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A variational iteration method for solving Troesch’s problem. (English) Zbl 1191.65101
Summary: Troesch’s problem is an inherently unstable two-point boundary value problem. A new and efficient algorithm based on the variational iteration method and variable transformation is proposed to solve Troesch’s problem. The underlying idea of the method is to convert the hyperbolic-type nonlinearity in the problem into polynomial-type nonlinearities by variable transformation, and the variational iteration method is then directly used to solve this transformed problem. Only the second-order iterative solution is required to provide a highly accurate analytical solution as compared with those obtained by other analytical and numerical methods.

65L10Boundary value problems for ODE (numerical methods)
Full Text: DOI
[1] Weibel, E. S.: On the confinement of a plasma by magnetostatic fields, Phys. fluids 2, No. 1, 52-56 (1959)
[2] 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, 21-33 (1966)
[3] Gidaspow, D.; Baker, B. S.: A model for discharge of storage batteries, J. electrochem. Soc. 120, 1005-1010 (1973)
[4] Roberts, S. M.; Shipman, J. S.: On the closed form solution of troesch’s problem, J. comput. Phys. 21, 291-304 (1976) · Zbl 0334.65062 · doi:10.1016/0021-9991(76)90026-7
[5] B.A. Troesch, Intrinsic difficulties in the numerical solution of a boundary value problem, Internal Report NN--142, TRW Inc., Redondo Beach, California, 1960. · Zbl 0117.19604
[6] Tsuda, T.; Ichida, K.; Kiyono, T.: Monte Carlo path-integral calculations for two-point boundary-value problems, Numer. math. 10, 110-116 (1967) · Zbl 0149.11402 · doi:10.1007/BF02174142
[7] Roberts, S. M.; Shipman, J. S.: Solution of troesch’s two-point boundary value problem by a combination of techniques, J. comput. Phys. 10, 232-241 (1972) · Zbl 0247.65052 · doi:10.1016/0021-9991(72)90063-0
[8] Miele, A.; Aggarwal, A. K.; Tietze, J. L.: Solution of two-point boundary-value problems with Jacobian matrix characterized by large positive eigenvalues, J. comput. Phys. 15, 117-133 (1974) · Zbl 0303.65075 · doi:10.1016/0021-9991(74)90080-1
[9] Vemuri, V.; Raefsky, A.: On a method of solving sensitive boundary value problems, J. franklin. Inst. 307, 217-243 (1979) · Zbl 0401.65052 · doi:10.1016/0016-0032(79)90049-8
[10] Jones, D. J.: Solutions of troesch’s and other two-point boundary-value problems by shooting techniques, J. comput. Phys. 12, 429-434 (1973) · Zbl 0264.65046 · doi:10.1016/0021-9991(73)90165-4
[11] Kubicek, M.; Hlavacek, V.: Solution of troesch’s two-point boundary value problem by shooting technique, J. comput. Phys. 17, 95-101 (1975) · Zbl 0301.65047 · doi:10.1016/0021-9991(75)90066-2
[12] Troesch, B. A.: A simple approach to a sensitive two-point boundary value problem, J. comput. Phys. 21, 279-290 (1976) · Zbl 0334.65063 · doi:10.1016/0021-9991(76)90025-5
[13] Chang, S. H.: Numerical solution of troesch’s problem by simple shooting method, Appl. math. Comput (2010) · Zbl 1191.65100
[14] Chiou, J. P.; Na, T. Y.: 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
[15] Scott, M. R.: On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, Numerical solutions of boundary-value problems for ordinary differential equations, 89-146 (1975) · Zbl 0335.65032
[16] Snyman, J. A.: Continuous and discontinuous numerical solutions to the troesch problem, J. comput. Appl. math. 5, 171-175 (1979) · Zbl 0419.65047 · doi:10.1016/0377-0427(79)90002-5
[17] Deeba, E.; Khuri, S. A.; Xie, S.: An algorithm for solving boundary value problems, J. comput. Phys. 159, 125-138 (2000) · Zbl 0959.65091 · doi:10.1006/jcph.2000.6452
[18] Khuri, S. A.: A numerical algorithm for solving troesch’s problem, Int. J. Comput. math. 80, 493-498 (2003) · Zbl 1022.65084 · doi:10.1080/0020716022000009228
[19] Momani, S.; Abuasad, S.; Odibat, Z.: Variational iteration method for solving nonlinear boundary value problems, Appl. math. Comput. 183, 1351-1358 (2006) · Zbl 1110.65068 · doi:10.1016/j.amc.2006.05.138
[20] Feng, X.; Mei, L.; He, G.: An efficient algorithm for solving troesch’s problem, Appl. math. Comput. 189, 500-507 (2007) · Zbl 1122.65373 · doi:10.1016/j.amc.2006.11.161
[21] Chang, S. H.; Chang, I. L.: A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. math. Comput. 195, 799-808 (2008) · Zbl 1132.65062 · doi:10.1016/j.amc.2007.05.026
[22] He, J. H.: A new approach to nonlinear partial differential equations, Commun. nonlinear sci. Numer. simul. 2, No. 4, 230-235 (1997) · Zbl 0923.35046 · doi:10.1016/S1007-5704(97)90029-0
[23] He, J. H.: Variational iteration method -- a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear mech. 34, 699-708 (1999) · Zbl 05137891
[24] He, J. H.: Variational iteration method for autonomous ordinary systems, Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[25] Strauss, M.; Ring, T. A.; Bowen, H. K.: Osmotic pressure for concentrated suspensions of polydisperse particles with thick double layers, J. colloid interface sci. 118, No. 2, 326-334 (1987)