In this paper, we present the application of the double exponential non-classical sinc method for solving a specific class of singularly perturbed two-point boundary value problems. This method is particularly effective for problems with singularities, infinite domains, or boundary layers. We discuss the convergence and error estimation of our approach. Using three illustrative examples, we demonstrate the superior performance of our method compared to existing approaches.
[1] O. Abu Arqub, Z. Abo-Hammour, S. Momani, and N. Shawagfeh, Solving singular two-point boundary value problems using continuous genetic algorithm, In Abstract and Applied Analysis, Wiley Online Library, 2012 (2012), 205391.
[2] A. Alipanah, K. Mohammadi, and M. Shiralizadeh, Numerical solution of third-order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation, Results in Applied Mathematics, 20 (2013), 100403.
[3] A. Eftekhari, Spectral poly-sinc collocation method for solving a singular nonlinear bvp of reaction-diffusion with michaelis-menten kinetics in a catalyst/biocatalyst, Iranian Journal of Mathematical Chemistry, 14 (2023), 77–96.
[4] A. Eftekhari and A. Saadatmandi, De sinc-collocation method for solving a class of second-order nonlinear bvps, Mathematics Interdisciplinary Research, 6 (2021), 11–22.
[5] J. E. Flaherty and W. Mathon, Collocation with polynomial and Tension Splines for singularly-perturbed boundary value problems, SIAM Journal on Scientific and Statistical Computing, 1 (1980), 260–289.
[6] G. Soujanya and K. Phnaeendra, Numerical integration method for singular singularly perturbed two-point boundary value problems, Procedia Engineering, 127 (2015), 545–552.
[7] A. Hamad, M. Tadi, and M. Radenkovic, A numerical method for singular boundary-value problems, Journal of Applied Mathematics and Physics, 2 (2014), 882.
[8] M. More and H. Takahasi, Double exponential formulas for numerical integration, Research Institute for Mathematical Sciences, 9 (1974), 721–741.
[9] A. Alipanah, K. Mohammadi, and M. Ghasemi, A non-classical sinc-collocation method for the solution of singular boundary value problems arising in physiology, International Journal of Computer Mathematics, 99 (2022), 1941– 1967.
[10] R. Mohanty and N. Jha, A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems, Applied Mathematics and Computation, 168 (2005), 704–716.
[11] M. Mori, A. Nurmuhammad, and M. Muhammad, DE-sinc method for second order singularly perturbed boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 41–63.
[12] N. Jha, R. K. Mohanty, and D. J. Evans, Spline in compression method for the numerical solution of singularly perturbed two-point singular boundary-value problems, International Journal of Computer Mathematics, 81 (2004), 615–627.
[13] J. Rashidinia, A. Barati, and M. Nabati, Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems, Numerical Algorithms, 66 (2014), 643–662.
[14] S. Roberts, A boundary value technique for singular perturbation problems, Journal of Mathematical Analysis and Applications, 87 (1982), 489–508.
[15] U. Sakai and R. A. Manabu, A class of simple exponential B-splines and their application to numerical solution to singular perturbation problems, Numerische Mathematik, 55 (1987), 493–500.
[16] B. Shizgal, A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, Journal of Computational Physics, 41 (1981), 309–328.
[17] B. D. Shizgal and H. Chen, The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions, The Journal of chemical physics, 104 (1996), 4137–4150.
[18] A. Shokri and M. M. Khalsaraei, A new efficient high order fourstep multi derivative method for the numerical solution of second-order ivps with oscillating solutions, Comput. Math. Methods, 2020.
[19] M. Sugihara, Optimality of the double exponential formula–functional analysis approach, Numerische Mathematik, 75 (1997), 379–395.
Mahamad haji, R. and AliPanah, A. (2026). Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method. Computational Methods for Differential Equations, 14(1), 247-260. doi: 10.22034/cmde.2025.64105.2885
MLA
Mahamad haji, R. , and AliPanah, A. . "Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method", Computational Methods for Differential Equations, 14, 1, 2026, 247-260. doi: 10.22034/cmde.2025.64105.2885
HARVARD
Mahamad haji, R., AliPanah, A. (2026). 'Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method', Computational Methods for Differential Equations, 14(1), pp. 247-260. doi: 10.22034/cmde.2025.64105.2885
CHICAGO
R. Mahamad haji and A. AliPanah, "Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method," Computational Methods for Differential Equations, 14 1 (2026): 247-260, doi: 10.22034/cmde.2025.64105.2885
VANCOUVER
Mahamad haji, R., AliPanah, A. Efficient numerical solution of singularly perturbed two-point boundary value problems using double exponential non-classical sinc-collocation method. Computational Methods for Differential Equations, 2026; 14(1): 247-260. doi: 10.22034/cmde.2025.64105.2885
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