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A novel method for nonlinear singular fourth order four-point boundary value problems. (English) Zbl 1228.65134
Summary: A novel method is proposed for solving nonlinear singular fourth order four-point boundary value problems (BVPs) by combining advantages of the homotopy perturbed method (HPM) and the reproducing kernel method (RKM). Some numerical examples are presented to illustrate the strength of the method.
MSC:
65L99Numerical methods for ODE
References:
[1]Chen, S. H.; Ni, W.; Wang, C. P.: Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Applied mathematics letters 19, 161-168 (2006) · Zbl 1096.34009 · doi:10.1016/j.aml.2005.05.002
[2]Karaca, I. Y.: Fourth-order four-point boundary value problem on time scales, Applied mathematics letters 21, 1057-1063 (2008) · Zbl 1170.34309 · doi:10.1016/j.aml.2008.01.001
[3]Zhang, X. G.; Liu, L. S.: Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator, Journal of mathematical analysis and applications 336, 1414-1423 (2007) · Zbl 1125.34018 · doi:10.1016/j.jmaa.2007.03.015
[4]Bai, C. Z.; Yang, D. D.; Zhu, H. B.: Existence of solutions for fourth order differential equation with four-point boundary conditions, Applied mathematics letters 20, 1131-1136 (2007) · Zbl 1140.34308 · doi:10.1016/j.aml.2006.11.013
[5]Ma, D. X.; Yang, X. Z.: Upper and lower solution method for fourth-order four-point boundary value problems, Journal of computational and applied mathematics 223, 543-551 (2009) · Zbl 1181.65106 · doi:10.1016/j.cam.2007.10.051
[6]Zhang, Q.; Chen, S. H.; Lv, J. H.: Upper and lower solution method for fourth-order four-point boundary value problems, Journal of computational and applied mathematics 196, 387-393 (2006) · Zbl 1102.65084 · doi:10.1016/j.cam.2005.09.007
[7]Zhong, Y. L.; Chen, S. H.; Wang, C. P.: Existence results for a fourth-order ordinary differential equation with a four-point boundary condition, Applied mathematics letters 21, 465-470 (2008) · Zbl 1141.34305 · doi:10.1016/j.aml.2007.03.029
[8]Agarwal, R. P.; Kiguradze, I.: On multi-point boundary value problems for linear ordinary differential equations with singularities, Journal of mathematical analysis and applications 297, 131-151 (2004) · Zbl 1058.34012 · doi:10.1016/j.jmaa.2004.05.002
[9]Geng, F. Z.: Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Applied mathematics and computation 215, 2095-2102 (2009) · Zbl 1178.65085 · doi:10.1016/j.amc.2009.08.002
[10]Geng, F. Z.; Cui, M. G.: Solving nonlinear multi-point boundary value problems by combining homotopy perturbation and variational iteration methods, International journal of nonlinear sciences and numerical simulation 10, No. 5, 597-600 (2009)
[11]Tatari, M.; Dehghan, M.: The use of the Adomian decomposition method for solving multipoint boundary value problems, Physica scripta 73, 672-676 (2006)
[12]Li, X. Y.; Wu, B. Y.: Reproducing kernel method for singular fourth order four-point boundary value problems, Bulletin of the malaysian mathematical sciences society 34, 147-151 (2011) · Zbl 1219.34032 · doi:http://math.usm.my/bulletin/html/vol34_1_13.html
[13]Geng, F. Z.: A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Applied mathematics and computation 213, 163-169 (2009) · Zbl 1166.65358 · doi:10.1016/j.amc.2009.02.053
[14]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 · doi:10.1016/j.amc.2007.03.016
[15]Geng, F. Z.; Cui, M. G.: Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the korean mathematical society 45, No. 3, 77-87 (2008) · Zbl 1154.34012 · doi:10.4134/JKMS.2008.45.3.631
[16]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 · doi:10.1016/j.jmaa.2006.05.011
[17]Geng, F. Z.; Cui, M. G.: Analytical approximation to solutions of singularly perturbed boundary value problems, Bulletin of the malaysian mathematical sciences society 33, 221-232 (2010) · Zbl 1194.34113 · doi:http://math.usm.my/bulletin/html/vol33_2_6.html
[18]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 · doi:10.1016/j.cam.2006.04.037
[19]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 · doi:10.1016/j.amc.2006.11.005
[20]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 · doi:10.1016/j.nonrwa.2006.06.004
[21]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, No. 2–3, 393-399 (2003)
[22]Cui, M. G.; Lin, Y. Z.: Nonlinear numerical analysis in reproducing kernel space, (2009)
[23]Berlinet, Alain; Thomas-Agnan, Christine: Reproducing kernel Hilbert space in probability and statistics, (2004)
[24]Du, J.; Cui, M. G.: Constructive approximation of solution for fourth-order nonlinear boundary value problems, Mathematical methods in the applied sciences 32, 723-737 (2009) · Zbl 1170.34015 · doi:10.1002/mma.1064
[25]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 · doi:10.1016/j.cam.2008.02.010
[26]He, J. H.: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999)
[27]He, J. H.: A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International journal of non-linear mechanics 35, No. 1, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[28]He, J. H.: Homotopy perturbation method: a new nonlinear analytical technique, Applied mathematics and computation 135, No. 1, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[29]He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied mathematics and computation 151, No. 1, 287-292 (2004) · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[30]He, J. H.: Comparison of homotopy perturbation method and homotopy analysis method, Applied mathematics and computation 156, No. 2, 527-539 (2004) · Zbl 1062.65074 · doi:10.1016/j.amc.2003.08.008
[31]He, J. H.: Homotopy perturbation method for bifurcation of nonlinear problems, International journal of nonlinear sciences and numerical simulation 6, No. 2, 207-208 (2005)
[32]He, J. H.: Application of homotopy perturbation method to nonlinear wave equations, Chaos soliton and fractal 26, No. 3, 695-700 (2005) · Zbl 1072.35502 · doi:10.1016/j.chaos.2005.03.006
[33]He, J. H.: Homotopy perturbation method for solving boundary value problems, Physics letters A 350, No. 1–2, 87-88 (2006) · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[34]J.H. He, Non-perturbative methods for strongly nonlinear problems, Disertation, de-Verlag in GmbH, Berlin, 2006.
[35]He, J. H.: Some asymptotic methods for strongly nonlinear equation, International journal of modern physics B 20, No. 10, 1141-1149 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[36]He, J. H.: Addendum new interpretation of homotopy pertirbation method, International journal of modern physics B 20, No. 18, 2561-2568 (2006)
[37]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, No. 2, 185-190 (2007)
[38]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, No. 3, 353-358 (2007)
[39]Ghorbani, A.; Saberi-Nadjafi, J.: He’s homotopy perturbation method for calculating Adomian polynomials, International journal of nonlinear sciences and numerical simulation 8, No. 2, 229-232 (2007)
[40]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, No. 4, 483-492 (2007)
[41]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, No. 4, 493-504 (2007)
[42]Yıldırım, A.: An algorithm for solving the fractional nonlinear schröinger equation by means of the homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 10, No. 4, 445-450 (2009)
[43]Beléndez, A.; Pascual, C.; Belndez, T.; Hernndez, A.: Application of a modified he’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities, Nonlinear analysis. Real world applications 10, No. 1, 416-427 (2009)
[44]Inc, M.; Cavlak, E.: Hes homotopy perturbation method for solving coupled- KdV equations, International journal of nonlinear sciences and numerical simulation 10, No. 3, 333-340 (2009)
[45]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, No. 1, 24-34 (2007) · Zbl 1120.65108 · doi:10.1016/j.cam.2006.07.030