A simple, efficient and accurate new Lie--group shooting method for solving nonlinear boundary value problems

[1] S. Abbasbandy, B. Azarnavid, and M.S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, J. Comput. Appl. Math. 279 (2015) 293–305.

[2] S. Abbasbandy and E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 3830–3846.

[3] S.J. Liao, A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int. J. Nonlinear Mech. 42 (2007) 819–830.

[4] H. Xu and S.J. Liao, Dual solutions of boundary layer flow over an upstream moving plate, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 350–358.

[5] C.-S. Liu, Cone of non-linear dynamical system and group preserving schemes, Int. J. Non–Linear Mech. 36 (2001) 1047–1068.

[6] H.-C. Lee, C.-K. Chen, and C.-I. Hung, A modified group–preserving scheme for solving the initial value problems of stiff ordinary differential equations, Appl. Math. Comput. 133 (2002) 445–459.

[7] C.-S. Liu, Two-dimensional bilinear oscillator: group–preserving scheme and steady–state motion under harmonic loading, Int. J. Non–Linear Mech. 38 (2003) 1581–1602.

[8] C.-S. Liu, Group preserving scheme for backward heat conduction problems, Int. J. Heat Mass Tran. 47 (2004) 2567–2576.

[9] S.-Y. Zhang and Z.-C. Deng, Group preserving schemes for nonlinear dynamic systems based on RKMK methods, Appl. Math. Comput. 175 (2006) 497–507.

 [10] C.-S. Liu, New integrating methods for Time–Varying linear systems and Lie–group computations, CMES 20 (2007) 157–175.

[11] C.-S. Liu, Solving an Inverse Sturm–Liouville Problem by a Lie–Group method, Bound. Value Prob. 2008 (2008) 18 pages.

[12] S. Abbasbandy and M.S. Hashemi, Group preserving scheme for the Cauchy problem of the Laplace equation, Engin. Anal. Bound. Elem. 35 (2011) 1003–1009.

[13] C.-S. Liu and C.-W. Chang, A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains, Engin. Anal. Bound. Elem. 36 (2012) 211–219.

[14] C.-S. Liu and S.N. Atluri, A GL(n, R) differential algebraic equation method for numerical differentiation of noisy signal, CMES, 92 (2013) 213–239.

[15] C.-S. Liu and W.-S. Jhao, The second Lie–Group SO(n, 1) used to solve ordinary differential equations, J. Math. Res. 6 (2014) 18–37.

[16] S. Abbasbandy, R.A. Van Gorder, M. Hajiketabi, and M. Mesrizadeh, Existence and numerical simulation of periodic travelling wave solutions to the Casimir equation for the Ito system, Commun. Nonlinear Sci. Numer. Simulat. 27 (2015) 254–262.

[17] C.-S. Liu, The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions, Comput. Model. Eng. Sci. 13 (2006) 149–163.

[18] C.-S. Liu, Efficient shooting methods for the second order ordinary differential equations, Comput. Model. Eng. Sci. 15 (2006) 69–86.

[19] C.-S. Liu, The Lie-group shooting method for singularly perturbed two-point boundary value problems, Comput. Model. Eng. Sci. 15 (2006) 179–196.

[20] S. Abbasbandy, R.A. Van Gorder, and M. Hajiketabi, The Lie–group Shooting Method for Radial Symmetric Solutions of the Yamabe Equation, CMES, 104 (2015) 329–351.

[21] S. Abbasbandy, M.S. Hashemi, and C.-S. Liu, The Lie–group shooting method for solving the Bratu equation, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 4238–4249.

[22] C.-S. Liu, The Lie–group shooting method for solving multi–dimensional nonlinear boundary value problems, J.Optim. Theory Appl. 152 (2012) 468–495.

[23] C.-S. Liu, A Lie–group shooting method for computing eigenvalues and eigenfunctions of Sturm–Liouville problems, Comput. Model. Eng. Sci. 26 (2008) 157–68.

[24] C.W. Chang, J.R. Chang, and C.-S. Liu, The Lie–group shooting method for solving classical Blasius flat plate problem, Comput. Mat. Cont.7 (2008) 139–153.

[25] C.W. Chang, J.R. Chang, and C.-S. Liu, The Lie–group shooting method for boundary layer equations in fluid mechanics, J. Hyd. 18 (2006) 103–108.

[26] C.-S. Liu, C.W. Chang, and J.R. Chang, A new shooting method for solving boundary layer equation in fluid mechanics, Comput. Mod. Engin. Sci. 32 (2008) 1–15.

[27] C.-S. Liu, The Lie-group shooting method for boundary–layer problems with suction/injection/reverse flow conditions for power-law fluids, Int. J. Non-Linear Mech. 46 (2011) 1001–1008.

[28] C.-S. Liu, Computing the eigenvalues of the generalized Sturm–Liouville problems based on the Lie-group SL(2, R), J. Comput. Appl. Math. 236 (2012) 4547–4560.

[29] C.-S. Liu, The Lie-group shooting method for solving multi–dimensional nonlinear boundary value problems, J. Opt. Theo. Appl. 152 (2012) 468–495.

[30] C.-S. Liu, A Lie-group shooting method for reconstructing a past time-dependent heat source, Int. J. Heat Mass Tran. 55 (2012) 1773–1781.

[31] C.-S. Liu, Developing an SL(2, R) Lie-group shooting method for a singular φ–Laplacian in a nonlinear ODE, Communi. Nonlinear Sci. Num. Simu. 18 (2013) 2327–2339.

[32] C.-S. Liu, An SL (3, R) shooting method for solving the Falkner–Skan boundary layer equation, Int. Jo Non–Linear Mech. 49 (2013) 145–151.

[33] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu–type equations, Appl. Math. Comput. 166 (2005) 652–663.

[34] A. Mohsen, L.F. Sedeek, and S.A. Mohamed, New smoother to enhance multigrid–based methods for Bratu problem, Appl. Math. Comput. 204 (2008) 325–339.

[35] S. Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. Eng. J. 136 (2008) 144–150.

[36] E. Magyari, Exact analytical solution of a nonlinear reaction–diffusion model in porous catalysts, Chem. Eng. J. 143 (2008) 167–171.

[37] Y.P. Sun, S.B. Liu, and S. Keith, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by decomposition method, Chem. Eng. J. 102 (2004) 1–10.

[38] S. Abbasbandy, E. Magyari, and E. Shivanian, The homotopy analysis method for multiple solutions of nonlinear boundary value problems, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 3530–3536.

[39] Q. Yao, Successive Iteration and positive solution for nonlinear second–order three–point boundary value problems, Comput. Math. Appl. 50 (2005) 433–444.

[40] Q.Yao, Successive iteration of positive solution for a discontinuous third–order boundary value problem, Comput. Math. Appl. 53 (2007) 741–749.

[41] M. Kubicek and V. Hlavacek, Numerical Solution of Nonlinear Boundary Value Problems with Applications, Prentice–Hall, New York, 1983.