×

A new recursive formulation of the Tau method for solving linear Abel-Volterra integral equations and its application to fractional differential equations. (English) Zbl 1432.65202

The paper presents and discusses a numerical scheme for approximately solving a class of singular linear Volterra integral equations subject to certain conditions. The approach is based on the analysis of singularities of the exact solution and exploits this information via the selection of suitable generalized polynomials as Ansatz functions.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
Full Text: DOI

References:

[1] Brunner, H., Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6, 221-239 (1986) · Zbl 0634.65142 · doi:10.1093/imanum/6.2.221
[2] Ma, Jt; Jiang, Yj, On a graded mesh method for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math., 231, 807-814 (2009) · Zbl 1172.65071 · doi:10.1016/j.cam.2009.05.005
[3] Shen, J.; Sheng, Ct; Wang, Zq, Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels, J. Math. Study, 48, 315-329 (2015) · Zbl 1349.65721 · doi:10.4208/jms.v48n4.15.01
[4] Sohrabi, S.; Ranjbar, H.; Saei, M., Convergence analysis of the Jacobi-collocation method for nonlinear weakly singular Volterra integral equations, Appl. Math Comput., 299, 141-152 (2017) · Zbl 1411.65172
[5] Diogo, T.; Franco, Nb; Lima, P., high order product integration methods for a Volterra integral equation with logarithmic singular kernel, Commun. Pure Appl. Anal., 3, 217-235 (2004) · Zbl 1066.65162 · doi:10.3934/cpaa.2004.3.217
[6] Diogo, T., Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math., 229, 363-372 (2009) · Zbl 1168.65073 · doi:10.1016/j.cam.2008.04.002
[7] Yong, Zx, Jacobi spectral method for the second-kind Volterra integral equations with a weakly singular kernel, Appl. Math. Model., 39, 15, 4421-4431 (2015) · Zbl 1443.65448 · doi:10.1016/j.apm.2014.12.046
[8] Chen, Y.; Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79, 147-167 (2010) · Zbl 1207.65157 · doi:10.1090/S0025-5718-09-02269-8
[9] Lighthill, Jm, Contributions to the theory of the heat transfer through a laminar boundary layer, Proc. R. Soc. Lond., 202A, 359-377 (1950) · Zbl 0038.11504
[10] Gorenflo, R.; Vessella, S., Abel Integral Equations: Analysis and Applications (1991), Berlin: Springer, Berlin · Zbl 0717.45002 · doi:10.1007/BFb0084665
[11] De, S.; Mandal, Bn; Chakrabarti, A., Use of Abel integral equations in water wave scattering by two surface piercing barriers, Wave Motion, 47, 5, 279-288 (2010) · Zbl 1231.76030 · doi:10.1016/j.wavemoti.2009.12.002
[12] Kumar, S.; Kumar, A., Analytical solution of Abel integral equation arising in astrophysics via Laplace transform, J. Egypt. Math. Soc., 23, 102-107 (2015) · Zbl 1311.45001 · doi:10.1016/j.joems.2014.02.004
[13] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations Methods (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1059.65122
[14] Sadri, K.; Amini, A.; Cheng, C., A new operational method to solve Abel’s and generalized Abel’s integral equations, Int. J. Appl. Math. Comput., 317, 49-67 (2018) · Zbl 1426.65218
[15] Vanani, Sk; Soleyman, F., Tau approximate solution of weakly singular Volterra integral equations, Math. Comput. Model., 57, 494-502 (2013) · Zbl 1305.65247 · doi:10.1016/j.mcm.2012.07.004
[16] Sahu, Pk; Ray, Ss, A novel Legendre wavelet Petrov-Galerkin method for fractional Volterra integro-differential equations, Comput. Math. Appl. (2016) · doi:10.1016/j.camwa.2016.04.042
[17] Lanczos, C., Trigonometric interpolation of empirical and analytic functions, J. Math. Phys., 17, 123-199 (1938) · Zbl 0020.01301 · doi:10.1002/sapm1938171123
[18] Lanczos, C., Applied Analysis (1956), Englewood Cliffs: Prentice-hall, Englewood Cliffs · Zbl 0111.12403
[19] Ortiz, E., The Tau method, SIAM J. Numer. Anal., 6, 480-492 (1969) · Zbl 0195.45701 · doi:10.1137/0706044
[20] El-Daou, Mk; Al-Hamad, Km, Computation of the canonical polynomials and applications to some optimal control problems, Numer. Algorithms, 61, 545-566 (2012) · Zbl 1257.65039 · doi:10.1007/s11075-012-9550-5
[21] Ortiz, E.; Samara, L., An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27, 1, 15-25 (1981) · Zbl 0449.65053 · doi:10.1007/BF02243435
[22] Pour Mahmoud, J.; Rahimi Ardabili, My; Shahmorad, S., Numerical solution of the system of Fredholm integro-differential equations by the Tau method, Appl. Math. Comput, 168, 465-478 (2005) · Zbl 1082.65600
[23] Shahmorad, S., Numerical solution of the general form linear Fredholm-Volterra integro-differential equations by the Tau method with an error estimation, Appl. Math. Comput., 167, 2, 1418-1429 (2005) · Zbl 1082.65602
[24] Hosseini, Sa; Shahmorad, S.; Talati, F., A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations, Numer. Algorithms, 68, 511-529 (2015) · Zbl 1317.65252 · doi:10.1007/s11075-014-9858-4
[25] Bunchaft, Me, Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the tau method, Math. Comput., 66, 218, 609-621 (1997) · Zbl 0866.65045 · doi:10.1090/S0025-5718-97-00816-8
[26] Pinkus, A., Weierstrass and approximation theory, J. Approx. Theory., 107, 1-66 (2000) · Zbl 0968.41001 · doi:10.1006/jath.2000.3508
[27] Borwein, P.; Erdelyi, T.; Zhang, J., Müntz systems and orthogonal Müntz-Legendre polynomials, Trans. Am. Math. Soc., 342, 2, 523-542 (1994) · Zbl 0799.41015
[28] Milovanovic, Gv, Müntz orthogonal polynomials and their numerical evaluation, Applications and computation of orthogonal polynomials, Int. Ser. Numer. Math., 131, 179-194 (1999) · Zbl 0941.65013
[29] Mokhtary, P.; Ghoreishi, F.; Srivastava, Hm, The Müntz-Legendre Tau method for fractional differential equations, Appl. Math. Model., 40, 671-684 (2016) · Zbl 1446.65041 · doi:10.1016/j.apm.2015.06.014
[30] Abdalkhani, J., A numerical approach to the solution of Abel integral equations of the second kind with nonsmooth solution, J. Comput. Appl. Math., 29, 249-255 (1990) · Zbl 0699.65088 · doi:10.1016/0377-0427(90)90011-N
[31] Micula, S., An iterative numerical method for fractional integral equations of the second kind, J. Comput. Appl. Math., 339, 124-133 (2018) · Zbl 1464.65291 · doi:10.1016/j.cam.2017.12.006
[32] Ghoreishi, F.; Hosseini, Sm, The Tau method and a new preconditioner, J. Comput. Appl. Math., 163, 351-379 (2004) · Zbl 1046.65060 · doi:10.1016/j.cam.2003.04.001
[33] Matos, Jc; Matos, Jma; Rodrigues, Mj, Solving differential and integral equations with Tau method, Math. Comput. Sci., 12, 2, 197-205 (2018) · Zbl 1402.65078 · doi:10.1007/s11786-018-0334-8
[34] Diethelm, K., The Analysis of Fractional Differential Equations. Lectures Notes in Mathematics (2010), Berlin: Springer, Berlin · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2
[35] Zhao, J.; Xiao, J.; Ford, Nj, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms, 65, 4, 723-743 (2014) · Zbl 1298.65197 · doi:10.1007/s11075-013-9710-2
[36] Wang, Y.; Zhu, L.; Wang, Z., Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel, Adv. Differ. Equ. (2018) · Zbl 1446.45009 · doi:10.1186/s13662-018-1699-3
[37] El-Daou, Mk; Ortiz, E., Error analysis of the Tau method: dependence of the error on the degree and on the length of the interval of approximation, Comput. Math. Appl., 25, 7, 33-45 (1993) · Zbl 0772.65054 · doi:10.1016/0898-1221(93)90310-R
[38] Rudin, W., Principles of Mathematical Analysis (1976), New York: McGraw-Hill, New York · Zbl 0346.26002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.