Google Scholar


• [54] O. Tunç, C. Tunç, Ulam stabilities of nonlinear iterative integro-differential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 117 (2023), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [55] O. Tunç, C. Tunç, On Ulam stabilities of iterative Fredholm and Volterra integral equations with multiple time-varying delays, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 118 (2024), 20 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [56] J. Vigo-Aguiar, S. Natesan, An efficient numerical method for singular perturbation problems, J. Comput. Appl. Math., 192 (2006), 132–141
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [57] A.-M. Wazwaz, Linear and nonlinear integral equations, Higher Education Press, Beijing; Springer, Heidelberg (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [58] S. Wei, W. Chen, Y. Zhang, H. Wei, R. M. Garrard, A local radial basis function collocation method to solve the variableorder time fractional diffusion equation in a two-dimensional irregular domain, Numer. Methods Partial Differ. Equ., 34 (2018), 1209–1223
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [59] J. A. C. Weideman, Algorithms for parameter selection in the Weeks method for inverting the Laplace transform, SIAM J. Sci. Comput., 21 (1999), 111–128
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [60] G. Yao, Z. Yu, A localized meshless approach for modeling spatial-temporal calcium dynamics in ventricular myocytes, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 187–204
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [61] A. Younus, M. Asif, U. Atta, T. Bashir, T. Abdeljawad, Applications of fuzzy conformable Laplace transforms for solving fuzzy conformable differential equations, Soft Comput., 27 (2023), 8583–8597
  • View Article
  • Google Scholar