The BVPh (version 1.0) is a Mathematica package for highly nonlinear boundary-value/eigenvalue problems with singularity and/or multipoint boundary conditions. It is a combination of the homotopy analysis method (HAM) and the computer algebra system Mathematica, and provides us a convenient analytic tool to solve many nonlinear ordinary differential equations (ODEs) and even some nonlinear partial differential equations (PDEs). In this chapter, we briefly describe its scope, the basic mathematical formulas, and the choice of base functions, initial guess and the auxiliary linear operator, and so on, together with a simple users guide. As open resource, the BVPh 1.0 is given in the appendix of this chapter and free available (Accessed 25 Nov 2011, will be updated in the future) at


Initial Guess Computer Algebra System Linear Differential Operator Chebyshev Series Auxiliary Linear Operator 
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  1. Abbas, Z., Wang, Y., Hayat, T., Oberlack, M.: Hydromagnetic flow in a viscoelstic fluid due to the oscillatory stretching surface. Int. J. Nonlin. Mech. 43, 783–793 (2008).zbMATHCrossRefGoogle Scholar
  2. Abbasbandy, S.: The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A. 360, 109–113 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  3. Abbasbandy, S.: The application of homotopy analysis method to solve a generalized HirotaSatsuma coupled KdV equation. Phys. Lett. A. 361, 478–483 (2007).zbMATHCrossRefGoogle Scholar
  4. Abbasbandy, S.: Solitary wave equations to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method. Nonlinear Dynam. 52, 35–40 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  5. Abbasbandy, S., Magyari, E., Shivanian, E.: The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation. 14, 3530–3536 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  6. Abbasbandy, S., Parkes, E.J.: Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method. Chaos Soliton. Fract. 36, 581–591 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  7. Abbasbandy, S., Parkes, E.J.: Solitary-wave solutions of the DegasperisProcesi equation by means of the homotopy analysis method. Int. J. Comp. Math. 87, 2303–2313 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  8. Abbasbandy, S., Shivanian, E.: Predictor homotopy analysis method and its application to some nonlinear problems. Commun. Nonlinear Sci. Numer. Simulat. 16, 2456–2468 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  9. Abell, M.L., Braselton, J.P.: Mathematica by Example (3rd Edition). Elsevier Academic Press. Amsterdam (2004).Google Scholar
  10. Akyildiz, F.T., Vajravelu, K.: Magnetohydrodynamic flow of a viscoelastic fluid. Phys. Lett. A. 372, 3380–3384 (2008).zbMATHCrossRefGoogle Scholar
  11. Akyildiz, F.T., Vajravelu, K., Mohapatra, R.N., Sweet, E., Van Gorder, R.A.: Implicit differential equation arising in the steady flow of a Sisko fluid. Applied Mathematics and Computation. 210, 189–196 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  12. Alizadeh-Pahlavan, A., Aliakbar, V., Vakili-Farahani, F., Sadeghy, K.: MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method. Commun. Nonlinear Sci. Numer. Simulat. 14, 473–488 (2009).CrossRefGoogle Scholar
  13. Alizadeh-Pahlavan, A., Borjian-Boroujeni, S.: On the analytical solution of viscous fluid flow past a flat plate. Physics Letters A. 372, 3678–3682 (2008).CrossRefGoogle Scholar
  14. Allan, F.M.: Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput. 190, 6–14 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  15. Allan, F.M.: Construction of analytic solution to chaotic dynamical systems using the homotopy analysis method. Chaos, Solitons and Fractals. 39, 1744–1752 (2009).zbMATHCrossRefGoogle Scholar
  16. Allan, F.M., Syam, M.I.: On the analytic solutions of the nonhomogeneous Blasius problem. J. Comp. Appl. Math. 182, 362–371 (2005).MathSciNetzbMATHCrossRefGoogle Scholar
  17. Battles, Z., Trefethen, L.N.: An extension of Matlab to continuous functions and operators. SIAM J. Sci. Comp. 25, 1743–1770 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  18. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. DOVER Publications, Inc. New York (2000).Google Scholar
  19. Cai, W.H.: Nonlinear Dynamics of Thermal-Hydraulic Networks. PhD dissertation, University of Notre Dame (2006).Google Scholar
  20. Cheng, J.: Application of the Homotopy Analysis Method in Nonlinear Mechanics and Finance. PhD dissertation, Shanghai Jiao Tong University (2008).Google Scholar
  21. Gao, L.M.: Analysis of the Propagation of Surface Acoustic Waves in Functionally Graded Material Plate. PhD dissertation, Tongji University (2007).Google Scholar
  22. Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A. 361, 316–322 (2007).zbMATHCrossRefGoogle Scholar
  23. Jiao, X.Y.: Approximate Similarity Reduction and Approximate Homotopy Similarity Reduction of Several Nonlinear Problems. PhD dissertation, Shanghai Jiao Tong University (2009).Google Scholar
  24. Jiao, X.Y., Gao, Y., Lou, S.Y.: Approximate homotopy symmetry method-Homotopy series solutions to the sixth-order Boussinesq equation. Science in China (G). 52, 1169–1178 (2009).CrossRefGoogle Scholar
  25. Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and theMATLAB PSE. ACM TOMS. 27, 299–316 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kumari, M., Nath, G.: Unsteady MHD mixed convection flow over an impulsively stretched permeable vertical surface in a quiescent fluid. Int. J. Non-Linear Mech. 45, 310–319 (2010).CrossRefGoogle Scholar
  27. Kumari, M., Pop, I., Nath, G.: Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest. Int. J. Non-Linear Mech. 45, 463–473 (2010).CrossRefGoogle Scholar
  28. Li, Y.J., Nohara, B.T., Liao, S.J.: Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J. Mathematical Physics 51, 063517 (2010). doi:10.1063/1.3445770.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Liang, S.X.: Symbolic Methods for Analyzing Polynomial and Differential Systems. PhD dissertation, University of Western Ontario (2010).Google Scholar
  30. Liang, S.X., Jeffrey, D.J.: Comparison of homotopy analysis method and homotopy perturbation method through an evalution equation. Commun. Nonlinear Sci. Numer. Simulat. 14, 4057–4064 (2009a).MathSciNetzbMATHCrossRefGoogle Scholar
  31. Liang, S.X., Jeffrey, D.J.: An efficient analytical approach for solving fourth order boundary value problems. Computer Physics Communications. 180, 2034–2040 (2009b).MathSciNetzbMATHCrossRefGoogle Scholar
  32. Liang, S.X., Jeffrey, D.J.: Approximate solutions to a parameterized sixth order boundary value problem. Computers and Mathematics with Applications. 59, 247–253 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  33. Liao, S.J.: The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD dissertation, Shanghai Jiao Tong University (1992).Google Scholar
  34. Liao, S.J.: A kind of approximate solution technique which does not depend upon small parameters (II)-An application in fluid mechanics. Int. J. Nonlin. Mech. 32, 815–822 (1997).zbMATHCrossRefGoogle Scholar
  35. Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlin. Mech. 34, 759–778 (1999a).zbMATHCrossRefGoogle Scholar
  36. Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mech. 385, 101–128 (1999b).MathSciNetzbMATHCrossRefGoogle Scholar
  37. Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 (2003a).MathSciNetzbMATHCrossRefGoogle Scholar
  38. Liao, S.J.: Beyond Perturbation-Introduction to the Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton (2003b).CrossRefGoogle Scholar
  39. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  40. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Tran. 48, 2529–2539 (2005).zbMATHCrossRefGoogle Scholar
  41. Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud. Appl. Math. 117, 2529–2539 (2006).CrossRefGoogle Scholar
  42. Liao, S.J.: Notes on the homotopy analysis method-Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat. 14, 983–997 (2009a).zbMATHCrossRefGoogle Scholar
  43. Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus. 61A, 550–564 (2009b).Google Scholar
  44. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 15, 2003–2016 (2010).zbMATHCrossRefGoogle Scholar
  45. Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411–425 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  46. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–355 (2007).MathSciNetCrossRefGoogle Scholar
  47. Liu, Y.P.: Study on Analytic and Approximate Solution of Differential Equations by Symbolic Computation. PhD Dissertation, East China Normal University (2008).Google Scholar
  48. Liu, Y.P., Li, Z.B.: The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation. Chaos Soliton. Fract. 39, 1–8 (2009).CrossRefGoogle Scholar
  49. Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963).CrossRefGoogle Scholar
  50. Mahapatra, T. R., Nandy, S.K., Gupta, A.S.: Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface. Applied Mathematics and Computation. 215, 1696–1710 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  51. Marinca, V., Herisanu, N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass. 35, 710–715 (2008).CrossRefGoogle Scholar
  52. Marinca, V., Herisanu, N.: An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plat. Appl. Math. Lett. 22, 245–251 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  53. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall/CRC Press, Boca Raton (2003).zbMATHGoogle Scholar
  54. Molabahrami, A., Khani, F.: The homotopy analysis method to solve the Burgers-Huxley equation. Nonlin. Anal. B. 10, 589–600 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  55. Motsa, S.S., Sibanda, P., Shateyi, S.: A new spectral homotopy analysis method for solving a nonlinear second order BVP. Commun. Nonlinear Sci. Numer. Simulat. 15, 2293–2302 (2010a).MathSciNetzbMATHCrossRefGoogle Scholar
  56. Motsa, S.S., Sibanda, P., Auad, F.G., Shateyi, S.: A new spectral homotopy analysis method for the MHD Jeffery-Hamel problem. Computer & Fluids. 39, 1219–1225 (2010b).CrossRefGoogle Scholar
  57. Niu, Z., Wang, C.: A one-step optimal homotopy analysis method for nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 15, 2026–2036 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  58. Pandey, R.K., Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space — time fractional advection-dispersion equation. Computer Physics Communications. 182, 1134–1144 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  59. Pirbodaghi, T., Ahmadian, M.T., Fesanghary, M.: On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications. 36, 143–148 (2009).CrossRefGoogle Scholar
  60. Sajid, M.: Similar and Non-similar Analytic Solutions for Steady Flows of Differential Type Fluids. PhD dissertation, Quaid-I-Azam University (2006).Google Scholar
  61. Sajid, M., Hayat, T.: Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. Nonlinear Anal. B. 9, 2296–2301 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  62. Shampine, L.F., Gladwell, I., Thompson, S.: Solving ODEs with MATLAB. Cambridge University Press. Cambridge (2003).Google Scholar
  63. Shampine, L.F., Reichelt, M.W., Kierzenka, J.: Solving boundary value problems for ordinary differential equations in MATLAB with BVP4c. Available at tutorial. Accessed 15 April 2011.Google Scholar
  64. Shidfar, A., Babaei, A., Molabahrami, A.: Solving the inverse problem of identifying an unknown source term in a parabolic equation. Computers and Mathematics with Applications. 60, 1209–1213 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  65. Shidfar, A., Molabahrami, A.: A weighted algorithm based on the homotopy analysis method-Application to inverse heat conduction problems. Commun. Nonlinear Sci. Numer. Simulat. 15, 2908–2915 (2010).MathSciNetzbMATHCrossRefGoogle Scholar
  66. Siddheshwar, P.G.: A series solution for the Ginzburg-Landau equation with a timeperiodic coefficient. Applied Mathematics. 3, 542–554 (2010). Online available at Accessed 15 April 2011.CrossRefGoogle Scholar
  67. Singh, O.P., Pandey, R.K., Singh, V.K.: An analytic algorithm of LaneEmden type equations arising in astrophysics using modified homotopy analysis method. Computer Physics Communications. 180, 1116–1124 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  68. Song, H., Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media. J. Coastal Res. 50, 292–295 (2007).Google Scholar
  69. Tao, L., Song, H., Chakrabarti, S.: Nonlinear progressive waves in water of finite depth-An analytic approximation. Coastal Engineering. 54, 825–834 (2007).CrossRefGoogle Scholar
  70. Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. in Comp. Sci. 1, 9–19 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  71. Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer. Physics of Fluids. 21, 106104 (2009).CrossRefGoogle Scholar
  72. Turkyilmazoglu, M.: A note on the homotopy analysis method. Appl. Math. Lett. 23, 1226–1230 (2010a).MathSciNetzbMATHCrossRefGoogle Scholar
  73. Turkyilmazoglu, M.: Series solution of nonlinear two-point singularly perturbed boundary layer problems. Computers and Mathematics with Applications. 60, 2109–2114 (2010b).MathSciNetzbMATHCrossRefGoogle Scholar
  74. Turkyilmazoglu, M.: An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. ASME J. Appl. Mech. 78, 021005 (2011a).CrossRefGoogle Scholar
  75. Turkyilmazoglu, M.: Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int. J. Thermal Sciences. 50, 831–842 (2011b).CrossRefGoogle Scholar
  76. Turkyilmazoglu, M.: An analytic shooting-like approach for the solution of nonlinear boundary value problems. Math. Comp. Modelling. 53, 1748–1755 (2011c).MathSciNetzbMATHCrossRefGoogle Scholar
  77. Turkyilmazoglu, M.: Some issues on HPM and HAM methods — A convergence scheme. Math. Compu. Modelling. 53, 1929–1936 (2011d).MathSciNetzbMATHCrossRefGoogle Scholar
  78. Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A. 372, 6060–6065 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  79. Van Gorder, R.A., Vajravelu, K.: On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations-A general approach. Commun. Nonlinear Sci. Numer. Simulat. 14, 4078–4089 (2009).zbMATHCrossRefGoogle Scholar
  80. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks. Applied Mathematics and Computation. 216, 1513–1523 (2010a).MathSciNetzbMATHCrossRefGoogle Scholar
  81. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces. Commun. Nonlinear Sci. Numer. Simulat. 15, 1494–1500 (2010b).zbMATHCrossRefGoogle Scholar
  82. Van Gorder, R.A., Vajravelu, K.: Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/ absorption. Applied Mathematics and Computation. 217, 5810–5821 (2011).MathSciNetzbMATHCrossRefGoogle Scholar
  83. Wu, Y.Y.: Analytic Solutions for Nonlinear Long Wave Propagation. PhD dissertation, University of Hawaii (2009).Google Scholar
  84. Wu, Y., Cheung, K.F.: Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations. Int. J. Numer. Meth. Fl. 57, 1649–1668 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  85. Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion. 46, 1–14 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  86. Xu, H., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy-based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls. Physics of Fluids. 22, 053601 (2010). doi:10.1063/1.3392770.CrossRefGoogle Scholar
  87. Yabushita, K., Yamashita, M., Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method. J. Phys. A — Math. Theor. 40, 8403–8416 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  88. Zand, M.M., Ahmadian, M.T: Application of homotopy analysis method in studying dynamic pull-in instability of microsystems. Mechanics Research Communications. 36, 851–858 (2009).zbMATHCrossRefGoogle Scholar
  89. Zand, M.M., Ahmadian, M.T., Rashidian, B.: Semi-analytic solutions to nonlinear vibrations of microbeams under suddenly applied voltages. J. Sound and Vibration. 325, 382–396 (2009).CrossRefGoogle Scholar
  90. Zhao, J., Wong, H.Y.: A closed-form solution to American options under general diffusions (2008). Available at SSRN: Accessed 15 April 2011.Google Scholar
  91. Zhu, J.: Linear and Non-linear Dynamical Analysis of Beams and Cables and Their Combinations. PhD dissertation, Zhejiang University (2008).Google Scholar
  92. Zhu, S.P.: A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield. ANZIAM J. 47, 477–494 (2006a).MathSciNetzbMATHCrossRefGoogle Scholar
  93. Zhu, S.P.: An exact and explicit solution for the valuation of American put options. Quant. Financ. 6, 229–242 (2006b).zbMATHCrossRefGoogle Scholar
  94. Zou, L.: A Study of Some NonlinearWater Wave Problems Using Homotopy Analysis Method. PhD dissertation, Dalian University of Technology (2008).Google Scholar
  95. Zou, L., Zong, Z., Wang, Z., He, L.: Solving the discrete KdV equation with homotopy analysis method. Phys. Lett. A. 370, 287–294 (2007).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Shijun Liao
    • 1
  1. 1.Shanghai Jiao Tong UniversityShanghaiChina

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