An $ {\varepsilon} $-approximate solution of BVPs based on improved multiscale orthonormal basis

Yingchao Zhang; Yuntao Jia; Yingzhen Lin; Yingchao Zhang; Yuntao Jia; Yingzhen Lin

doi:10.3934/math.2024282

AIMS Mathematics

2024, Volume 9, Issue 3: 5810-5826. doi: 10.3934/math.2024282

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Research article

An $ {\varepsilon} $-approximate solution of BVPs based on improved multiscale orthonormal basis

Zhuhai Campus, Beijing Institute of Technology, Zhuhai 519085, China; zhych0314@163.com

Received: 26 December 2023 Revised: 21 January 2024 Accepted: 24 January 2024 Published: 31 January 2024
MSC : 65L10, 65L20

In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking $ {\varepsilon} $-approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of $ u^{(m)}(x) $. In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields extremely accurate approximation to the exact solution. Meanwhile, compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective and convenient.
- Legendre polynomials,
- $ {\varepsilon} $-approximate solution,
- numerical solution
Citation: Yingchao Zhang, Yuntao Jia, Yingzhen Lin. An $ {\varepsilon} $-approximate solution of BVPs based on improved multiscale orthonormal basis[J]. AIMS Mathematics, 2024, 9(3): 5810-5826. doi: 10.3934/math.2024282

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Abstract

In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking $ {\varepsilon} $-approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of $ u^{(m)}(x) $. In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields extremely accurate approximation to the exact solution. Meanwhile, compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective and convenient.

References

[1]	M. Moshlinsky, Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol. Soc. Mat. Mex., 7 (1950), 10–25.
[2]	S. Timoshenko, Theory of elastic stability, McGraw-Hill, 1996.
[3]	T. Y. Na, Computational methods in engineering boundary value problems, Academic Press, 1979.
[4]	X. Wu, W. Kong, C. Li, Sinc collocation method with boundary treatment for two-point boundary value problems, J. Comput. Appl. Math., 196 (2006), 229–240. https://doi.org/10.1016/j.cam.2005.09.003 doi: 10.1016/j.cam.2005.09.003
[5]	Siraj-ul-Islam, I. Aziz, B. Šarler, The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Math. Comput. Modell., 52 (2010), 1577–1590. https://doi.org/10.1016/j.mcm.2010.06.023 doi: 10.1016/j.mcm.2010.06.023
[6]	E. Ideona, P. Oja, Quadratic/linear rational spline collocation for linear boundary value problems, Appl. Numer. Math., 125 (2018), 143–158. https://doi.org/10.1016/j.apnum.2017.11.005 doi: 10.1016/j.apnum.2017.11.005
[7]	M. Lakestani, M. Dehgan, The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal B-spline wavelets, Int. J. Comput. Math., 83 (2006), 685–694. https://doi.org/10.1080/00207160601025656 doi: 10.1080/00207160601025656
[8]	P. Roul, V. M. K. P. Goura, A Bessel collocation method for solving Bratus problem, J. Math. Chem., 58 (2020), 1601–1614. https://doi.org/10.1007/s10910-020-01147-w doi: 10.1007/s10910-020-01147-w
[9]	P. Roul, K. Thula, V. M. K. P. Goura, An optimal sixth-order quartic B-spline collocation method for solving Bratu and Lane-Emden type problems, Math. Methods Appl. Sci., 42 (2019), 2613–2630. https://doi.org/10.1002/mma.5537 doi: 10.1002/mma.5537
[10]	P. Roul, V. M. K. P. Goura, A sixth order optimal B-spline collocation method for solving Bratus problem, J. Math. Chem., 58 (2020), 967–988. https://doi.org/10.1007/s10910-020-01105-6 doi: 10.1007/s10910-020-01105-6
[11]	S. Tanaka, S. Sadamoto, S. Okazawa, Nonlinear thin-plate bending analyses using the Hermite reproducing kernel approximation, Int. J. Comput. Methods, 9 (2012), 1240012. https://doi.org/10.1142/S0219876212400129 doi: 10.1142/S0219876212400129
[12]	W. Jiang, M. Cui, Solving nonlinear singular pseudoparabolic equations with nonlocal mixed conditions in the reproducing kernel space, Int. J. Comput. Math., 87 (2010), 3430–3442. https://doi.org/10.1080/00207160903082397 doi: 10.1080/00207160903082397
[13]	M. Xu, E. Tohidi, J. Niu, Y. Fang, A new reproducing kernel-based collocation method with optimal convergence rate for some classes of BVPs, Appl. Math. Comput., 432 (2022), 127343. https://doi.org/10.1016/j.amc.2022.127343 doi: 10.1016/j.amc.2022.127343
[14]	X. Y. Li, B. Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math., 243 (2013), 10–15. https://doi.org/10.1016/j.cam.2012.11.002 doi: 10.1016/j.cam.2012.11.002
[15]	F. Z. Geng, S. P. Qian, A new reproducing kernel method for linear nonlocal boundary value problems, Appl. Math. Comput., 248 (2014), 421–425. https://doi.org/10.1016/j.amc.2014.10.002 doi: 10.1016/j.amc.2014.10.002
[16]	F. Z. Geng, M. G. Cui, Multi-point boundary value problem for optimal bridge design, Int. J. Comput. Math., 87 (2010), 1051–1056. https://doi.org/10.1080/00207160903023573 doi: 10.1080/00207160903023573
[17]	Y. Jia, M. Xu, Y. Lin, D. Jiang, An efficient technique based on least-squares method for fractional integro-differential equations, Alex. Eng. J., 64 (2023), 97–105. https://doi.org/10.1016/j.aej.2022.08.033 doi: 10.1016/j.aej.2022.08.033
[18]	M. Xu, L. Zhang, E. Tohidi, An efficient method based on least-squares technique for interface problems, Appl. Math. Lett., 136 (2022), 108475. https://doi.org/10.1016/j.aml.2022.108475 doi: 10.1016/j.aml.2022.108475
[19]	M. Xu, L. Zhang, E. Tohidi, A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems, Appl. Numer. Math., 162 (2021), 124–136. https://doi.org/10.1016/j.apnum.2020.12.015 doi: 10.1016/j.apnum.2020.12.015
[20]	M. Moustafa, Y. H. Youssri, A. G. Atta, Explicit Chebyshev-Galerkin scheme for the time-fractional diffusion equation, Int. J. Mod. Phys. C, 35 (2023), 0025. https://doi.org/10.1142/S0129183124500025 doi: 10.1142/S0129183124500025
[21]	G. A. Ahmed, Y. H. Youssri, Shifted second-Kind Chebyshev spectral collocation-based technique for time-fractional KdV-Burgers equation, Iran. J. Math. Chem., 4 (2023), 207–224. https://doi.org/10.22052/IJMC.2023.252824.1710 doi: 10.22052/IJMC.2023.252824.1710
[22]	M. Abdelhakem, D. Abdelhamied, M. El-kady, Y. H. Youssri, Enhanced shifted Tchebyshev operational matrix of derivatives: two spectral algorithms for solving even-order BVPs, J. Appl. Math. Comput., 69 (2023), 3893–3909. https://doi.org/10.1007/s12190-023-01905-4 doi: 10.1007/s12190-023-01905-4
[23]	Y. H. Youssri, M. I. Ismail, A. G. Atta, Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions, Phys. Scr., 99 (2024), 015251. https://doi.org/10.1088/1402-4896/ad1700 doi: 10.1088/1402-4896/ad1700
[24]	R. M. Hafez, Y. H. Youssri, A. G. Atta, Jacobi rational operational approach for time-fractional sub-diffusion equation on a semi-infinite domain, Contemp. Math., 4 (2023), 853–876. https://doi.org/10.37256/cm.4420233594 doi: 10.37256/cm.4420233594
[25]	Y. Zheng, Y. Lin, Y. Shen, A new multiscale algorithm for solving second order boundary value problems, Appl. Numer. Math., 156 (2020), 528–541. https://doi.org/10.1016/j.apnum.2020.05.020 doi: 10.1016/j.apnum.2020.05.020
[26]	Y. Zhang, L. Mei, Y. Lin, A new method for high-order boundary value problems, Bound. Value Probl., 2021 (2021), 48. https://doi.org/10.1186/s13661-021-01527-4 doi: 10.1186/s13661-021-01527-4
[27]	Y. Zhang, L. Mei, Y. Lin, A novel method for nonlinear boundary value problems based on multiscale orthogonal base, Int. J. Comput. Methods, 18 (2021), 2150036. https://doi.org/10.1142/S0219876221500365 doi: 10.1142/S0219876221500365
[28]	Y. Zhang, H. Sun. Y. Jia, Y. Lin, An algorithm of the boundary value problem based on multiscale orthogonal compact base, Appl. Math. Lett., 101 (2020), 106044. https://doi.org/10.1016/j.aml.2019.106044 doi: 10.1016/j.aml.2019.106044
[29]	B. Wu, Y. Lin. Application of the reproducing kernel space, Science Press, 2012
[30]	X. Luo, L. Liu, Solving two-point boundary value problem with the cubic B-spline interpolation method, Henan Sci., 26 (2008), 1–4. https://doi.org/10.13537/j.issn.1004-3918.2008.04.002 doi: 10.13537/j.issn.1004-3918.2008.04.002

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