New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. (English) Zbl 1205.65216

Geng, Fazhan; Cui, Minggen

J. Comput. Appl. Math. 233, No. 2, 165-172 (2009).

This paper is concerned with the approximate solution of second order differential equations \( u''(t) + \sigma u'(t) + f(t, u(t)) = 0,\) \(t \in [0,1],\) where \( \sigma \) is a non zero constant and \( f: [0,1] \times {\mathbb R} \to {\mathbb R} \) is a sufficiently smooth function, supplemented with linear integral boundary conditions of type: \( u(0) - \mu_1 u'(0) = \int_0^1 h_1(s) u(s) ds,\) \( u(1) + \mu_2 u'(1) = \int_0^1 h_2(s) u(s) ds, \) with positive constants \( \mu_j\) and given smooth functions \( h_j(t)\).
The proposed approach starts establishing an homotopy defined by family of differential equations \( H(u,p) \equiv u'' + \sigma u' + p f(t,u) = 0\), with the parameter \( p \in [0,1]\) so that for \( p=0\) gives a linear equation such that with the boundary conditions has a unique solution \( u = u_0(t)\) easily computed and for the parameter value \( p=1\) is the desired solution of the non linear problem. Now by using \( p\) as a small parameter the solution of \( H(u,p)=0\) can be written as an asymptotic series \( u=u_0+ p u_1 + \dots \) where the successive \( u_j\) can be computed recursively as a solution linear problems and then the solution for \( p=1\) is approximated by the \((m+1)\)-sum \( u = \sum_{j=0}^m u_j\). For solving each linear boundary value problem of \( u_j\) the authors propose a reproducing kernel Hilbert space method. Two numerical experiments are presented to show the behaviour of the method depending on the terms \(m\) of the series and the number of grid points in the interval \([0,1].\)

Reviewer: Manuel Calvo (Zaragoza)

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

Keywords:

second order ordinary differential equations; integral boundary conditions; perturbation methods; reproducing kernel Hilbert space method (RKHSM); numerical experiments; homotopy perturbation method (HPM)

References:

[1]	Cahlon, B.; Kulkarni, D.M.; Shi, P., Stepwise stability for the heat equation with a nonlocal constraint, SIAM journal of numerical analysis, 32, 571-593, (1995) · Zbl 0831.65094
[2]	Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quarterly of applied mathematics, 21, 155-160, (1963) · Zbl 0173.38404
[3]	Kamynin, N.I., A boundary value in the theory of the heat conduction with non-local boundary condition, USSR computational mathematics and mathematical physics, 4, 33-59, (1964) · Zbl 0206.39801
[4]	Choi, Y.S.; Chan, K.Y., A parabolic equation with nonlocal boundary conditions arising from eletrochemistry, Nonlinear analysis, 18, 317-331, (1992) · Zbl 0757.35031
[5]	Shi, P., Weak solution to evolution problem with a nonlocal constraint, SIAM journal of analysis, 24, 46-58, (1993) · Zbl 0810.35033
[6]	Samarski, A.A., Some problems in the modern theory of differential equation, Differential’nye uravneniya, 16, 1221-1228, (1980) · Zbl 0484.93075
[7]	Cakir, M.; Amiraliyer, G.M., A finite difference method for the singularly perturbed problem with nonlocal boundary condition, Applied mathematics and computation, 160, 539-549, (2005) · Zbl 1068.65100
[8]	Borovykh, Natalia, Stability in the numerical solution of the heat equation with nonlocal boundary conditions, Applied numerical mathematics, 42, 17-27, (2002) · Zbl 1003.65102
[9]	Boucherif, A., Second order boundary value problems with integral boundary condition, Nonlinear analysis, 70, 364-371, (2009) · Zbl 1169.34310
[10]	Dehghan, M., Fully implicit finite differences methods for two-dimensional diffusion with a non-local boundary condition, Journal of computational and applied mathematics, 106, 255-269, (1999) · Zbl 0931.65091
[11]	Dehghan, M., Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition, Applied mathematics and computation, 49, 331-349, (1999) · Zbl 0949.65085
[12]	Dehghan, M., Grank – nicolson finite difference method for two-dimensional diffusion with an integral condition, Applied mathematics and computation, 124, 17-27, (2001) · Zbl 1024.65076
[13]	Dehghan, M., A new ADI technique for two-dimensional parabolic equation with an integral condition, Computers and mathematics with applications, 43, 1477-1488, (2002) · Zbl 1001.65094
[14]	Dehghan, M., Numerical solution of a non-local boundary value problem with neumann’s boundary conditions, Communications in numerical methods in engineering, 19, 1-12, (2003) · Zbl 1014.65072
[15]	He, J.H., Homotopy perturbation technique, Computational methods in applied mechanics and engineering, 178, 257-262, (1999) · Zbl 0956.70017
[16]	He, J.H., Comparison of homotopy perturbation method and homotopy analysis method, Applied mathematics and computation, 156, 2, 527-539, (2004) · Zbl 1062.65074
[17]	He, J.H., Homotopy perturbation method for bifurcation of nonlinear problems, International journal of nonlinear sciences and numerical simulation, 6, 2, 207-208, (2005)
[18]	J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, de-Verlag in GmbH, Berlin, 2006
[19]	He, J.H., Some asymptotic methods for strongly nonlinear equation, International journal of modern physics B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[20]	He, J.H., Addendum new interpretation of homotopy perturbation method, International journal of modern physics B, 20, 18, 2561-2568, (2006)
[21]	Rana, M.A.; Siddiqui, A.M.; Ghori, Q.K.; Qamar, R., Application of he’s homotopy perturbation method to sumudu transform, International journal of nonlinear sciences and numerical simulation, 8, 2, 185-190, (2007)
[22]	Yusufoǧlu, E., Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, International journal of nonlinear sciences and numerical simulation, 8, 3, 353-358, (2007)
[23]	Ghorbani, A.; Saberi-Nadjafi, J., He’s homotopy perturbation method for calculating Adomian polynomials, International journal of nonlinear sciences and numerical simulation, 8, 2, 229-232, (2007)
[24]	Beléndez, A.; Pascual, C.; Márquez, A.; Méndez, D.I., Application of he’s homotopy perturbation method to the relativistic (an)harmonic oscillator. I: comparison between approximate and exact frequencies, International journal of nonlinear sciences and numerical simulation, 8, 4, 483-492, (2007)
[25]	Beléndez, A.; Pascual, C.; Méndez, D.I.; álvarez, M.L.; Neipp, C., Application of he’s homotopy perturbation method to the relativistic (an)harmonic oscillator. II: A more accurate approximate solution, International journal of nonlinear sciences and numerical simulation, 8, 4, 493-504, (2007)
[26]	Xu, L., He’s homotopy perturbation method for a boundary layer equation in unbounded domain, Computers, mathematics with applications, 54, 7-8, 1067-1070, (2007) · Zbl 1267.76089
[27]	Ganji, D.D.; Sadighi, A., Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of computational and applied mathematics, 207, 1, 24-34, (2007) · Zbl 1120.65108
[28]	Abbasbandy, S., Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method, Applied mathematics and computation, 172, 1, 431-438, (2006) · Zbl 1088.65043
[29]	Abbasbandy, S., Homotopy perturbation method for quadratic Riccati differential equation and comparison with adomians decomposition method, Applied mathematics and computation, 172, 1, 485-490, (2006) · Zbl 1088.65063
[30]	Daniel, A., Reproducing kernel spaces and applications, (2003), Springer · Zbl 1021.00005
[31]	Berlinet, A.; Thomas-Agnan, C., Reproducing kernel Hilbert space in probability and statistics, (2004), Kluwer Academic Publishers · Zbl 1145.62002
[32]	Xie, S.S.; Heo, S.; Kim, S.; Woo, G.; Yi, S., Numerical solution of one-dimensional burgers’ equation using reproducing kernel function, Journal of computational and applied mathematics, 214, 417-434, (2008) · Zbl 1140.65069
[33]	Yao, H.M.; Lin, Y.Z., Solving singular boundary-value problems of higher even-order, Journal of computational and applied mathematics, 223, 703-713, (2009) · Zbl 1181.65108
[34]	Cui, M.G.; Geng, F.Z., Solving singular two-point boundary value problem in reproducing kernel space, Journal of computational and applied mathematics, 205, 6-15, (2007) · Zbl 1149.65057
[35]	Geng, F.Z.; Cui, M.G., Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied mathematics and computation, 192, 389-398, (2007) · Zbl 1193.34017
[36]	Geng, F.Z.; Cui, M.G., Solving a nonlinear system of second order boundary value problems, Journal of mathematical analysis and applications, 327, 1167-1181, (2007) · Zbl 1113.34009
[37]	Cui, M.G.; Geng, F.Z., A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied mathematics and computation, 188, 1389-1401, (2007) · Zbl 1118.35348
[38]	Cui, M.G.; Chen, Z., The exact solution of nonlinear age-structured population model, Nonlinear analysis. real world applications, 8, 1096-1112, (2007) · Zbl 1124.35030
[39]	Li, C.L.; Cui, M.G., The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied mathematics and computation, 143, 393-399, (2003) · Zbl 1034.47030
[40]	Li, C.L.; Cui, M.G., How to solve the equation aubu+cu=f, Applied mathematics and computation, 133, 643-653, (2002) · Zbl 1051.47009