On solving the Troesch problem for large sensitivity parameter values using exact derivative evaluations. (English) Zbl 1497.65111
Summary: A simple, straightforward, accurate, and efficient numerical method for the solution of the Troesch’s problem is presented in this paper. The Troesch’s problem is a well-known nonlinear two-point boundary value problem and it involves a parameter \(\lambda \), which is known as the sensitivity parameter. Obtaining numerical solutions for large values of \(\lambda\) has been known to be a challenging problem for several decades now. We solve the Troesch’s problem by using Taylor series and shooting. While we use higher-order derivatives in the Taylor expansion, we do not use finite difference formulas or lengthy analytical expressions for evaluating them. On the other hand, we use exact derivative evaluations which are obtained by using recursive formulas derived from the governing differential equation itself. Thus, our method avoids round-off effects and the use of symbolic manipulation systems, and requires much less computational effort when compared to other existing methods for producing results of comparable accuracy. We also find that our numerical results are in excellent agreement with the several approximate solutions obtained previously. Additionally, it is important to emphasize that we have been able to obtain numerical solutions for the Troesch’s problem with \(\lambda \approx 1000\), which only few researchers have been able to accomplish by using more complicated approaches.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L04 | Numerical methods for stiff equations |
Keywords:
Troesch’s problem; sensitivity parameter; hyperbolic nonlinearity; polynomial nonlinearity; stiffness; Taylor series; recursive formulas for the evaluation of exact Taylor coefficients; shootingReferences:
| [1] | Ben-Romdhane, M.; Temimi, H., A novel computational method for solving Troesch’s problem with high-sensitivity parameter, Int. J. Comput. Methods Eng. Sci. Mech., 18, 4-5, 230-237 (2017) · Zbl 07871461 · doi:10.1080/15502287.2017.1339137 |
| [2] | Bendtsen, C., Stauning, O.: FADBAD: A Flexible C++ Package for Automatic Differentiation. Technical Report, IMM Department of Mathematical Modelling, Technical University of Denmark, (1996) |
| [3] | Berz, M.; Makino, K.; Shamseddine, K.; Hoffstätter, GH; Wan, W., COSY INFINITY and Its Applications in Nonlinear Dynamics (1996), Computational Differentiation: Techniques, Applications, and Tools, SIAM Publications, Philadelphia, PA, Computational Differentiation |
| [4] | Cash, J.R., Wright, M.H., Mazzia, F.: TWPBVP: Test Set for BVP Solvers. https://archimede.dm.uniba.it/ bvpsolvers/testsetbvpsolvers/?page_id=488 |
| [5] | Chang, S-H, Numerical solution of Troesch’s problem by simple shooting method, Appl. Math. Comput., 216, 11, 3303-3306 (2010) · Zbl 1191.65100 |
| [6] | Charpentier, I.; Utke, J., Rapsodia: User Manual (2009), Argonne National Laboratory: Technical Report, Argonne National Laboratory · Zbl 1166.65009 |
| [7] | Chiou, JP; Na, TY, On the solution of Troesch’s nonlinear two-point boundary value problem using an initial value method, J. Comput. Phys., 19, 311-316 (1975) · Zbl 0318.65036 · doi:10.1016/0021-9991(75)90080-7 |
| [8] | Deeba, E.; Khuri, SA; Xie, S., An algorithm for solving boundary value problems, J. Comput. Phys., 159, 2, 125-138 (2000) · Zbl 0959.65091 · doi:10.1006/jcph.2000.6452 |
| [9] | Feng, X.; Mei, L.; He, G., An efficient algorithm for solving Troesch’s problem, Appl. Math. Comput., 189, 1, 500-507 (2007) · Zbl 1122.65373 · doi:10.1016/j.amc.2006.11.161 |
| [10] | Gear, CW, Numerical Initial Value Problems in Ordinary Differential Equations (1971), New Jersey: Prentice Hall, New Jersey · Zbl 1145.65316 |
| [11] | Gidaspow, D.; Baker, BS, A model for discharge of storage batteries, J. Electrochem. Soc., 120, 8, 1005 (1973) · doi:10.1149/1.2403617 |
| [12] | Griewank, A.; Juedes, D.; Utke, J., Algorithm 755: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++, ACM Transactions on Mathematical Software, 22, 131-167 (1996) · Zbl 0884.65015 · doi:10.1145/229473.229474 |
| [13] | Hashemi, MS; Abbasbandy, S., A geometric approach for solving troesch’s problem, Bulletin of the Malaysian Mathematical Sciences Society, 40, 1, 97-116 (2017) · Zbl 1357.65099 · doi:10.1007/s40840-015-0260-8 |
| [14] | Inc, M.; Akgül, A., The reproducing kernel Hilbert space method for solving Troesch’s problem, Journal of the Association of Arab Universities for Basic and Applied Sciences, 14, 1, 19-27 (2013) · doi:10.1016/j.jaubas.2012.11.005 |
| [15] | Ismail, MS; Al-Basyoni, KS, A Logarithmic Finite Difference Method for Troesch’s Problem, Appl. Math., 9, 5, 550 (2018) · doi:10.4236/am.2018.95039 |
| [16] | Kafri, HQ; Khuri, SA; Sayfy, A., A new approach based on embedding Green’s functions into fixed-point iterations for highly accurate solution to Troesch’s problem, Int. J. Comput. Methods Eng. Sci. Mech., 17, 2, 93-105 (2016) · Zbl 07871423 · doi:10.1080/15502287.2016.1157646 |
| [17] | Khuri, SA, A numerical algorithm for solving Troesch’s problem, Int. J. Comput. Math., 80, 4, 493-498 (2003) · Zbl 1022.65084 · doi:10.1080/0020716022000009228 |
| [18] | Khuri, SA; Sayfy, A., Troesch’s problem: A B-spline collocation approach, Math. Comput. Model., 54, 9-10, 1907-1918 (2011) · Zbl 1235.65086 · doi:10.1016/j.mcm.2011.04.030 |
| [19] | Makarov, VL; Dragunov, DV, An efficient approach for solving stiff nonlinear boundary value problems, J. Comput. Appl. Math., 345, 452-470 (2019) · Zbl 1398.65151 · doi:10.1016/j.cam.2018.06.025 |
| [20] | Markin, V.S., Chernenko, A.A., Chizmadehev, Y.A., Chirkov, Y.G.: Aspects of the theory of gas porous electrodes. Fuel Cells: Their Electrochemical Kinetics, pages 22-33, (1966) |
| [21] | Mirmoradi, SH; Hosseinpour, I.; Ghanbarpour, S.; Barari, A., Application of an approximate analytical method to nonlinear Troesch’s problem, Appl. Math. Sci., 3, 32, 1579-1585 (2009) · Zbl 1334.76106 |
| [22] | Momani, S.; Abuasad, S.; Odibat, Z., Variational iteration method for solving nonlinear boundary value problems, Appl. Math. Comput., 183, 2, 1351-1358 (2006) · Zbl 1110.65068 · doi:10.1016/j.amc.2006.05.138 |
| [23] | Moore, RE, Methods and Applications of Interval Analysis (1979), Philadelphia, PA: SIAM Publications, Philadelphia, PA · Zbl 0417.65022 · doi:10.1137/1.9781611970906 |
| [24] | Nabati, M.; Jalalvand, M., Solution of Troesch’s problem through double exponential Sinc-Galerkin method, Computational Methods for Differential Equations, 5, 2, 141-157 (2017) · Zbl 1424.65109 |
| [25] | Rall, LB, Lecture Notes in Computer Science (1981), Berlin: Springer, Berlin · Zbl 0473.68025 |
| [26] | Roberts, SM; Shipman, JS, On the closed form solution of Troesch’s problem, J. Comput. Phys., 21, 3, 291-304 (1976) · Zbl 0334.65062 · doi:10.1016/0021-9991(76)90026-7 |
| [27] | Saadatmandi, A.; Abdolahi-Niasar, T., Numerical solution of Troesch’s problem using Christov rational functions, Computational Methods for Differential Equations, 3, 4, 247-257 (2015) · Zbl 1412.65061 |
| [28] | Johannes Arnoldus Snyman, Continuous and discontinuous numerical solutions to the Troesch problem, J. Comput. Appl. Math., 5, 3, 171-175 (1979) · Zbl 0419.65047 · doi:10.1016/0377-0427(79)90002-5 |
| [29] | Temimi, H.; Ben-Romdhane, M.; Ansari, AR; Shishkin, GI, Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh, Calcolo, 54, 1, 225-242 (2017) · Zbl 1369.65093 · doi:10.1007/s10092-016-0184-1 |
| [30] | Temimi, H.; Kurkcu, H., An accurate asymptotic approximation and precise numerical solution of highly sensitive Troesch’s problem, Appl. Math. Comput., 235, 253-260 (2014) · Zbl 1337.65093 · doi:10.1016/j.amc.2014.03.022 |
| [31] | Troesch, BA, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys., 21, 3, 279-290 (1976) · Zbl 0334.65063 · doi:10.1016/0021-9991(76)90025-5 |
| [32] | Troesch, BA.: Intrinsic Difficulties in the numerical solution of a boundary value problem. Continental Graphics, (1970) |
| [33] | von Hippel, GM, New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95, Comput. Phys. Commun., 176, 710-711 (2007) · Zbl 1196.65217 · doi:10.1016/j.cpc.2007.03.008 |
| [34] | Weibel, ES, On the confinement of a plasma by magnetostatic fields, The Physics of Fluids, 2, 1, 52-56 (1959) · doi:10.1063/1.1724391 |
| [35] | Weibel, E.S., Landshoff, R.K.M.: The plasma in magnetic field. In: A symposium on magneto hydrodynamics. Stanford University Press, Stanford, pages 60-67, (1958) |
| [36] | Zarebnia, M.; Sajjadian, M., The sinc-Galerkin method for solving Troesch’s problem, Math. Comput. Model., 56, 9-10, 218-228 (2012) · Zbl 1255.65153 · doi:10.1016/j.mcm.2011.11.071 |
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.
