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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 1/2020

04.07.2019

Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations

verfasst von: G. Chandhini, K. S. Prashanthi, V. Antony Vijesh

Erschienen in: BIT Numerical Mathematics | Ausgabe 1/2020

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Abstract

In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and uniqueness for fractional order initial value problem is given. This convergence analysis also proves quadratic convergence of the generalized quasilinearization method. Both the schemes are compared in terms of accuracy and convergence and it is found that IRBF scheme handles inherent RBF ill-condition better than corresponding DRBF method. Variety of numerical examples are provided and compared with other available results to confirm the efficiency of the schemes.
Vorheriger Artikel Perturbation analysis of an eigenvector-dependent nonlinear eigenvalue problem with applications
Nächster Artikel Numerical upscaling of discrete network models

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Literatur
1.
Antunes, P.R.S., Ferreira, R.A.C.: An augmented-RBF method for solving fractional Sturm–Liouville eigenvalue problems. SIAM J. Sci. Comput. 37(1), A515–A535 (2015)CrossRefMathSciNetMATH
2.
Antunes, P.R.S., Ferreira, R.A.C.: Analysis of a class of boundary value problems depending on left and right Caputo fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 48, 398–413 (2017)CrossRefMathSciNet
3.
4.
Bagley, R.L., Torvik, J.: Fractional calculus—a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983)CrossRefMATH
5.
Chen, W., Fu, Z.J., Chen, C.S.: Recent Advances in Radial Basis Function Collocation Methods. Springer Briefs in Applied Sciences and Technology. Springer, Heidelberg (2014)CrossRef
6.
7.
Dehghan, M., Nikpour, A.: Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Appl. Math. Model. 37(18–19), 8578–8599 (2013)CrossRefMathSciNetMATH
8.
Denton, Z., Ng, P., Vatsala, A.: Quasilinearization method via lower and upper solutions for Riemann–Liouville fractional differential equation. Nonlinear Dyn. Syst. Theory 11(3), 239–252 (2011)MathSciNetMATH
9.
Devi, J.V., McRae, F.A., Drici, Z.: Generalized quasilinearization for fractional differential equations. Comput. Math. Appl. 59, 1057–1062 (2010)CrossRefMathSciNetMATH
10.
Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5(1), 1–6 (1997)MathSciNetMATH
11.
Diethelm, K.: The Analysis of Fractional Differential Equations. CRC Press, Boca Raton (2010)CrossRefMATH
12.
Diethelm, K.: Increasing the efficiency of shooting methods for terminal value problems of fractional order. J. Comput. Phys. 293, 135–141 (2015)CrossRefMathSciNetMATH
13.
Drábek, P., Milota, J.: Methods of nonlinear analysis. In: Applications to Differential Equations. Birkhäuser, Basel (2013)
14.
Esmaeili, S., Shamsi, M., Dehghan, M.: Numerical solution of fractional differential equations via a Volterra integral equation approach. Cent. Eur. J. Phys. 11(10), 1470–1481 (2013)
15.
Fakhr Kazemi, B., Ghoreishi, F.: Error estimate in fractional differential equations using multiquadratic radial basis functions. J. Comput. Appl. Math. 245, 133–147 (2013)CrossRefMathSciNetMATH
16.
Fasshauer, G.F.: Meshfree approximation methods with MATLAB. In: Interdisciplinary Mathematical Sciences, vol. 6. World Scientific, Singapore (2007)
17.
Firoozjaee, M., Yousefi, S., Jafari, H., Baleanu, D.: On a numerical approach to solve multi-order fractional differential equations with initial/boundary conditions. J. Comput. Nonlinear Dyn. 10(6), 061025(1-6) (2015)
18.
Franke, R.: Scattered data interpolation: tests of some methods. Math. Comput. 38(157), 181–200 (1982)MathSciNetMATH
19.
Garrappa, R.: Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simul. 110, 96–112 (2015)CrossRefMathSciNet
20.
Ghehsareh, H.R., Heydari Bateni, S., Zaghian, A.: A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation. Eng. Anal. Bound. Elem. 61, 52–60 (2015)CrossRefMathSciNetMATH
21.
Hamarsheh, M., Ismail, A.M., et al.: Analytical approximation for fractional order logistic equation. Int. J. Pure Appl. Math. 115(2), 225–245 (2017)CrossRef
22.
Huber, S.E., Trummer, M.R.: Radial basis functions for solving differential equations: ill-conditioned matrices and numerical stability. Comput. Math. Appl. 71(1), 319–327 (2016)CrossRefMathSciNet
23.
Jaishankar, A., McKinley, G.H.: Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2149), 20120284 (2013)CrossRefMathSciNetMATH
24.
Jang, B.: Efficient analytic method for solving nonlinear fractional differential equations. Appl. Math. Model. 38(5), 1775–1787 (2014)CrossRefMathSciNetMATH
25.
Kansa, E.J.: Multiquadrics–a scattered data approximation scheme with applications to computational fluid-dynamics—II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8), 147–161 (1990)CrossRefMathSciNetMATH
26.
Kazem, S.: An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl. Math. Modell. 37(3), 1126–1136 (2013)CrossRefMathSciNetMATH
27.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)CrossRefMATH
28.
Lakshmikantham, V., Carl, S., Heikkilä, S.: Fixed point theorems in ordered Banach spaces via quasilinearization. Nonlinear Anal. 71(7–8), 3448–3458 (2009)CrossRefMathSciNetMATH
29.
Li, C., Yi, Q., Chen, A.: Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316, 614–631 (2016)CrossRefMathSciNetMATH
30.
31.
Magin, R., Ovadia, M.: Modeling the cardiac tissue electrode interface using fractional calculus. J. Vib. Control 14(9–10), 1431–1442 (2008)CrossRefMATH
32.
Mai-Duy, N.: Solving high order ordinary differential equations with radial basis function networks. Int. J. Numer. Methods Eng. 62(6), 824–852 (2005)CrossRefMathSciNetMATH
33.
Mai-Duy, N., Tran-Cong, T.: Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw. 14(2), 185–199 (2001)CrossRefMATH
34.
35.
Pang, G., Chen, W., Fu, Z.: Space-fractional advection-dispersion equations by the Kansa method. J. Comput. Phys. 293, 280–296 (2015)CrossRefMathSciNetMATH
36.
Parand, K., Abbasbandy, S., Kazem, S., Rad, J.: A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation. Commun. Nonlinear Sci. Numer. Simul. 16(3), 4250–4258 (2011)CrossRefMathSciNetMATH
37.
Piret, C., Hanert, E.: A radial basis functions method for fractional diffusion equations. J. Comput. Phys. 238, 71–81 (2013)CrossRefMathSciNetMATH
38.
Qu, H., Liu, X.: A numerical method for solving fractional differential equations by using neural network. Adv. Math. Phys. pp. Art. ID 439526, 1–12 (2015)
39.
Saeed, U., ur Rehman, M.: Haar wavelet-quasilinearization technique for fractional nonlinear differential equations. Appl. Math. Comput. 220, 630–648 (2013)
40.
Schneider, W.: Completely monotone generalised Mittag-Leffler functions. Expos. Math. 14, 3–16 (1996)MATH
41.
Sheng, H., Chen, Y., Qiu, T.: Fractional processes and fractional-order signal processing. In: Signals and Communication Technology. Springer, London (2012)
42.
Shu, C., Ding, H., Yeo, K.: Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 192(7–8), 941–954 (2003)CrossRefMATH
43.
Tien, C.M.T., Mai-Duy, N., Tran, C.D., Tran-Cong, T.: A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations. Comput. Math. Appl. 72(9), 2364–2387 (2016)CrossRefMathSciNetMATH
44.
Vijesh, V.A., Kumar, K.: Wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems. Appl. Math. Comput. 266, 1163–1176 (2015)MathSciNetMATH
45.
46.
Wright, G.B., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212(1), 99–123 (2006)CrossRefMathSciNetMATH
Metadaten
Titel
Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations
verfasst von
G. Chandhini
K. S. Prashanthi
V. Antony Vijesh
Publikationsdatum
04.07.2019
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 1/2020
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-019-00766-3

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