The reproducing kernel method for nonlinear fourth-order BVPs

Shiyv Wang; Xueqin Lv; Songyan He; Shiyv Wang; Xueqin Lv; Songyan He

doi:10.3934/math.20231294

AIMS Mathematics

2023, Volume 8, Issue 11: 25371-25381. doi: 10.3934/math.20231294

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The reproducing kernel method for nonlinear fourth-order BVPs

Shiyv Wang ¹,
Xueqin Lv ^{2
,
,},
Songyan He ^{3
,
,}

1.
School of Mathematics and Sciences, Harbin Normal University, Harbin 150025, China
2.
College of Basic Science, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China
3.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received: 08 December 2022 Revised: 09 February 2023 Accepted: 01 August 2023 Published: 31 August 2023
MSC : 35G30, 65L10

Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space $ W_{2}^{5}[0, 1] $. Its approximate solution is obtained by truncating the n-term of the exact solution and using the $ \varepsilon $-best approximate method. Meanwhile, the approximate solution $ u^{(i)}_{n}(x) $ converges uniformly to the exact solution $ u^{(i)}(x), (i, 0, 1, 2, 3, 4) $. The validity and accuracy of this method are verified by some examples.
- reproducing kernel space,
- nonlinear fourth-order BVPs,
- $ \varepsilon $-best approximate solution
Citation: Shiyv Wang, Xueqin Lv, Songyan He. The reproducing kernel method for nonlinear fourth-order BVPs[J]. AIMS Mathematics, 2023, 8(11): 25371-25381. doi: 10.3934/math.20231294

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Abstract

Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space $ W_{2}^{5}[0, 1] $. Its approximate solution is obtained by truncating the n-term of the exact solution and using the $ \varepsilon $-best approximate method. Meanwhile, the approximate solution $ u^{(i)}_{n}(x) $ converges uniformly to the exact solution $ u^{(i)}(x), (i, 0, 1, 2, 3, 4) $. The validity and accuracy of this method are verified by some examples.

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