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Searching the least value method for solving fourth-order nonlinear boundary value problems. (English) Zbl 1189.65149
Summary: This paper obtains a searching least value (SLV) method for a class of fourth-order nonlinear boundary value problems is investigated. The argument is based on the reproducing kernel space $W_{5}[0,1]$. The approximate solutions $u_n(x)$ and $u_n^{(k)}(x)$ are uniformly convergent to the exact solution $u(x)$ and $u_n^{(k)}(x)$, respectively. Numerical results verify that the method is quite accurate and efficient for this kind of problem.
##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B15 Nonlinear boundary value problems for ODE
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##### References:
 [1] Aftabizadeh, A. R.: Existence and uniqueness theorems for fourth-order boundary value problems, J. math. Anal. appl. 116, 415-426 (1986) · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3 [2] Yang, Y.: Fourth-order two-point boundary value problem, Proc. amer. Math. soc. 104, 175-180 (1988) · Zbl 0671.34016 · doi:10.2307/2047481 [3] Del Pino, M. A.; Manasevich, R. F.: Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. amer. Math. soc. 112, 81-86 (1991) · Zbl 0725.34020 · doi:10.2307/2048482 [4] De Coster, C.; Fabry, C.; Munyamarere, F.: Nonresonance conditions for fourth-order nonlinear boundary value problems, Int. J. Math. math. Sci. 17, 725-740 (1994) · Zbl 0810.34017 · doi:10.1155/S0161171294001031 [5] Gupta, C. P.: Existence and uniqueness results for some fourth order fully quasilinear boundary value problem, Appl. anal. 36 (1990) · Zbl 0713.34025 · doi:10.1080/00036819008839930 [6] Ma, R.; Wang, H.: On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal. 59, 225-231 (1995) · Zbl 0841.34019 · doi:10.1080/00036819508840401 [7] Cabada, A.: The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. math. Anal. appl. 185, 302-320 (1994) · Zbl 0807.34023 · doi:10.1006/jmaa.1994.1250 [8] Ma, R.; Zhang, J.; Fu, S.: The method of lower and upper solutions for fourth-order two-point boundary value problems, J. math. Anal. appl. 215, 415-422 (1997) · Zbl 0892.34009 · doi:10.1006/jmaa.1997.5639 [9] Bai, Z.: The method of lower and upper solutions for a bending of an elastic beam equation, J. math. Anal. appl. 248, 195-202 (2000) · Zbl 1016.34010 · doi:10.1006/jmaa.2000.6887 [10] Liu, B.: Positive solutions of fourth-order two point boundary value problems, Appl. math. Comput. 148, 407-420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3 [11] Jiang, D.; Liu, Huizhao; Xu, Xiaojie: Nonresonant singular fourth-order boundary value problems, Appl. math. Lett. 18, 69-75 (2005) · Zbl 1074.34019 · doi:10.1016/j.aml.2003.05.016 [12] Li, Y.: Positive solutions of fourth-order boundary value problems with two parameters, J. math. Anal. appl. 281, 477-484 (2003) · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8 [13] Chai, G.: Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear anal. 66, 870-880 (2007) · Zbl 1113.34008 · doi:10.1016/j.na.2005.12.028