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Iterated He’s homotopy perturbation method for quadratic Riccati differential equation. (English) Zbl 1089.65072
Summary: The iterated homotopy perturbation method of {\it J. He} [Int. J. Non-Linear Mech. 35, No. 1, 37--43 (2000; Zbl 1068.74618)] is proposed to solving a quadratic Riccati differential equation. Comparisons are made between Adomian’s decomposition method, the exact solution, and the proposed method. The results reveal that the method is very effective and simple.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
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References:
[1] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[2] Adomian, G.; Rach, R.: On the solution of algebraic equations by the decomposition method. Math. anal. Appl. 105, 141-166 (1985) · Zbl 0552.60060
[3] El-Shahed, M.: Application of he’s homotopy perturbation method to Volterra’s integro-differential equation. Int. J. Nonlinear sci. Numer. simul. 6, No. 2, 163-168 (2005)
[4] 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
[5] Hillermeier, C.: Generalized homotopy approach to multiobjective optimization. Int. J. Optim. theory appl. 110, No. 3, 557-583 (2001) · Zbl 1064.90041
[6] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. methods appl. Mech. eng. 167, No. 1-2, 57-68 (1998) · Zbl 0942.76077
[7] He, J. H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. methods appl. Mech. eng. 167, No. 1-2, 69-73 (1998) · Zbl 0932.65143
[8] He, J. H.: An approximate solution technique depending upon an artificial parameter. Commun. nonlinear sci. Simulat. 3, No. 2, 92-97 (1998) · Zbl 0921.35009
[9] He, J. H.: Variational iteration method: a kind of nonlinear analytical technique: some examples. Int. J. Non-linear mech. 34, No. 4, 699-708 (1999) · Zbl 05137891
[10] He, J. H.: Homotopy perturbation technique. Comput. methods appl. Mech. eng. 178, No. 3/4, 257-262 (1999) · Zbl 0956.70017
[11] He, J. H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Nonlinear mech. 35, No. 1, 37-43 (2000) · Zbl 1068.74618
[12] He, J. H.: A review on some new recently developed nonlinear analytical techniques. Int. J. Nonlinear sci. Numer. simul. 1, No. 1, 51-70 (2000) · Zbl 0966.65056
[13] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl. math. Comput. 114, No. 2-3, 115-123 (2000) · Zbl 1027.34009
[14] He, J. H.: Bookkeeping parameter in perturbation methods. Int. J. Nonlinear sci. Numer. simul. 2, No. 3, 257-264 (2001) · Zbl 1072.34508
[15] He, J. H.: Modified lindsted-Poincarè methods for some strongly nonlinear oscillations, part III: Double series expansion. Int. J. Nonlinear sci. Numer. simul. 2, No. 4, 317-320 (2001) · Zbl 1072.34507
[16] He, J. H.: Modified Lindstedt-Poincarè methods for some strongly non-linear oscillations, part I: Expansion of a constant. Int. J. Nonlinear mech. 37, No. 2, 309-314 (2002) · Zbl 1116.34320
[17] He, J. H.: Modified Lindstedt-Poincarè methods for some strongly non-linear oscillations, part II: a new transformation. Int. J. Nonlinear mech. 37, No. 2, 315-320 (2002) · Zbl 1116.34321
[18] He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. math. Comput. 151, 287-292 (2004) · Zbl 1039.65052
[19] He, J. H.: Comparison of homotopy perturbtion method and homotopy analysis method. Appl. math. Comput. 156, 527-539 (2004) · Zbl 1062.65074
[20] He, J. H.: Asymptotology by homotopy perturbtion method. Appl. math. Comput. 156, 591-596 (2004) · Zbl 1061.65040
[21] He, J. H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear sci. Numer. simul. 6, No. 2, 207-208 (2005)
[22] He, J. H.; Wan, Y. Q.; Guo, Q.: An iteration formulation for normalized diode characteristics. Int. J. Circ. theor. Appl. 32, No. 6, 629-632 (2004) · Zbl 1169.94352
[23] Liu, H. M.: Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincarè method. Chaos, solitons & fractals 23, No. 2, 577-579 (2005) · Zbl 1078.34509
[24] Nayfeh, A. H.: Problems in perturbation. (1985) · Zbl 0573.34001
[25] Wang, Q.; Chen, Y.; Zhang, H. Q.: A new Riccati equation rational expansion method and its application to (2+1)-dimensional Burgers equation. Chaos, solitons & fractals 25, No. 5, 1019-1028 (2005) · Zbl 1070.35073
[26] Xie, F.; Gao, X.: Exact travelling wave solutions for a class of nonlinear partial differential equations. Chaos, solitons & fractals 19, No. 5, 1113-1117 (2004) · Zbl 1068.35146