Numerical treatment of non-monotonic blow-problems based on some non-local transformations. (English) Zbl 07878308
Summary: We consider the numerical treatment of blow-up problems having non-monotonic singular solutions that tend to infinity at some point in the domain. The use of standard numerical methods for solving problems with blow-up solutions can lead to significant errors. The reason being that solutions of such problems have singularities whose positions are unknown in advance. To be able to integrate such non-monotonic blow-up problems, we describe and use a method of non-local transformations. To show the efficiency of the method, we present a comparison of exact and numerical solutions in addition to some comparison with the work of other authors.
MSC:
| 65Lxx | Numerical methods for ordinary differential equations |
Keywords:
nonlinear differential equations; blow-up solutions; non-local transformations; numerical integrationReferences:
| [1] | G. Acosta, G. Duran, J.D. Rossi, An adaptive time step procedure for a parabolic problem with blow-up, Computing 68 (2002), 343-373. · Zbl 1005.65102 · doi:10.1007/s00607-002-1449-x |
| [2] | B. Attili, Numerical solution of a third-order Falkner-Skan-like equation arising in boundary layer theory, Chaos 30 (2020), 073119. |
| [3] | J. Baris, P. Baris, and B. Ruchlewicz, Blow-up solutions of quadratic differential systems, J. Math. Sciences 149 (2008), 1369-1375. · doi:10.1007/s10958-008-0070-8 |
| [4] | J. Butcher, Numerical Methods for Ordinary Differential Equations, third ed., John Wiley and Sons Ltd, 2016. · Zbl 1354.65004 |
| [5] | C.-H. Cho, On the convergence of numerical blow-up time for a second order nonlinear ordinary differential equation, Appl. Math. Letters 24 (2011), 49-54. · Zbl 1209.65082 · doi:10.1016/j.aml.2010.08.011 |
| [6] | P.G. Dlamini, M. Khumalo, On the computation of blow-up solutions for semilinear ODEs and parabolic PDEs, Math. Problems in Eng. 2012 (2012), Article ID 162034. · Zbl 1264.65208 |
| [7] | U. Elias and H. Gingold, Critical points at infinity and blow-up of solutions of autonomous polynomial differential systems via compactification, J. Math. Anal. Appl. 318 (2006), 305-322. · Zbl 1100.34023 · doi:10.1016/j.jmaa.2005.06.002 |
| [8] | C. Hirota, K. Ozawa, Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations - An application to the blow-up problems of partial differential equations, J. Comput. Applied Math. 193 (2006), 614-637. · Zbl 1101.65095 · doi:10.1016/j.cam.2005.04.069 |
| [9] | C. Muriel and J.L. Romero, Non-local transformations and linearization of second-order ordinary differential equations, J. Physics A: Math. Theor. 43 (2010), 434025. · Zbl 1217.34059 |
| [10] | N.R. Nassif, N. Makhoul-Karamb, and Y. Soukiassian, Computation of blowing-up solutions for second-order differential equations using re-scaling techniques, J. Comput. Applied Math. 227 (2009), 185-195. · Zbl 1165.65040 · doi:10.1016/j.cam.2008.07.020 |
| [11] | M. Stuart, M.S. Floater, On the computation of blow-up, European J. Applied Math. 1 (1990), 47-71. · Zbl 0701.76038 · doi:10.1017/S095679250000005X |
| [12] | A. Takayasu, K. Matsue, T. Sasaki, K. Tanaka, M. Mizuguchi, S. Oishi, Numerical validation of blow-up solutions of ordinary differential equations. J. Comput. Applied Math. 314 (2017), 10-29. · Zbl 1376.34035 · doi:10.1016/j.cam.2016.10.013 |
| [13] | A. Polyanina and I. Shingarevad, Nonlinear problems with blow-up solutions: Numerical integration based on differential and non-local transformations, and differential constraints, Applied Math. and Computation 336 (2018), 107-137. · Zbl 1427.65112 · doi:10.1016/j.amc.2018.04.071 |
| [14] | A. Polyanina and I. Shingarevad, Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations, Applied Mathematics Letters 76 (2018), 123-129. · Zbl 1377.65084 · doi:10.1016/j.aml.2017.08.009 |
| [15] | Y.C. Zhou, Z.W. Yang, H.Y. Zhang, and Y.Wang, Theoretical analysis for blow-up behaviors of differential equations with piece-wise constant arguments, Applied Math. Computation 274 (2016), 353-361. · Zbl 1410.34184 · doi:10.1016/j.amc.2015.10.080 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
