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Numerical solutions of highly oscillatory integrals. (English) Zbl 1139.65019
Summary: An application of J.-H. He’s homotopy perturbation method (HPM) [ibid. 135, No. 1, 73–79 (2003; Zbl 1030.34013)] is proposed to numerical solution of highly oscillatory integrals. To apply the HPM to the oscillatory integrals, we assume that the oscillatory function has not critical point at the endpoints of integration region. The results reveal that the method is very effective and simple.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures
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##### References:
 [1] Abbasbandy, S., Application of he’s homotopy perturbation method for Laplace transform, Chaos soliton fract., 30, 1206-1212, (2006) · Zbl 1142.65417 [2] Changbum, Chun, Integration using he’s homotopy perturbation method, Chaos soliton fract., 30, 1130-1134, (2007) · Zbl 1142.65464 [3] A. Molabahrami, F. Khani, S. Hamedi-Nezhad, Soliton solutions of the two-dimensional KdV Burgers equation by homotopy perturbation method, Phys. Lett. A (2007) doi:10.1016/j.physleta.2007.05.101, in press. · Zbl 1209.65113 [4] Filon, L.N.G., On a quadrature formula for trigonometric integrals, Proc. roy. soc. Edinburgh, 49, 38-47, (1958) · JFM 55.0946.02 [5] He, J.-H., Homotopy perturbation technique, Comput. methods appl. mech. eng., 178, 3/4, 230-235, (1999) [6] Watson, L.T., Globally convergent homotopy methods: a tutorial, Appl. math. comput., 13BK, 369-396, (1989) · Zbl 0689.65033 [7] Watson, Layne T.; Scott, Melvin R., Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method, Appl. math. comput., 24, 3X-357, (1987) · Zbl 0635.65099 [8] Watson, Layne T., Engineering applications of the chow – yorke algorithm, Appl. math. comput., 9, 111-133, (1981) · Zbl 0481.65029 [9] Watson, Layne T.; Haftka, Raphael T., Modern homotopy methods in optimization, Comput. meth. appl. mech. eng., 74, 289-305, (1989) · Zbl 0693.65046 [10] Wang, Y.; Bernstein, D.S.; Watson, L.T., Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order $$H^2 / H^\infty$$ modeling estimation and control, Appl. math. comput., 123, 155-185, (2001) · Zbl 1028.93011 [11] He, J.-H., A coupling method of homotopy technique and perturbation technique for nonlinear problem, Int. J. nonlinear mech., 35, 1, 37-43, (2000) · Zbl 1068.74618 [12] He, J.-H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. math. comput., 135, 73-79, (2003) · Zbl 1030.34013 [13] He, J.-H., Periodic solutions and bifurcations of delay-differential equations, Phys. lett. A, 347, 4-6, 228-230, (2005) · Zbl 1195.34116 [14] He, J.-H., The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. math. comput., 151, 1, 287-292, (2004) · Zbl 1039.65052 [15] He, J.-H., Homotopy perturbation method for bifurcation of non-linear problems, Int. J. non-linear sci. numer. simul., 6, 2, 207-208, (2005) · Zbl 1401.65085 [16] He, J.-H., Application of homotopy perturbation method to non-linear wave equations, Chaos soliton fract., 26, 3, 695-700, (2005) · Zbl 1072.35502 [17] He, J.-H., Limit cycle and bifurcation of nonlinear problems, Chaos soliton fract., 26, 3, 827-833, (2005) · Zbl 1093.34520 [18] He, J.-H., New interpretation of homotopy perturbation method, Int. J. mod. phys. B, 20, 18, 2561-2568, (2006) [19] He, J.-H., Some asymptotic methods for strongly nonlinear equations, Int. J. mod. phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039 [20] Levin, D., Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. comput., 38, 531-538, (1982) · Zbl 0482.65013 [21] Kim, K.; Cools, R.; Ixaru, L.G.R., Extended quadrature rules for oscillatory integrands, Appl. numer. math., 46, 59-73, (2003) · Zbl 1025.41016 [22] Nayfeh, A.H., Problems in perturbation, (1985), John Wiley New York · Zbl 0139.31904 [23] Wu, X.-Y.; Xia, J.-L., Two low accuracy methods for stiff systems, Appl. math. comput., 123, 141-153, (2001) · Zbl 1024.65053 [24] Xiang, On the filon and Levin methods for highly oscillatory integral $$\int_a^b f(x) \operatorname{e}^{\operatorname{i} w \varphi(x)} \operatorname{d} x$$, J. comput. appl. math., (2006)
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