本研究應用同倫分析方法於非線性熱傳遞問題，有別於其它近似解析法，同倫分析法是一種有效且易於使用的計算方法，它提供了一個簡單的作法可調節和控制近似解的收斂區域，在完全不需要疊代的情況下，便能得到無窮冪級數解。
本文的研究包含了三個主題，首先是考量具有熱擴散與質量擴散因素的邊界層自然對流，除了無因次速度場、溫度場和濃度場的圖解說明外，普朗特數、舒密特數和浮力參數對流場的影響也有詳細的討論。其次是對四種不同幾何外型的縱向散熱片(矩形、三角形、凸拋物線形、凹拋物線形)進行熱傳分析，假設熱傳導係數與熱對流係數皆為溫度的函數，與四階Runge-Kutta作比較後證實近似解有極高的精確性，同時也探討散熱片效率和最佳化尺寸。第三個研究主題是在受制溫度週期性變化條件下，分析可變熱傳導矩形散熱片之暫態熱傳，研究各物理參數對於散熱片表面無因次溫度分佈情形的影響。

In this research, the homotopy analysis method is applied to investigate the non-linear heat transfer problems. Unlike all other analytic methods, this approximate analytical method is a powerful and easy-to-use tool for non-linear problems and it provides us with a simple way to adjust and control the convergence region of solution series. Without the need of iteration, the obtained solution is in the form of an infinite power series.
This study contains three themes. The first topic is about the natural convection boundary layer flow with effects of thermal and mass diffusion. The non-dimensional velocity, temperature and concentration fields are well illustrated. And the impact of the Prandtl number, Schmidt number and the buoyancy parameter on the flow are widely discussed in detail. Next, heat transfer problem with four different shape of longitudinal fins (rectangular, triangular and parabolic profiles) is analyzed. Both the thermal conductivity and heat transfer coefficient are assumed to be functions of temperature. The obtained solution has high accuracy when comparing it with other-generated by the 4th-order Runge-Kutta method. The fin efficiency and the optimum fin length are also investigated in detail. The third research topic is about the transient heat transfer processes under periodical temperature variation condition occurring in a convective rectangular fin with variable thermal conductivity. The effects of the physical applicable parameters on the non-dimensional temperature distribution along the fin surface are widely discussed.

[1] Cole J.D., Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, Massachusetts, 1968.
[2] Nayfeh A.H., Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1975.
[3] Nayfeh A.H., Problem in Perturbation, John Wiley & Sons, New York, 1985.
[4] Lyapunov A.M., General problem on stability of motion, Taylor & Francis, London, 1992.
[5] Karmishin A.V., Zhukov A.T., Kolosov V.G., Methods of dynamics calculation and testing for thin-walled structures, Mashinostroyenie, Moscow, 1990.
[6] Adomian G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston and London, 1994.
[7] Adomian G., "Solution of physical problems by decomposition", Computers & Mathematics with Applications, 27, 145-154, 1994.
[8] Adomian G., "Explicit solutions of nonlinear partial differential equations", Applied Mathematics and Computation, 88, 117-126, 1997.
[9] Adomian G., "Solutions of nonlinear PDE", Applied Mathematics Letters, 11, 121-123, 1998.
[10] Adomian G., "Solution of the Thomas-Fermi equation", Applied Mathematics Letters, 11, 131-133, 1998.
[11] Adomian G., Rach R., "Modified decomposition solution of linear and nonlinear boundary-value-problems", Nonlinear Analysis-Theory Methods & Applications, 23, 615-619, 1994.
[12] Liao S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, in, Shanghai Jiao Tong University, 1992.
[13] Liao S.J., Beyond Perturbation: Introduction to homotopy analysis method, Chapman & Hall, Boca Raton, 2003.
[14] Kojima M., Nishino H., Arima N., "A PL homotopy for finding all the roots of a polynomial", Mathematical Programming, 16, 37-62, 1979.
[15] Li T.Y., Sauer T., "Homotopy method for generalized eigenvalue problems Ax= ΛBx", Linear Algebra and its Applications, 91, 65-74, 1987.
[16] Liao S.J., "An approximate solution technique not depending on small parameters: A special example", International Journal of Non-Linear Mechanics, 30, 371-380, 1995.
[17] Liao S.J., "A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics", International Journal of Non-Linear Mechanics, 32, 815-822, 1997.
[18] Liao S.J., "A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate", Journal of Fluid Mechanics, 385, 101-128, 1999.
[19] Liao S.J., Campo A., "Analytic solutions of the temperature distribution in Blasius viscous flow problems", Journal of Fluid Mechanics, 453, 411-425, 2002.
[20] Howarth L., "On the Solution of the Laminar Boundary Layer Equations", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 164, 547-579, 1938.
[21] Yao B., "Series solution of the temperature distribution in the Falkner–Skan wedge flow by the homotopy analysis method", European Journal of Mechanics - B/Fluids, 28, 689-693, 2009.
[22] White F.M., Viscous Fluid Flow, McGraw-Hill, New York, 1991.
[23] Ifidon E.O., "An Application of Homotopy Analysis to the Viscous Flow Past a Circular Cylinder", Journal of Applied Mathematics, 2009, 1-17, 2009.
[24] Liao S.J., "An analytic approximation of the drag coefficient for the viscous flow past a sphere", International Journal of Non-Linear Mechanics, 37, 1-18, 2002.
[25] Zhang T.T., Jia L., Wang Z.C., Li X., "The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels", Physics Letters A, 372, 3223-3227, 2008.
[26] Siddiqui A.M., Ahmed M., Islam S., Ghori Q.K., "Homotopy analysis of Couette and Poiseuille flows for fourth grade fluids", Acta Mechanica, 180, 117-132, 2005.
[27] Xu H., "An explicit analytic solution for free convection about a vertical flat plate embedded in a porous medium by means of homotopy analysis method", Applied Mathematics and Computation, 158, 433-443, 2004.
[28] Ziabakhsh Z., Domairry G., "Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method", Communications in Nonlinear Science and Numerical Simulation, 14, 1868-1880, 2009.
[29] Motsa S.S., Shateyi S., Makukula Z., "Homotopy analysis of free convection boundary layer flow with heat and mass transfer", Chemical Engineering Communications, 198, 783-795, 2011.
[30] Abbasbandy S., "The application of homotopy analysis method to nonlinear equations arising in heat transfer", Physics Letters A, 360, 109-113, 2006.
[31] Abbasbandy S., "Homotopy analysis method for heat radiation equations", International Communications in Heat and Mass Transfer, 34, 380-387, 2007.
[32] Domairry G., Nadim N., "Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation", International Communications in Heat and Mass Transfer, 35, 93-102, 2008.
[33] Domairry G., Fazeli M., "Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity", Communications in Nonlinear Science and Numerical Simulation, 14, 489-499, 2009.
[34] 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.
[35] Kargarnovin M.H., Jafari-Talookolaei R.A., "Application of the homotopy method for the analytic approach of the nonlinear free vibration analysis of the simple end beams using four engineering theories", Acta Mechanica, 212, 199-213, 2009.
[36] Chen Y.M., Liu J.K., "A study of homotopy analysis method for limit cycle of van der Pol equation", Communications in Nonlinear Science and Numerical Simulation, 14, 1816-1821, 2009.
[37] Sun Q., "Solving the Klein–Gordon equation by means of the homotopy analysis method", Applied Mathematics and Computation, 169, 355-365, 2005.
[38] Abbasbandy S., Zakaria F.S., "Soliton solutions for the fifth-order KdV equation with the homotopy analysis method", Nonlinear Dynamics, 51, 83-87, 2007.
[39] Song L., Zhang H., "Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation", Physics Letters A, 367, 88-94, 2007.
[40] Zou L., Zong Z., Wang Z., He L., "Solving the discrete KdV equation with homotopy analysis method", Physics Letters A, 370, 287-294, 2007.
[41] Alomari A.K., Noorani M.S.M., Nazar R., "Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method", Communications in Nonlinear Science and Numerical Simulation, 14, 1196-1207, 2009.
[42] Liao S.J., Cheung K.F., "Homotopy analysis of nonlinear progressive waves in deep water", Journal of Engineering Mathematics, 45, 105-116, 2003.
[43] Tan Y., Abbasbandy S., "Homotopy analysis method for quadratic Riccati differential equation", Communications in Nonlinear Science and Numerical Simulation, 13, 539-546, 2008.
[44] Inc M., "On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method", Physics Letters A, 365, 412-415, 2007.
[45] Liao S.J., "An explicit analytic solution to the Thomas–Fermi equation", Applied Mathematics and Computation, 144, 495-506, 2003.
[46] Alomari A.K., Noorani M.S.M., Nazar R., "On the homotopy analysis method for the exact solutions of Helmholtz equation", Chaos, Solitons & Fractals, 41, 1873-1879, 2009.
[47] Inc M., "On numerical solution of Burgers' equation by homotopy analysis method", Physics Letters A, 372, 356-360, 2008.
[48] He J.H., "Homotopy perturbation technique", Computer Methods in Applied Mechanics and Engineering, 178, 257-262, 1999.
[49] He J.H., "Comparison of homotopy perturbation method and homotopy analysis method", Applied Mathematics and Computation, 156, 527-539, 2004.
[50] Liao S.J., "Comparison between the homotopy analysis method and homotopy perturbation method", Applied Mathematics and Computation, 169, 1186-1194, 2005.
[51] Hassan H.N., El-Tawil M.A., "A new technique of using homotopy analysis method for solving high-order nonlinear differential equations", Mathematical Methods in the Applied Sciences, 34, 728-742, 2011.
[52] Sami Bataineh A., Noorani M.S.M., Hashim I., "On a new reliable modification of homotopy analysis method", Communications in Nonlinear Science and Numerical Simulation, 14, 409-423, 2009.
[53] Bataineh A.S., Noorani M.S.M., Hashim I., "Modified homotopy analysis method for solving systems of second-order BVPs", Communications in Nonlinear Science and Numerical Simulation, 14, 430-442, 2009.
[54] Bataineh A.S., Noorani M.S.M., Hashim I., "Approximate solutions of singular two-point BVPs by modified homotopy analysis method", Physics Letters A, 372, 4062-4066, 2008.
[55] Yabushita K., Yamashita M., Tsuboi K., "An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method", Journal of Physics A: Mathematical and Theoretical, 40, 8403-8416, 2007.
[56] Wu Y., Cheung K.F., "Homotopy solution for nonlinear differential equations in wave propagation problems", Wave Motion, 46, 1-14, 2009.
[57] Wu Y., Cheung K.F., "Two-parameter homotopy method for nonlinear equations", Numerical Algorithms, 53, 555-572, 2010.
[58] Marinca V., Herişanu N., Nemeş I., "Optimal homotopy asymptotic method with application to thin film flow", Central European Journal of Physics, 6, 648-653, 2008.
[59] Marinca V., Herişanu N., "Application of Optimal Homotopy Asymptotic Method for solving nonlinear equations arising in heat transfer", International Communications in Heat and Mass Transfer, 35, 710-715, 2008.
[60] Marinca V., Herişanu N., Bota C., Marinca B., "An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate", Applied Mathematics Letters, 22, 245-251, 2009.
[61] Niu Z., Wang C., "A one-step optimal homotopy analysis method for nonlinear differential equations", Communications in Nonlinear Science and Numerical Simulation, 15, 2026-2036, 2010.
[62] Liao S.J., "An optimal homotopy-analysis approach for strongly nonlinear differential equations", Communications in Nonlinear Science and Numerical Simulation, 15, 2003-2016, 2010.
[63] Liao S.J., "Notes on the homotopy analysis method: Some definitions and theorems", Communications in Nonlinear Science and Numerical Simulation, 14, 983-997, 2009.
[64] Liao S.J., Tan Y., "A general approach to obtain series solutions of nonlinear differential equations", Studies in Applied Mathematics, 119, 297-354, 2007.
[65] Lau W., Tan C.W., "Errors in one-dimensional heat transfer analysis in straight and annular fins", ASME Journal of Heat Transfer, 549-551, 1973.
[66] Irey R.K., "Errors in the one-dimensional fin solution", ASME Journal of Heat Transfer, 175-176, 1968.
[67] Aziz A., Hug S.M.E., "Perturbation solution for convecting fin with variable thermal conductivity", ASME Journal of Heat Transfer, 300-301, 1975.
[68] Krane R.J., "Discussion on a previously published paper by A. Aziz and S.M.E. Hug", ASME Journal of Heat Transfer, 98, 685-686, 1976.
[69] Chiu C.H., Chen C.K., "A decomposition method for solving the convective longitudinal fins with variable thermal conductivity", International Journal of Heat and Mass Transfer, 45, 2067-2075, 2002.
[70] Mueller Jr D.W., Abu-Mulaweh H.I., "Prediction of the temperature in a fin cooled by natural convection and radiation", Applied Thermal Engineering, 26, 1662-1668, 2006.
[71] Coşkun S.B., Atay M.T., "Fin efficiency analysis of convective straight fins with temperature dependent thermal conductivity using variational iteration method", Applied Thermal Engineering, 28, 2345-2352, 2008.
[72] Aziz A., Na T.Y., "Transient response of fins by coordinate perturbation expansion", International Journal of Heat and Mass Transfer, 23, 1695-1698, 1980.
[73] Ünal H.C., "The effect of the boundary condition at a fin tip on the performance of the fin with and without internal heat generation", International Journal of Heat and Mass Transfer, 31, 1483-1496, 1988.
[74] Kim S., Huang C.H., "A series solution of the fin problem with a temperature-dependent thermal conductivity", Journal of Physics D: Applied Physics., 39, 4894-4901, 2006.
[75] Kim S., Huang C.H., "A series solution of the non-linear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient", Journal of Physics D: Applied Physics, 40, 2979, 2007.
[76] Kim S., Moon J.H., Huang C.H., "An approximate solution of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient", Journal of Physics D: Applied Physics, 40, 4382, 2007.
[77] Joneidi A.A., Ganji D.D., Babaelahi M., "Differential Transformation Method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity", International Communications in Heat and Mass Transfer, 36, 757-762, 2009.
[78] Moitsheki R.J., Harley C., "Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient", Pramana-Journal of physics, 77, 519-532, 2011.
[79] Moitsheki R.J., "Steady one-dimensional heat flow in a longitudinal triangular and parabolic fin", Communications in Nonlinear Science and Numerical Simulation, 16, 3971-3980, 2011.
[80] Harley C., Moitsheki R.J., "Numerical investigation of the temperature profile in a rectangular longitudinal fin", Nonlinear Analysis: Real World Applications, 13, 2343-2351, 2012.
[81] Rajabi A., "Homotopy perturbation method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity", Physics Letters A, 364, 33-37, 2007.
[82] Inc M., "Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity", Mathematics and Computers in Simulation, 79, 189-200, 2008.
[83] Khani F., Ahmadzadeh Raji M., Hamedi-Nezhad S., "A series solution of the fin problem with a temperature-dependent thermal conductivity", Communications in Nonlinear Science and Numerical Simulation, 14, 3007-3017, 2009.
[84] Khani F., Aziz A., "Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient", Communications in Nonlinear Science and Numerical Simulation, 15, 590-601, 2010.
[85] Odibat Z.M., "A study on the convergence of homotopy analysis method", Applied Mathematics and Computation, 217, 782-789, 2010.
[86] Liao S.J., "On the homotopy analysis method for nonlinear problems", Applied Mathematics and Computation, 147, 499-513, 2004.
[87] Padé H., Cabannes H. Pad´e approximants method and its applications to mechanics, Springer-Verlag, New York, 1976.
[88] Wazwaz A.M., "Analytical approximations and Padé approximants for Volterra's population model", Applied Mathematics and Computation, 100, 13-25, 1999.
[89] Wazwaz A.M., "The modified decomposition method and Padé approximants for solving the Thomas–Fermi equation", Applied Mathematics and Computation, 105, 11-19, 1999.
[90] Tsai P.Y., Chen C.K., "Application of the hybrid Laplace Adomian decomposition method to the nonlinear oscillatory systems", Journal of The Chinese Society of Mechanical Engineers, 30, 493-501, 2009.
[91] Tsai P.Y., Chen C.K., "Free vibration of the nonlinear pendulum using hybrid Laplace Adomian decomposition method", International Journal for Numerical Methods in Biomedical Engineering, 27, 262-272, 2011.
[92] Allan F.M., "Derivation of the Adomian decomposition method using the homotopy analysis method", Applied Mathematics and Computation, 190, 6-14, 2007.
[93] Wazwaz A.M., "A new algorithm for calculating adomian polynomials for nonlinear operators", Applied Mathematics and Computation, 111, 33-51, 2000.
[94] Ostrach S., An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, in, NACA report, 1953.
[95] Gupta P.S., Gupta A.S., "Heat and mass transfer on a stretching sheet with suction or blowing", The Canadian Journal of Chemical Engineering, 55, 744-746, 1977.
[96] Merkin J.H., "A note on the similarity solutions for free convection on a vertical plate", Journal of Engineering Mathematics, 19, 189-201, 1985.
[97] Ghotbi A.R., Bararnia H., Domairry G., Barari A., "Investigation of a powerful analytical method into natural convection boundary layer flow", Communications in Nonlinear Science and Numerical Simulation, 14, 2222-2228, 2009.
[98] Hellums J.D., Churchill S.W., "Transient and steady state free and natural convection numerical solutions", AlChE Journal, 8, 690-692, 1962.
[99] Saville D.A., Churchill S.W., "Simultaneous heat and mass transfer in free convection boundary layers", AlChE Journal, 16, 268-273, 1970.
[100] Gebhart B., Pera L., "The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion", International Journal of Heat and Mass Transfer, 14, 2025-2050, 1971.
[101] Ingham D.B., "Singular and non-unique solutions of the boundary-layer equations for the flow due to free convection near a continuously moving vertical plate", Journal of Applied Mathematics and Physics, 37, 559-572, 1986.
[102] Hussain S., Hossain M.A., Wilson M., "Natural convection flow from a vertical permeable flat plate with variable surface temperature and species concentration", Engineering Computations, 17, 789-812, 2000.
[103] Blasius H., "Grenzschichten in Flüssigkeiten mit kleiner Reibung", Z. Math. Phys., 56, 1-37, 1908.
[104] Kraus A.D., Aziz A., Welty J., Extended Surface Heat Transfer, John Wiley & Sons, 2001.
[105] Yang J.W., "Periodic heat transfer in straight fins", ASME Journal of Heat Transfer, 94, 310-314, 1972.
[106] Aziz A., "Periodic heat transfer in annular fins", ASME Journal of Heat Transfer, 97, 302-303, 1975.
[107] Yang Y.T., Chien S.K., Chen C.K., "A double decomposition method for solving the periodic base temperature in convective longitudinal fins", Energy Conversion and Management, 49, 2910-2916, 2008.