Document Type : Original Article


University of Lagos, Nigeria


The developments of approximate analytical solutions to nonlinear differential equations have been achieved through the use of various approximate analytical and semi-analytical methods. These methods provide different analytical expressions which give difference values for the same input data and variables. However, under some certain conditions, the methods provide similar analytical expressions, thereby give the same values for the same input data and variables. Therefore, in this work, the conditions of similar analytical solutions by homotopy perturbation, differential transformation and Taylor series methods for linear and nonlinear differential equations are investigated. From the analysis, it is established that if some specific values or functions are assigned to the auxiliary parameters in the homotopy perturbation method, the approximate analytical solutions provided by homotopy perturbation method is entirely similar to the approximate analytical solutions given by differential transformation and Taylor series methods. Also, it is found that the results of Taylor series method when expansion is at the center, is exactly the same to the results of homotopy perturbation and differential transformation methods. It is hoped that this work will great assist and enhance the understanding of mathematical solutions providers and enthusiasts as it provides better insight into finding analytical solutions to linear and nonlinear differential equations.


[1] Stern R. H., Rasmussen, H. (1996). Left ventricular ejection: Model solution by collocation, an approximate analytical method. Comput Boil Med. 26, 255–61.
[2] Vaferi, B., Salimi, V.  Baniani, D. D., Jahanmiri, A.,  Khedri, S. (2012). Prediction of transient pressure response in the petroleum reservoirs using orthogonal collocation. J Petrol Sci and Eng 98-99, 156-163.
[3] Hatami, M.,  Hasanpour, A., Ganji. D. D. (2013). Heat transfer study through porous fins (Si3N4 and AL) with temperature-dependent heat generation. Energy Convers Manage 74, 9–16.
[4] Bouaziz, M. N. Aziz. A (2010). Simple and accurate solution for convective–radiative fin with temperature dependent thermal conductivity using double optimal linearization. Energy Convers Manage 51(2010), 76–82.
[5] Aziz, A., Bouaziz, M. N. (2011). A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity. Energy Convers Manage 52; 2876–2882.
[6] Shaoqin, G., Huoyuan, D.(2008). Negative norm least-squares methods for the incompressible magneto-hydrodynamic equations. Act Math Sci. 28B(3), 675–84.
[7] Hatami, M., Nouri, R. Ganji, D. D. (2013).  Forced convection analysis for MHD Al2O3–water nanofluid flow over a horizontal plate. J Mol Liq 187, 294–301.
[8] Hatami, M., Sheikholeslami, M., Ganji, D.D.(2014).  Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method. Powder Technol 253, 769–79.
[9] Hatami, M.,  Hatami, J.  Ganji. D. D. (2014).  Computer simulation of MHD blood conveying gold nanoparticles as a third grade non-Newtonian nanofluid in a hollow porous vessel. Comput Methods Programs Biomed 113, :632–41.
[10] Hatami M., Ganji, D. D. (2013). Thermal performance of circular convective–radiative porous fins with different section shapes and materials. Energy Convers Manage 76, :185–93.
[11] Hatami M., Ganji, D. D. (2014). Heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic field. J Mol Liq 190, 159–68.
[12] Hatami M., Ganji, D. D. (2014). Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods. Case Studies Therm Eng 2(2014), 14–22.
[13] Hatami, M., DomairryG (2014). Transient vertically motion of a soluble particle in a Newtonian fluid media. Powder Technol 253, 481–485.
[14] Domairry, M. Hatami, M. (2014). Squeezing Cu–water nanofluid flow analysis between parallel plates by DTM-Padé Method. J Mol Liq 193, 37–44.
[15] Ahmadi, A. R.,  A. M. Zahmtkesh, Hatami, M., Ganji, D. D. (2014). A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate. Powder Technol 258, 125–33
[16] Saedodin, S., Shahbabaei, M. (2013). Thermal analysis of natural convection in porous fins with homotopy perturbation method (HPM). Arabian Journal for Science and Engineering, 38, 2227{2231.
[17] Darvishi., M. T.,   Gorla R. S.R. Gorla, R. Aziz,  A. (2015). Thermal performance of a porous radial fin with natural convection and radiative heat losses. Thermal Science, 19(2), 669-678.
[18] Moradi., A. Hayat, T., Alsaedi, A. (2014). Convective-radiative thermal analysis of triangular fins with temperature-dependent thermal conductivity by DTM. Energy Conversion and Management, 77(2014), 70{77.
[19] Ha,. H., Ganji, D. D. Abbasi, M. (2005). Determination of temperature distribution for porous fin with temperature-dependent heat generation by homotopy analysis method. Journal of Applied Mechanical Engineering, 4(1), 1-5
[20] Sobamowo, M. G.,  Adeleye, O. A.,  Yinusa, A. A.(2017).  Analysis of convective-radiative porous fin with temperature-dependent internal heat generation and magnetic field using Homotopy Perturbation method. Journal of Computational and Applied Mechanics. 12(2), 127-145.
[21] He, J. H. (2006). Homotopy perturbation method for solving boundary value problems, Phys. Lett. A 350, 87-88.
[22] He, J. H. (2005). Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlin. Scne. Numer. Simul. 6 (2.2),  20-208.
[23] He, J. H. (2004). The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput. 151, 287{292.
[24] He, J. H. (2000). A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (2.1), 115-123.
[25] He, J. H. (1999). Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257-262.
[26] He, J. H. (2003). Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73-79
[27] Mohyud-Din, S. T., Noor, M. A. (2007). Homotopy perturbation method for solving fourth-order boundary value problems, Math. Prob. Eng. 1-15, Article ID 98602,
[28] Noor, M. A.  and Mohyud-Din, S. T. (2008). Homotopy perturbation method for solving sixth-order boundary value problems, Comput. Math. Appl. 55 (12) (2008), 2953-2972.
[29] Noor, M. A.  and S. T. Mohyud-Din, S. T. (2008). Homotopy perturbation method for nonlinear higher-order boundary value problems, Int. J. Nonlin. Sci. Num. Simul. 9 (2.4), 395-408.
[30] Biazar, J.,  Azimi, F. (2008). He’s homotopy perturbation method for solving Helmoltz equation. Int. J. Contemp. Math. Sci. 3, 739-744.
[3Biazar, J., Ghazvini (2009). Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Anal., Real World Appl. 10, 2633-2640.
[32] Sweilam, N. H. Khader, M. M. (2009). Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput. Math. Appl. 58, 2134-2141.
[33] Biazar, J., Ghazvini, H. (2008). Homotopy perturbation method for solving hyperbolic partial differential equations. Comput. Math. Appl. 56, 453-458.
[34] Junfeng, L. (2009). An analytical approach to the Sine-Gordon equation using the modified homotopy perturbation method. Comput. Math. Appl. 58, 2313-2319.
[35] Corliss, G.,  Chang, Y. F. (1982). Solving ordinary differential equations using Taylor series, ACM Trans. Math. Software 8 (2) (1982) 114–144.
[36] Chang, Y. -F.,  Corliss, G. (1994). ATOMFT: solving ODEs and DAEs using Taylor series, Comput. Math. Appl. 28 (10-12), 209–233.
[37] Pryce J. D. (1998). Solving high-index DAEs by Taylor series, Numer. Algorithms 19 (1–4), 195–211.
[38] Barrio, R. (2005). Performance of the Taylor series method for ODEs/DAEs, Appl. Math. Comput. 163 (2), 525–545.
[39] Nedialkov, N. S.,  Pryce, J. D. (2005). Solving differential-algebraic equations by Taylor series. I. Computing Taylor coefficients, BIT 45 (3), 561–591.
[40] Nedialkov, N. S.,  Pryce, J. D. (2007). Solving differential-algebraic equations by Taylor series. II. Computing the system Jacobian, BIT 47 (1), 121–135.
[41] Nedialkov, N. S.,  Pryce, J. D. (2008). Solving differential algebraic equations by Taylor series. III. The DAETS code, J. Numer. Anal. Ind. Appl. Math. 3 (1-2), 61–80.
[42] Jorba, Á., Zou., M. (2005). A software package for the numerical integration of ODEs by means of high-order Taylor methods, Exp. Math. 14 (1), 99–117.
[43] Makino, K.,  Berz, M. (2003). Taylor models and other validated functional inclusion methods, Int. J. Pure Appl. Math. 6 (3) (2003) 239–316.
[44] Barrio, R.  (2005). Performance of the Taylor series method for ODEs/DAEs. Appl. Math. Comput. 163, 525-545.
[45] Ren, Y, Zhang, B., Qiao, H. (1999).  A simple Taylor-series expansion method for a class of second kind integral equations. Journal of Computational and Applied Mathematics. 110(1), 15 15-24.
[46] Abbasbandy  S. and Bervillier, C. (2011). Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations, Appl. Math. Comput. 218 (2011) 2178.
[47] Kanwal, R. P. and  Liu, K. C. (1989). A Taylor expansion approach for solving integral equations, Int. J. Math. Ed. Sci. Technol 20 (1989) 411-414.
[48] Huang, L.,  Li, X. F. Zhao, Y. Duan. X. Y. (2011). Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl. 62, 1127-1134.
[49] Nedialkov, N. S. and Pryce, J. D. (2007). Solving differential-algebraic equations by Taylor series (II): computing the system Jacobian, BIT Numer. Math. 47, 121-135.
[50] Goldfine, A.  (1977). Taylor series methods for the solution of Volterra integral and integro-differential equations, Math. Comput. 31, 691-708.
[51] Zhou, J. K. (1986). Differential transformation and its applications for electrical circuits, in Chinese, Huarjung University Press, Wuuhahn, China.
[52] Jang, M. J.  and Chen, C. L. (1997). Analysis of the response of a strongly nonlinear damped system using a differential transformation technique, Appl. Math. Comput. 88, 137-151.
[53] Chen, C. -L., Liu, Y. -C (1998). Differential transformation technique for steady nonlinear heat conduction problems, Appl. Math. Comput. 95, 155-164.
[54] Yu, L. -T. and Chen, C. -K. (1998). The solution of the Blasius equation by the differential transformation method, Math. Comput. Model. 28, 101-111.
[55] Chen, C. -K.  and Chen, S. S. (2004). Application of the differential transformation method to a non-linear conservative system, Appl. Math. Comput. 154 (2004), 431-441.
[56] I. H. A.-H. Hassan. (2004). Differential transformation technique for solving higher-order initial value problems, Appl. Math. Comput. 154, 299-311.
[57] Yaghoobi, H. and Torabi. M. (2011). The application of differential transformation method to nonlinear equations arising in heat transfer, Int. Commun. Heat Mass 38, 815-820.
[58] Jang, M. J., Chen, C.-L, and Y.-C. Liu, Y.-C. (2001). Two-dimensional differential transform for partial differential equations. Appl. Math. Comput. 121, 261-270.
[59] Jang, M.-J., Chen, C. -L. Liy. Y.-C. (2000). On solving the initial-value problems using the differential transformation method, Appl. Math. Comput. 115, 145-160.
[60] Rashidi, M. M. (2009). The modified differential transform method for solving MHD boundary-layer equations, Comput. Phys. Commun. 180, 2210-2217.
[61] Erfani, E., Rashidi, M. M.,  Parsa, A. B.  (2010). The modified differential transform method for solving off-centered stagnation flow toward a rotating disc, Int. J. Comput. Meth. 7, 655-670.
[62] Gokdogan, A.,  Merdan, M., Yildirim, A. (2012). The modified algorithm for the differential transform method to solution of Genesio Systems. Comm Nonlinear Sci. 17, 45-51.
[63] Alomari, A. K. (2011). A new analytic solution for fractional chaotic dynamical systems using the differential transform method, Comput. Math. Appl. 61, 2528-2534.
[64] Arikoglu, A. and Ozkol, I. (2005). Solution of boundary value problems for integrodifferential equations by using differential transform method, Appl. Math. Comput. 168 (2005) 1145-1158.
[65] Ho, S. H., Chen, C. K. (1998). Analysis of general elastically end restrained nonuniform beams using differential transform, Appl. Math. Model. 22, 219-234.
[66] Chen, C. K., Ho, S. H. (1999). Solving partial differential equations by two-dimensional differential transform method, Appl. Math. Comput. 106, 171-179.
[67] Arikoglu, A. and Ozkol, I. (2006). Solution of difference equations by using differential transform method, Appl. Math. Comput. 174, 1216.
[68] Arikoglu, A. and Ozkol, I. (2006). Solution of differential-difference equations by using differential transform method, Appl. Math. Comput. 181, 153-162.