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Numerical Algorithms
Published in: Numerical Algorithms 2/2020

12-07-2019 | Original Paper

Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels

Authors: Junjie Ma, Huilan Liu

Published in: Numerical Algorithms | Issue 2/2020

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Abstract

We discuss the numerical solution to a class of weakly singular Volterra integral equations in this paper. Firstly, the fractional Lagrange interpolation is applied to deal with the singularity of the solution, and efficient fractional collocation boundary value methods are developed. Secondly, local convergence estimates are derived from examining the asymptotic property of the solution and the interpolation remainder. We find that the second kind Volterra integral equation with a weakly singular kernel can be efficiently solved on a uniform grid. Finally, several numerical examples are given to illustrate the performance of fractional collocation boundary value methods.
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Footnotes
1
In the remaining part, we will denote the various constant to be B for simplicity.
 
Literature
1.
Roberts, C.A., Lasseigne, D.G., Olmstead, W.E.: Volterra equations which model explosion in a diffusive medium. J. Integr. Eq. Appl. 5(4), 531–546 (1993)MathSciNetMATHCrossRef
2.
Olmstead, W.E., Roberts, C.A.: Explosion in a diffusive strip due to a source with local and nonlocal features. Methods Appl. Anal. 3(3), 345–357 (1996)MathSciNetMATH
3.
Brunner, H.: Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge University Press, Cambridge (2017)MATHCrossRef
4.
5.
6.
Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighboring solutions intersect? J. Integr. Eq. Appl. 24(1), 25–37 (2012)MathSciNetMATHCrossRef
7.
Esmaeili, S., Shamsi, M., Dehghan, M.: Numerical solution of fractional differential equations via a Volterra integral equation approach. Central Eur. J. Phys. 11(10), 1470–1481 (2013)
8.
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)MATHCrossRef
9.
Diogo, T.: Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations. J. Comput. Appl. Math. 229(2), 363–372 (2009)MathSciNetMATHCrossRef
10.
Zhao, J., Xiao, J., Ford, N.J.: Collocation methods for fractional integro-differential equations with weakly singular kernels. Numer. Algorithm. 65(4), 723–743 (2014)MathSciNetMATHCrossRef
11.
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)MATHCrossRef
12.
Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)MathSciNetMATHCrossRef
13.
Huang, C., Stynes, M.: A spectral collocation method for a weakly singular Volterra integral equation of the second kind. Adv. Comput. Math. 42(5), 1015–1030 (2016)MathSciNetMATHCrossRef
14.
Brunner, H., Linz, P.: Analytical and numerical methods for Volterra equations. Math. Comput. 48(178), 841 (1987)CrossRef
15.
Berrut, J.P., Hosseini, S.A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36(1), A105–A123 (2014)MathSciNetMATHCrossRef
16.
Li, M., Huang, C.: The linear barycentric rational quadrature method for auto-convolution Volterra integral equations. J. Sci. Comput. 78(1), 549–564 (2019)MathSciNetMATHCrossRef
17.
Lubich, C. h.: Runge-Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comput. 41(163), 87–102 (1983)MathSciNetMATHCrossRef
18.
Crisci, M.R., Jackiewicz, Z., Russo, E., Vecchio, A.: Global stability analysis of the Runge-Kutta methods for Volterra integral and integro-differential equations with degenerate kernels. Computing 45(4), 291–300 (1991)MathSciNetMATHCrossRef
19.
20.
van der Houwen, P.J., Riele, H.J.J.: Linear multistep methods for Volterra integral equations of the second kind. Queueing Systems (1982)
21.
Lubich, C. h.: Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45(172), 463–469 (1985)MathSciNetMATHCrossRef
22.
Garrappa, R.: On some explicit Adams multistep methods for fractional differential equations. J. Comput. Appl. Math. 229(2), 392–399 (2009)MathSciNetMATHCrossRef
23.
Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45(172), 417–437 (1985)MathSciNetMATHCrossRef
24.
Diogo, T., McKee, S., Tang, T.: Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc. R. Soc. Edinb. 124, 199–210 (1994)MathSciNetMATHCrossRef
25.
Brunner, H.: Nonpolynomial spline collocation for Volterra equations with weakly singular kernels. SIAM J. Numer. Anal. 20(6), 1106–1119 (1983)MathSciNetMATHCrossRef
26.
Cao, Y., Herdman, T., Xu, Y.: A hybrid collocation method for Volterra integral equations with weakly singular kernels. SIAM J. Numer. Anal. 41(1), 364–381 (2003)MathSciNetMATHCrossRef
27.
Rebelo, M., Diogo, T.: A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel. J. Comput. Appl. Math. 234(9), 2859–2869 (2010)MathSciNetMATHCrossRef
28.
Ford, N.J., Morgado, M.L., Rebelo, M.: Nonpolynomial collocation approximation of solutions to fractional differential equations. Fract. Calc. Appl. Anal. 16(4), 874–891 (2013)MathSciNetMATHCrossRef
29.
Ford, N.J., Morgado, M.L., Rebelo, M.: A nonpolynomial collocation method for fractional terminal value problems. J. Comput. Appl. Math. 275(1), 392–402 (2015)MathSciNetMATHCrossRef
30.
Pedas, A., Vainikko, G.: Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations. Computing 73(3), 271–293 (2004)MathSciNetMATHCrossRef
31.
Diogo, T., Lima, P.M., Pedas, A., Vainikko, G.: Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations. Appl. Numer. Math. 114, 63–76 (2017)MathSciNetMATHCrossRef
32.
Hou, D., Xu, C.: A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43(5), 911–944 (2017)MathSciNetMATHCrossRef
33.
Pedas, A., Tamme, E., Vikerpuur, M.: Smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems. J. Comput. Appl. Math. 317, 1–16 (2017)MathSciNetMATHCrossRef
34.
Cai, H., Chen, Y.: A fractional order collocation method for second kind Volterra integral equations with weakly singular kernels. J. Sci. Comput. 75(2), 970–992 (2018)MathSciNetMATHCrossRef
35.
Brugnano, L., Trigiante, D.: Solving differential problems by multistep initial and boundary value methods. Cordon and Breach Science Publishers (1998)
36.
Lopez, L., Trigiante, D.: Boundary value methods and BV-stability in the solution of initial value problems. Appl. Numer. Math. 11(1), 225–239 (1993)MathSciNetMATHCrossRef
37.
Chen, H., Zhang, C.: Boundary value methods for Volterra integral and integro-differential equations. Appl. Math. Comput. 218(6), 2619–2630 (2011)MathSciNetMATH
38.
Ma, J., Xiang, S.: A collocation boundary value method for linear Volterra integral equations. J. Sci. Comput. 71(1), 1–20 (2017)MathSciNetMATHCrossRef
39.
40.
Olver, F., Lozier, D., Boisvert, R., Clark, C.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)MATH
Metadata
Title
Fractional collocation boundary value methods for the second kind Volterra equations with weakly singular kernels
Authors
Junjie Ma
Huilan Liu
Publication date
12-07-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 2/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00777-9

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