Abstract
In this chapter we shall introduce the basic principles of the homotopy analysis method for nonlinear differential equations. In Section 2.1, we shall review the topological principle of homotopy and then discuss how this relates to the homotopy analysis method for constructing solutions to nonlinear differential equations. As with any perturbation method, we shall need a way to compute higher order corrections, and this is outlined in Section 2.2, where we discuss the higher order deformation equations. From here, we are then able to outline the general method of constructing series solutions to nonlinear differential equations and relevant initial or boundary value problems in Section 2.3. Often times when dealing with nonlinear differential equations, the question of whether solutions exist and are unique is of importance. Accordingly, in Section 2.4, we discuss the existence and uniqueness of solutions obtained via the homotopy analysis method. Throughout mathematics, there are often multiple ways to solve a given problem. In obtaining perturbation solutions to differential equations, there are multiple iterative routines one may employ. In Section 2.5, we compare the method of homotopy analysis to some other perturbation schemes, and highlight one primary benefit of the method — namely, that the homotopy analysis method solution does not require small model parameters. From here, we will be in a position to introduce more advanced methods which permit the control of convergence of solutions in Chapter 3 and advanced treatments for more complicated systems of equations in Chapter 4.
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References
S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall\CRC Press, Boca Raton, 2003.
S. J. Liao, On the proposed homotopy analysis techniques for nonlinear problems and its application, Ph.D. dissertation, Shanghai Jiao Tong University, 1992.
S. J. Liao, An explicit, totally analytic approximation of Blasius’ viscous flow problems, Int. J. Non-Linear Mech., 34 (1999) 759.
S. J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147 (2004) 499.
S. J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Studies in Applied Mathematics, 119 (2007) 297.
S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 983.
R.A. Van Gorder and K. Vajravelu, 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, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 4078.
S. Li and S.J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem, Applied Mathematics and Computation, 169 (2005) 854.
H. Xu and S.J. Liao, Dual solutions of boundary layer flow over an upstream moving plate, Communications in Nonlinear Science and Numerical Simulation, 13 (2008) 350.
S.J. Liao, Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method, Nonlinear Analysis: Real World Applications, 10 (2009) 2455.
S.J. Liao, Series solutions of unsteady boundary-layer flows over a stretching flat plate, Studies Applied Mathematics, 117 (2006) 239.
S.J. Liao, Finding multiple solutions of nonlinear problems by means of the homotopy analysis method, Journal of Hydrodynamics, Ser. B, 18 (2006) 54.
S. Abbasbandy and A. Shirzadi, Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems, Numerical Algorithms, 54 (2010) 521.
S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation, 169 (2005) 1186.
M.S.H. Chowdhury, I. Hashim and O. Abdulaziz, Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 371.
G. Domairry and N. Nadim, Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation, International Communications in Heat and Mass Transfer, 35 (2008) 93.
S. Liang and D. J. Jeffrey, Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 4057.
F. T. Akyildiz, D. A. Siginer, K. Vajravelu and R. A. Van Gorder, Analytical and numerical results for the Swift-Hohenberg equation, Applied Mathematics and Computation, 216 (2010) 221.
J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977) 319.
G.T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., 60 (1988) 2641.
W. Zimmerman, Propagating fronts near a Lifschitz point, Phys. Rev. Lett., 66 (1991) 1546.
G. Caginalp and P.C. Fife, Higher order phase field models and detailed anisotropy, Phys. Rev. B, 34 (1986) 4940.
R.A. Gardner and C.K.R.T. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. J., 38 (1989) 1197.
J. Chaparova, L.A. Peletier and S. Tersian, Existence and nonexistence of nontrivial solutions of semilinear sixth order ordinary differential equations, Applied Mathematics Letters, 17 (2004) 1207.
J. Chaparova, L.A. Peletier and S. Tersian, Existence and nonexistence of nontrivial solutions of semilinear fourth-and sixth-order ordinary differential equations, Adv. Diff. Eqns., 8 (2003) 1237.
S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics of the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 4 (2005) 1.
C.I. Christov and J. Pontes, Numerical scheme for Swift-Hohenberg equation with strict implementation of Lyapunov functional, Mathematical and Computer Modelling, 35 (2002) 87.
M.O. Caceres, About the stochastic multiplicative Swift-Hohenberg equation: The eigenvalue problem, Chaos, Solitons & Fractals, 6 (1995) 43.
L.A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 194 (2004) 95.
E. Atkinson, An Introduction to Numerical Analysis, John Wiley & Sons, Inc, 1989.
E.B. Saff and R.S. Varga (Eds.), Padé and Rational Approximation (Tampa, 1976), Acad. Press, 1977.
R. A. Van Gorder and K. Vajravelu, Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation / absorption, Applied Mathematics and Computation, 217 (2011) 5810.
H. A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Computational Materials Science, 38 (2007) 741.
A. R. A. Khaled and K. Vafai, The role of porous media in modeling flow and heat transfer in biological tissues, Int. J. Heat Mass Transfer, 46 (2003) 4989.
R. Nazar, M. Amin, D. Philip and I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Int. J. Non-Linear Mech., 39 (2004) 1227.
L. J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys., 21 (1970) 645.
U. Ascher, R. Mattheij and R. Russell, Numerical solution of boundary value problems or ordinary differential equations, SIAM Classics in Applied Mathematics, 1995.
U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1998.
R. A. Van Gorder and K. Vajravelu, Hydromagnetic stagnation point flow of a second grade fluid over a stretching sheet, Mechanics Research Communications, 37 (2010) 113.
H. Schlichting, Boundary Layer Theory, McGraw-Hill, 1960.
K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytech J., 326 (1911) 321.
L. Howarth, On the solution of the laminar boundary layer equations, Proc. Roy. Soc. London A, 164 (1938) 547.
K. Gersten, H.D. Papenfuss and J.F. Gross, Influence of the Prandtl number on secondorder heat transfer due to surface curvature at a three dimensional stagnation point, Int. J. Heat Mass Transfer, 21 (1978) 275.
S.J. Liao, A uniformly valid analytic solution of two-dimensional viscous flow over a semiinfinite flat plate, J. Fluid Mech., 385 (1999) 101.
P.D. Weidman and V. Putkaradze, Axisymmetric stagnation flow obliquely impinging on a circular cylinder, Eur. J. Mech. B/Fluids, 22 (2003) 123.
J.T. Stuart, The viscous flow near a stagnation point when the external flow has uniform vorticity, J. Aerospace Sci., 26 (1959) 124.
K.J. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn., 46 (1979) 310.
N. Takemitsu and Y. Matunobu, Unsteady stagnation-point flow impinging obliquely on an oscillating flat plate, J. Phys. Soc. Jpn., 47 (1979) 1347.
J.M. Dorrepaal, An exact solution of the Navier-Stokes equation which describes nonorthogonal stagnation-point flow in two dimensions, J. Fluid Mech., 163 (1986) 141.
J.M. Dorrepaal, Is two-dimensional oblique stagnation-point flow unique, Canad. Appl. Math. Quart., 8 (2000) 61.
F. Labropulu, J.M. Dorrepaal and O.P. Chandna, Oblique flow impinging on a wall with suction or blowing, Acta Mech., 115 (1996) 15.
B.S. Tilley and P.D. Weidman, Oblique two-fluid stagnation-point flow, Eur. J. Mech. B/Fluids, 17 (1998) 205.
J.N. Kapur and R.C. Srivastava, Similar solutions of the boundary layer equations for power law fluids, Z. Angew. Math. Phys., 14 (1963) 383.
M.K. Maiti, Axially-symmetric stagnation point flow of power law fluids, Z. Angew. Math. Phys., 16 (1965) 594.
S.R. Koneru and R. Manohar, Stagnation point flows of non-newtonian power law fluids, Z. Angew. Math. Phys., 19 (1968) 84.
Y.G. Sapunkov, Similarity solutions of boundary layer of non-Newtonian fluids in magnetohydrodynamics, Mech. Zidkosti I Gaza, 6 (1967) 77 (in Russian).
D.S. Djukic, Hiemenz magnetic flow of power-law fluids, Trans. ASME J. Appl. Mech., 41 (1974) 822.
T.R. Mahapatra and A.S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, Heat and Mass Transfer, 38 (2002) 517.
R. Nazar, N. Amin, D. Filip and I. Pop, Stagnation point flow of a micropolar fluid towards a stretching sheet, Int. J. Non-Linear Mech., 39 (2004) 1227.
Y.Y. Lok, N. Amin and I. Pop, Comments on: Steady two-dimensional oblique stagnationpoint flow towards a stretching surface, Fluid Dynamics Research, 39 (2007) 505.
M. Reza and A.S. Gupta, Steady two-dimensional oblique stagnation-point flow towards a stretching surface, Fluid Dyn. Res., 37 (2005) 334.
T.R. Mahapatra, S. Dholey and A.S. Gupta, Heat Transfer in oblique stagnation-point flow of an incompressible viscous fluid towards a stretching surface, Heat and Mass Transfer, 43 (2007) 767.
J. Paullet and P. Weidman, Analysis of stagnation point flow toward a stretching sheet, Int. J. Non-Linear Mech., 42 (2007) 1084.
Y.Y. Lok, N. Amin and I. Pop, Non-orthogonal stagnation point flow towards a stretching sheet, Int. J. Non-Linear Mech., 41 (2006) 622.
Y.Y. Lok, N. Amin and I. Pop, Mixed convection flow near a non-orthogonal stagnation point towards a stretching vertical plate, Int. J. Heat Mass Transfer, 50 (2007) 4855.
Y.Y. Lok, N. Amin, D. Campean and I. Pop, Steady mixed connection flow of a micropolar fluid near the stagnation point of a vertical surface, Int. J. Numer. Method. Heat Fluid Flow, 15 (2005) 654.
Y.Y. Lok, N. Amin and I. Pop, Mixed convection near a non-orthogonal stagnation point flow on a vertical plate with uniform surface heat flux, Acta Mech., 186 (2006) 99.
N. Ramachandran, T.S. Chen and B.F. Armaly, Mixed convection in stagnation flows adjacent to vertical surfaces, ASME J. Heat Transfer, 110 (1988) 373.
A. Ishak, K. Jafar, R. Nazar and I. Pop, MHD stagnation point flow towards a stretching sheet, Physica A, 388 (2009) 3377.
T.R. Mahapatra and A.S. Gupta, Magnetohydrodynamic stagnation-point flow towards a stretching sheet, Acta Mech., 152 (2001) 191.
T.R. Mahapatra, S.K. Nandy and A.S. Gupta, Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface, Int. J. Non-Linear Mech., 44 (2009) 124.
Z. Ziabakhsh, G. Domairry and H.R. Ghazizadeh, Analytical solution of the stagnationpoint flow in a porous medium by using the homotopy analysis method, Journal of the Taiwan Institute of Chemical Engineers, 40 (2009) 91.
Q. Wu, S. Weinbaum and Y. Andreopoulos, Stagnation-point flow in a porous medium, Chem. Eng. Sci., 60 (2005) 123.
S.A. Kechil and I. Hashim, Approximate analytical solution for MHD stagnation-point flow in porous media, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 1346.
R.S. Rivlin and J.L. Ericksen, Stress deformation relations for isotropic materials, J. Rat. Mech. Anal., 4 (1955) 323.
J.E. Dunn and K.R. Rajagopal, Fluids of differential type: Critical review and thermodynamic analysis, Int. J. Engng. Sci., 33 (1995) 689.
K.R. Rajagopal, A.S. Gupta and T.A. Na, A note on the Faulkner-Skan flows of a non-Newtonian fluid, Int. J. Non-Linear Mech., 18 (1983) 313.
K. Vajravelu and D. Rollins, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl., 158 (1991) 241.
M.S. Sarma and B.N. Rao, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl., 222 (1998) 268.
W.C. Troy, E.A. Overman, G.B. Ermentrout and J.P. Keener, Uniqueness of flow of a second order fluid past a stretching sheet, Quart. Appl. Math., 44 (1987) 753.
W.D. Chang, The nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math., 47 (1989) 365.
P.S. Lawrence and B.N. Rao, Reinvestigation of the nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math., 51 (1993) 401.
W.D. Chang, N.D. Kazarinoff and C. Lu, A new family of explicit solutions for the similarity equations modelling flow of a non-Newtonian fluid over a stretching sheet, Arch. Rat. Mech. Anal., 113 (1991) 191.
K. Vajravelu and T. Roper, Flow and heat transfer in a second grade fluid over a stretching sheet, Int. J. Non-Linear Mech., 34 (1999) 1031.
K. Vajravelu and D. Rollins, Hydromagnetic flow of a second grade fluid over a stretching sheet, Applied Mathematics and Computation, 148 (2004) 783.
A.D. Barinberg, A.B. Kapusta and B.V. Chekin, Magnitnaya Gidrodinamika (English translation), 11 (1975) 111.
T. Hayat, Z. Abbas and I. Pop, Mixed convection in the stagnation point flow adjacent to a vertical surface in a viscoelastic fluid, Int. J. Heat Mass Transfer, 51 (2008) 3200.
M. S. Abel and N. Mahesha, Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Applied Mathematical Modelling, 32 (2008) 1965.
M. Ayub, H. Zaman, M. Sajid and T. Hayat, Analytical solution of stagnation-point flow of a viscoelastic fluid towards a stretching surface, Communications in Nonlinear Science and Numerical Simulation, 13 (2008) 1822.
T. Hayat, Z. Abbas and M. Sajid, MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface, Chaos, Solitons & Fractals, 39 (2009) 840.
C.-H. Chen, Magneto-hydrodynamic mixed convection of a power-law fluid past a stretching surface in the presence of thermal radiation and internal heat generation/absorption, Int. J. Non-Linear Mech., 44 (2009) 596.
J.C. Misra and G.C. Shit, Biomagnetic viscoelastic fluid flow over a stretching sheet, Applied Mathematics and Computation, 210 (2009) 350.
K.V. Prasad, D. Pal, V. Umesh and N.S. P. Rao, The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 331.
A. Chakrabarti and A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Quart. Appl. Math., 37 (1979) 73.
K. Vajravelu and D. Rollins, Heat transfer in an electrically conducting fluid over a stretching sheet, Int. J. Non-Linear Mech., 27 (1992) 265.
H.I. Andersson, An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow, Acta Mech., 113 (1995) 241.
I. Pop and T.Y. Na, A note on MHD flow over a stretching permeable surface, Mech. Res. Commun., 25 (1998) 263.
M.S. Abel and M.M. Nandeppanavar, Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 2120.
H.S. Takhar, A.J. Chamkha and G. Nath, Unsteady three-dimensional MHD boundarylayer flow due to the impulsive motion of a stretching surface, Acta Mech., 146 (2001) 59.
P.D. McCormak and L.J. Crane, Physical Fluid Dynamics, Academic Press, New York, 1973.
P.S. Gupta and A.S. Gupta, Heat and mass transfer on a stretching sheet with suction and blowing, J. Chem. Eng., 55 (1977) 744.
P. Carragher and L.J. Crane, Heat transfer on a continuous stretching sheet, J. Appl. Math. Mech. (ZAMM), 62 (1982) 564.
B.K. Dutta, Heat transfer from a stretching sheet with uniform suction and blowing, Acta Mech., 78 (1989) 255.
K. Vajravelu and A. Hadjicolaou, Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream, Int. J. Eng. Sci., 35 (1997) 1237.
E. Magyari and B. Keller, Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface, J. Phys. D: Appl. Phys., 32 (1999) 577.
E. Magyari and B. Keller, Exact solutions for self-similar boundary-layer flows induced by permeable stretching surfaces, Eur. J. Mech. B/Fluids, 19 (2000) 109.
S.J. Liao and I. Pop, Explicit analytic solution for similarity boundary layer equations, Int. J. Heat Mass Transfer, 47 (2004) 75.
K.R. Rajagopal, T.Y. Na and A.S. Gupta, Flow of a visco-elastic fluid over a stretching sheet, Rheol. Acta, 23 (1984) 213.
K. Vajravelu and D. Rollins, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl., 158 (1991) 241.
J.B. McLeod and K.R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Ration. Mech. Anal., 98 (1987) 385.
W.C. Troy, E.A. Overman II, G.B. Ermen-Trout and J.P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math., 44 (1987) 753.
K. Vajravelu and J.R. Cannon, Fluid flow over a nonlinear stretching sheet, Applied Mathematics and Computation, 181 (2006) 609.
F. T. Akyildiz, D. A. Siginer, K. Vajravelu, J. R. Cannon and R. A. Van Gorder, Similarity solutions of the boundary layer equation for a nonlinearly stretching sheet, Mathematical Methods in the Applied Sciences, 33 (2010) 601.
R. A. Van Gorder and K. Vajravelu, A general class of coupled nonlinear differential equations arising in self-similar solutions of convective heat transfer problems, Applied Mathematics and Computation, 217 (2010) 460.
H. Xu, S.J. Liao and G.X. Wu, A family of new solutions on the wall jet, Eur. J. Mech. B/Fluid, 27 (2008) 322.
S.J. Liao, A new branch of boundary layer flows over a permeable stretching plate, Int. J. Non-linear Mech., 42 (2007) 819.
S.J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat Mass Transfer, 48 (2005) 2529.
S. Abbasbandy, E. Magyari and E. Shivanian, The homotopy analysis method for multiple solutions of nonlinear boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 3530.
R.A. Van Gorder and K. Vajravelu, Multiple solutions for hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet, Quart. Appl. Math., 69 (2011) 405.
R. A. Van Gorder, High-order nonlinear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet, Applied Mathematics and Computation, 216 (2010) 2177.
T.C. Chiam, Stagnation-point flow towards a stretching plate, J. Phys. Soc. Jpn., 63 (1994) 2443.
T.R. Mahapatra and A.S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, Heat and Mass Transfer, 38 (2002) 517.
M. Reza and A.S. Gupta, Steady two-dimensional oblique stagnation-point flow towards a stretching surface, Fluid Dyn. Res., 27 (2005) 334.
J.T. Stuart, The viscous flow near a stagnation point when the external flow has uniform vorticity, J. Aerosp. Sci., 26 (1959) 124.
K.J. Tamada, Two-dimensional stagnation point flow impinging obliquely on a plane wall, J. Phys. Soc. Jpn., 46 (1979) 310.
J.M. Dorrepaal, An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions, J. Fluid Mech., 163 (1986) 141.
F. Labropulu, J.M. Dorrepaal and O.P. Chandna, Oblique flow impinging on a wall with suction or blowing, Acta Mech., 115 (1996) 15.
M. Amaouche and D. Boukari, Influence of thermal convection on nonorthogonal stagnation point flow, Int. J. Thermal Sci., 42 (2003) 303.
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Vajravelu, K., van Gorder, R.A. (2012). Principles of Homotopy Analysis. In: Nonlinear Flow Phenomena and Homotopy Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32102-3_2
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