Galoitica: Journal of Mathematical Structures and Applications
Journal DOI
2834-5568ISSN (Online)
Volume 10 , Issue 2 , PP: 52-60, 2024 | Cite this article as | XML | Html | PDF | Full Length Article
Ahmed R. Khlefha 1 *
Doi: https://doi.org/10.54216/GJMSA.0100205
This research dealt with the study of the boundary values associated with differential equations, which are of the non-linear and third-order type. A new algorithm was created that uses exponential spline functions to study and address boundary value problems of a general nature. We have demonstrated that the numerical approach used, which was built using exponential spline functions, gives us good and accurate results for these problems, which have been compared to existing numerical methods. We found that the proposed method is accurate and effective compared to other Spalline methods.
Exponential spline functions , boundary value problems, absolute errors
[1] Al-Said EA. Numerical solutions for system of third-order boundary value problems. Int J Comput Math 2001;78:111–21. https://doi.org/10.1080/00207160108805100.
[2] Noor MA, Al-Said EA. Finite-difference method for a system of third-order boundary-value problems. J Optim Theory Appl 2002;112:627–37. https://doi.org/10.1023/a:1017972217727.
[3] Noor MA, Al-Said E, Noor KI. Finite difference method for solving a system of third-order boundary value problems. J Appl Math 2012;2012:1–10. https://doi.org/10.1155/2012/351764.
[4] Mamatha K, Phaneendra K. Solution of convection-diffusion problems using fourth order adaptive cubic spline method. J Math Comput Sci 2020;10:817–32. https://doi.org/10.28919/jmcs/4485.
[5] Mohanty RK, Kumar R, Setia N. Cubic spline approximation based on half-step discretization for 2D quasilinear elliptic equations. Int J Comput Methods Eng Sci Mech 2020;22:45–59. https://doi.org/10.1080/15502287.2020.1849444.
[6] Khandelwal P, Khan A, Sultana T. Discrete cubic spline technique for solving one-dimensional Bratu’s problem. Asian-European J Math 2022;15:2250011. https://doi.org/10.1142/s1793557122500115.
[7] Bica AM, Curilă D, Satmari Z. Improved error estimate and applications of the complete quartic spline. Gen Math 2019;27:71–83. https://doi.org/10.2478/gm-2019-0016.
[8] Wen W, Deng S, Liu T, Duan S, Huang F. An improved quartic B-spline based explicit time integration algorithm for structural dynamics. Eur J Mech 2022;91:104407. https://doi.org/10.1016/j.euromechsol.2021.104407.
[9] Vaid MK, Arora G. Quintic B-spline technique for numerical treatment of third order singular perturbed delay differential equation. Int J Math Eng Manag Sci 2019;4:1471. https://doi.org/10.33889/ijmems.2019.4.6-116.
[10] Huang W-F, Ren Y-X, Wang Q, Jiang X. High resolution finite volume scheme based on the quintic spline reconstruction on non-uniform grids. J Sci Comput 2018;74:1816–52.
[11] El Hajaji A, Serghini A, Melliani S, El Ghordaf J, Hilal K. A quintic spline collocation method for solving time-dependent convection-diffusion problems. Tatra Mt Math Publ 2021;80:15–34. https://doi.org/10.2478/tmmp-2021-0029.
[12] Khan A, Sultana T. Parametric quintic spline solution of third-order boundary value problems. Int J Comput Math 2012;89:1663–77. https://doi.org/10.1080/00207160.2012.689480.
[13] Sultana T, Khan A, Khandelwal P. A new non-polynomial spline method for solution of linear and non-linear third order dispersive equations. Adv Differ Equations 2018;2018:1–14. https://doi.org/10.1186/s13662-018-1763-z.
[14] Wakijira YA, Duressa GF, Bullo TA. Quintic non-polynomial spline methods for third order singularly perturbed boundary value problems. J King Saud Univ 2018;30:131–7. https://doi.org/10.1016/j.jksus.2017.01.008.
[15] Kumara Swamy D, Siva Prasad E, Soujanya G. Solution of convection-diffusion problems with singularity using variable mesh spline of third order. J Math Comput Sci 2020;10:1399–408.
[16] Chaurasia A, Gupta Y, Srivastava PC. A smooth approximation for non-linear second order boundary value problems using composite non-polynomial spline functions. Stud Univ Babes-Bolyai, Math 2020;65.
[17] Hamasalh FK, Headayat MA. The applications of non-polynomial spline to the numerical solution for fractional differential equations. AIP Conf. Proc., vol. 2334, AIP Publishing; 2021.
[18] Al-Said EA, Noor MA. Cubic splines method for a system of third-order boundary value problems. Appl Math Comput 2003;142:195–204.
[19] Said EA. A cubic spline method for solving a system of third order boundary value problems. Int J Pure Appl Math 2008;43:677–86.
[20] Khan MA, Tirmizi IA, Twizell EH. Non polynomial spline approach to the solution of a system of third-order boundary-value problems. Appl Math Comput 2005;168:152–63.
[21] Tirmizi IA, Khan MA. Quartic non-polynomial spline approach to the solution of a system of third-order boundary-value problems. J Math Anal Appl 2007;335:1095–104. https://doi.org/10.1016/j.jmaa.2007.02.046.
[22] Li L, Gao J. Non-polynomial spline method for a system of third order obstacle boundary value problems. J Math Sci Adv Appl 2014;30:49–69.
[23] Khan A, Sultana T. Non-polynomial quintic spline solution for the system of third order boundary-value problems. Numer Algorithms 2012;59:541–59. https://doi.org/10.1007/s11075-011-9503-4.
[24] Noor MA, Al-Said EE. Quartic splines solutions of third-order obstacle problems. Appl Math Comput 2004;153:307–16.
[25] Khan A, Aziz T. The numerical solution of third-order boundary-value problems using quintic splines. Appl Math Comput 2003;137:253–60. https://doi.org/10.1016/s0096-3003(02)00051-6.
[26] Salama AA, Mansour AA. Fourth-order finite-difference method for third-order boundary-value problems. Numer Heat Transf Part B 2005;47:383–401.
