Homotopy analysis method for quadratic Riccati differential equation. (English) Zbl 1132.34305

Summary: The quadratic Riccati differential equation is solved by means of an analytic technique, namely the homotopy analysis method. Comparisons are made between Adomian’s decomposition method, homotopy perturbation method and the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple.


34A45 Theoretical approximation of solutions to ordinary differential equations
Full Text: DOI


[1] Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with adomian’s decomposition method, Appl math comput, 172, 485-490, (2006) · Zbl 1088.65063
[2] Abbasbandy, S., Iterated he’s homotopy perturbation method for quadratic Riccati differential equation, Appl math comput, 175, 581-589, (2006) · Zbl 1089.65072
[3] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers. Dordrecht · Zbl 0802.65122
[4] Adomian, G.; Rach, R., On the solution of algebraic equations by the decomposition method, Math anal appl, 105, 141-166, (1985) · Zbl 0552.60060
[5] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int J eng sci, 41, 2091-2103, (2003) · Zbl 1211.76076
[6] El-Tawil, M.A.; Bahnasawi, A.A.; Abdel-Naby, A., Solving Riccati differential equation using adomian’s decomposition method, Appl math comput., 157, 503-514, (2004) · Zbl 1054.65071
[7] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int J eng sci, 42, 123-135, (2004) · Zbl 1211.76009
[8] Hayat, T.; Khan, M.; Ayub, M., Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J math anal appl, 298, 225-244, (2004) · Zbl 1067.35074
[9] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech, 168, 213-232, (2004) · Zbl 1063.76108
[10] Hayat, T.; Khan, M., Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear dyn, 42, 395-405, (2005) · Zbl 1094.76005
[11] Hayat, T.; Khan, M.; Ayub, M., On non-linear flows with slip boundary condition, Zamp, 56, 1012-1029, (2005) · Zbl 1097.76007
[12] He, J.-H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int J nonlinear mech, 35, 1, 37-43, (2000) · Zbl 1068.74618
[13] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[14] Liao, S.J., An approximate solution technique which does not depend upon small parameters: a special example, Int J nonlinear mech, 30, 371-380, (1995) · Zbl 0837.76073
[15] Liao, S.J., An approximate solution technique which does not depend upon small parameters (part 2): an application in fluid mechanics, Int J nonlinear mech, 32, 5, 815-822, (1997) · Zbl 1031.76542
[16] Liao, S.J., An explicit totally analytic approximation of Blasius viscous flow problems, Int J non-linear mech, 34, 4, 759-778, (1999) · Zbl 1342.74180
[17] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall CRC Press, Boca Raton
[18] Liao, S.J., On the homotopy anaylsis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[19] Liao, S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl math comput, 169, 1186-1194, (2005) · Zbl 1082.65534
[20] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142
[21] Liao, S.J.; Pop, I., Explicit analytic solution for similarity boundary layer equations, Int J heat mass transfer, 47, 75-78, (2004) · Zbl 1045.76008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.