A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

doi:10.1007/s42967-022-00206-0

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1406-1421.doi: 10.1007/s42967-022-00206-0

• ORIGINAL PAPERS • Previous Articles Next Articles

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

Hongfang Bai¹, Ieng Tak Leong²

1 Department of Mathematics, College of Medical Information Engineering, Guangdong Pharmaceutical University, Guangzhou 510006, Guangdong, China;
2 Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

Received:2022-01-19 Revised:2022-06-08 Published:2023-12-16
Contact: Hongfang Bai,E-mail:byebyenever@163.com;Ieng Tak Leong,E-mail:itleong@um.edu.mo E-mail:byebyenever@163.com;itleong@um.edu.mo

Abstract

Abstract: In this paper, the weak pre-orthogonal adaptive Fourier decomposition (W-POAFD) method is applied to solve fractional boundary value problems (FBVPs) in the reproducing kernel Hilbert spaces (RKHSs) W₀⁴[0, 1] and W¹[0, 1]. The process of the W-POAFD is as follows: (i) choose a dictionary and implement the pre-orthogonalization to all the dictionary elements; (ii) select points in [0, 1] by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively; (iii) express the analytical solution as a linear combination of these determined dictionary elements. Convergence properties of numerical solutions are also discussed. The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.

Key words: Weak pre-orthogonal adaptive Fourier decomposition(W-POAFD), Weak maximal selection principle, Fractional boundary value problems(FBVPs), Reproducing kernel Hilbert space(RKHS)

CLC Number:

Hongfang Bai, Ieng Tak Leong. A Sparse Kernel Approximate Method for Fractional Boundary Value Problems[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1406-1421.

TrendMD

References

1. Babolian, E., Javadi, S., Moradi, E.: RKM for solving Bratu-type differential equations of fractional order. Math. Methods Appl. Sci. 39, 1548-1557 (2016)
2. Cardone, A., Conte, D., Paternoster, B.: Two-step collocation methods for fractional differential equations. Discrete Contin. Dyn. Syst. Ser. B 23, 2709-2725 (2018)
3. Cui, M.G., Lin, Y.Z.: Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publisher, New York (2009)
4. Ghaziani, R.K., Fardi, M., Ghasemi, M.: Solving multi-order fractional differential equations by reproducing kernel Hilbert space method. Comput. Methods Differ. Equ. 4, 170-190 (2016)
5. Jafari, H., Khalique, C.M., Ramezani, M., Tajadodi, H.: Numerical solution of fractional differential equations by using fractional B-spline. Centr. Eur. J. Phys. 11, 1372-1376 (2013)
6. Li, X.Y., Wu, B.Y.: Approximate analytical solutions of nonlocal fractional boundary value problems. Appl. Math. Model. 39, 1717-1724 (2015)
7. Milici, C., Drăgănescu, G., Machado, J.T.: Introduction to Fractional Differential Equations. Springer Nature Switzerland AG, Switzerland (2019)
8. Odibat, Z., Momani, S.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl. 58, 2199-2208 (2009)
9. Qian, T.: Two-dimensional adaptive Fourier decomposition. Math. Methods Appl. Sci. 39, 2431- 2448 (2016)
10. Qian, T.: A novel Fourier theory on non-linear phases and applications. Adv. Math. China 47, 321- 347 (2018)
11. Qian, T.: Reproducing kernel sparse representations in relation to operator equations. Complex Anal. Oper. Theory 14, 1-15 (2020)
12. Qian, T., Wang, Y.B.: Adaptive Fourier series—a variation of greedy algorithm. Adv. Comput. Math. 34, 279-293 (2011)
13. Saha Raya, S., Bera, R.K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167, 561-571 (2005)
14. Sakar, M.G.: Iterative reproducing kernel Hilbert spaces method for Riccati differential equation. J. Comput. Appl. Math. 309, 163-174 (2017)
15. Saldir, O., Sakar, M.G., Erdogan, F.: Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate. Comput. Appl. Math. 38, 198 (2019)
16. Zurigat, M., Momani, S., Odibat, Z., Alawneh, A.: The homotopy analysis method for handling systems of fractional differential equations. Appl. Math. Model. 34, 24-35 (2010)

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 0

Recommended Articles

Metrics

Comments