×

Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets. (English) Zbl 1459.65243

Summary: We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss-Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval \([- 1, 1]\), by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss-Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme.

MSC:

65R20 Numerical methods for integral equations
45D05 Volterra integral equations
65T60 Numerical methods for wavelets
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Allaei, SS; Yang, Z-W; Brunner, H., Collocation methods for third-kind VIEs, IMA J. Numer. Anal., 37, 3, 1104-1124 (2017) · Zbl 1433.65345
[2] Vainikko, G., Cordial Volterra integral equations I, Numer. Funct. Anal. Optim., 30, 9-10, 1145-1172 (2009) · Zbl 1195.45004 · doi:10.1080/01630560903393188
[3] Vainikko, G., Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim., 31, 1-3, 191-219 (2010) · Zbl 1194.65152 · doi:10.1080/01630561003666234
[4] Nemati, S., Lima, P.M.: Numerical solution of a third-kind Volterra integral equation using an operational matrix technique. In: 2018 European Control Conference (ECC), Limassol, pp 3215-3220 (2018)
[5] Sidi Ammi, M. R., Torres, D. F. M.: Analysis of fractional integro-differential equations of thermistor type. In: Kochubei, A., Luchko, Y. (eds.) Handbook of Fractional Calculus with Applications, Vol. 1, De Gruyter, Berlin, Boston, 327-346 (2019)
[6] Grossmann, A.; Morlet, J., Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math Anal., 15, 4, 723-736 (1984) · Zbl 0578.42007 · doi:10.1137/0515056
[7] Grossmann, A.; Morlet, J.; Paul, T., Transforms associated to square integrable group representations, J. Math. Phys., 26, 2473-2479 (1985) · Zbl 0571.22021 · doi:10.1063/1.526761
[8] Daubechies, I.; Lagarias, JC, Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal., 23, 4, 1031-1079 (1992) · Zbl 0788.42013 · doi:10.1137/0523059
[9] Mallat, S., A theory of multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11, 7, 674-693 (1989) · Zbl 0709.94650 · doi:10.1109/34.192463
[10] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl Math., 41, 7, 909-996 (1988) · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[11] Li, X., Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun. Nonlinear Sci. Numer. Simul., 17, 10, 3934-3946 (2012) · Zbl 1250.65094 · doi:10.1016/j.cnsns.2012.02.009
[12] Chen, YM; Yi, MX; Yu, CX, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, J. Comput. Sci., 3, 5, 367-373 (2012) · doi:10.1016/j.jocs.2012.04.008
[13] Li, Y., Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul., 15, 9, 2284-2292 (2010) · Zbl 1222.65087 · doi:10.1016/j.cnsns.2009.09.020
[14] Jafari, H.; Yousefi, SA; Firoozjaee, MA; Momani, S.; Khalique, CM, Application of Legendre wavelets for solving fractional differential equations, Comput. Math. Appl., 62, 3, 1038-1045 (2011) · Zbl 1228.65253 · doi:10.1016/j.camwa.2011.04.024
[15] Rahimkhani, P.; Ordokhani, Y.; Babolian, E., Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math., 309, 493-510 (2017) · Zbl 1468.65089 · doi:10.1016/j.cam.2016.06.005
[16] Nemati, S.; Lima, PM; Sedaghat, S., Legendre wavelet collocation method combined with the Gauss-Jacobi quadrature for solving fractional delay-type integro-differential equations, Appl. Numer. Math., 149, 99-112 (2020) · Zbl 1464.65125 · doi:10.1016/j.apnum.2019.05.024
[17] Boggess, A.; Narcowich, FJ, A First Course in Wavelets with Fourier Analysis (2009), NJ: John Wiley & Sons, Inc., Hoboken, NJ · Zbl 1185.42001
[18] Walnut, D.F.: An introduction to wavelet analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston MA (2002) · Zbl 0989.42014
[19] Shen, J.; Tang, T.; Wang, L-L, Spectral Methods, Springer Series in Computational Mathematics, vol. 41 (2011), Heidelberg: Springer, Heidelberg · Zbl 1227.65117
[20] Pang, G.; Chen, W.; Sze, KY, Gauss-Jacobi-type quadrature rules for fractional directional integrals, Comput Math. Appl., 66, 5, 597-607 (2013) · Zbl 1350.65019 · doi:10.1016/j.camwa.2013.04.020
[21] Canuto, C.; Hussaini, MY; Quarteroni, A.; Zang, TA, Spectral Methods, Scientific Computation (2006), Berlin: Springer-Verlag, Berlin · Zbl 1093.76002 · doi:10.1007/978-3-540-30726-6
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.