×

Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations. (English) Zbl 1141.65397

Summary: This paper compares the homotopy perturbation method with the sine-cosine wavelet method for solving linear integro-differential equations.
From the computational viewpoint, the homotopy perturbation method is more efficient and easier than the sine-cosine wavelet method.

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
65T60 Numerical methods for wavelets
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] He, J. H., Asymptotology of homotopy perturbation method, Appl. Math. Comput., 156, 591-596 (2004) · Zbl 1061.65040
[2] He, J. H., Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons Fractals, 26, 827-833 (2005) · Zbl 1093.34520
[3] He, J. H., Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. Numer. Simul., 6, 2, 207-208 (2005) · Zbl 1401.65085
[4] He, J. H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons Fractals, 19, 4, 847-851 (2005) · Zbl 1135.35303
[5] Liu, H. M., Variational approach to nonlinear electrochemical system, Chaos, Solitons Fractals, 23, 2, 573-576 (2005)
[6] Liu, H. M., Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Int. J. Nonlinear Sci. Numer. Simul., 5, 1, 95-96 (2004)
[7] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167, 1-2, 57-68 (1998) · Zbl 0942.76077
[8] He, J. H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 167, 1-2, 69-73 (1998) · Zbl 0932.65143
[9] He, J. H., Variational iteration method: a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34, 4, 699-708 (1999) · Zbl 1342.34005
[10] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114, 2-3, 115-123 (2000) · Zbl 1027.34009
[11] He, J. H.; Wan, Y. Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Int. J. Circ. Theor. Appl., 32, 6, 629-632 (2004) · Zbl 1169.94352
[12] He, J. H.; Wu, X. H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons Fractals, 29, 108-113 (2006) · Zbl 1147.35338
[13] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillators, Part III: Double series expansion, Int. J. Nonlinear Sci. Numer. Simul., 2, 4, 317-320 (2001) · Zbl 1072.34507
[14] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillators, Part I: Expansion of constant, Int. J. Nonlinear Mech., 37, 2, 309-314 (2002) · Zbl 1116.34320
[15] He, J. H., Modified Lindstedt-Poincare methods for some strongly nonlinear oscillators, Part II: A new transformation, Int. J. Nonlinear Mech., 37, 2, 315-320 (2002) · Zbl 1116.34321
[16] Liu, H. M., Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method, Chaos, Solitons Fractals, 23, 2, 577-579 (2005) · Zbl 1078.34509
[17] El-Shahed, M., Application of He’s homotopy perturbation method to Volterra’s integro-differential equation, Int. J. Nonlinear Sci. Numer. Simul., 6, 2, 163-168 (2005) · Zbl 1401.65150
[18] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B, 20, 1141-1199 (2006) · Zbl 1102.34039
[19] He, J. H., A review on some new recently developed nonlinear analytical techniques, Int. J. Nonlinear Sci. Numer. Simul., 1, 1, 51-70 (2000) · Zbl 0966.65056
[20] Hillermeier, C., Generalized homotopy approach to multiobjective optimization, Int. J. Optim. Theory Appl., 110, 3, 557-583 (2001) · Zbl 1064.90041
[21] He, J. H., Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons Fractals, 26, 695-700 (2005) · Zbl 1072.35502
[22] He, J. H., An approximate solution technique depending upon an artificial parameter, Commun. Nonlinear Sci. Simul., 3, 2, 92-97 (2001)
[23] He, J. H., Homotopy perturbation technique, Comput Methods Appl. Mech. Engng., 178, 3-4, 257-262 (1999) · Zbl 0956.70017
[24] 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
[25] He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl. Math. Comput., 151, 287-292 (2004) · Zbl 1039.65052
[26] He, J. H., Comparison of homotopy perturbation method and homotopy analysis method, Appl. Math. Comput., 156, 527-539 (2004) · Zbl 1062.65074
[27] Tavassoli Kajani, M.; Ghasemi, M.; Babolian, E., Numerical Solution of linear integro-differential equation by using sine-cosine wavelets, Appl. Math. Comput., 180, 569-574 (2006) · Zbl 1102.65137
[28] Razzaghi, M.; Yousefi, S., Sine-Cosine wavelets operational matrix of integration and its applications in the calculus of variations, Int. J. Syst. Sci., 33, 805-810 (2002) · Zbl 1012.65063
[29] Gu, J. S.; Jiang, W. S., The haar wavelets operational matrix of integration, Int. J. Syst. Sci., 27, 623-628 (1996) · Zbl 0875.93116
[30] A.M. Wazwaz, A First Course in Integral Equations, New Jersey, 1997; A.M. Wazwaz, A First Course in Integral Equations, New Jersey, 1997 · Zbl 0924.45001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.