Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis. (English) Zbl 1369.65174
Summary: In this paper, a spectral tau method based on Legendre Wavelet basis is proposed. For this purpose we present a stable operational tau method based on Legendre Wavelet basis. This method provides an efficient approximate solution for weakly singular Volterra integral equations by using reduced set of matrix operations. An error estimation of the tau method is also introduced. Finally we demonstrate the validity and applicability of the method by numerical examples.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45D05 | Volterra integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 65T60 | Numerical methods for wavelets |
Keywords:
weakly singular Volterra integral equations; tau method; Legendre wavelet basis; error estimation; numerical examplesReferences:
| [1] | DOI: 10.1016/j.apnum.2014.02.013 · Zbl 1291.65375 · doi:10.1016/j.apnum.2014.02.013 |
| [2] | DOI: 10.1016/j.cam.2003.08.047 · Zbl 1038.65144 · doi:10.1016/j.cam.2003.08.047 |
| [3] | DOI: 10.1016/0168-9274(92)90018-9 · Zbl 0761.65103 · doi:10.1016/0168-9274(92)90018-9 |
| [4] | Canuto C., Spectral Methods. Fundamentals in Single Domains (2006) · Zbl 0675.65117 |
| [5] | DOI: 10.1023/A:1018994802659 · Zbl 0915.65134 · doi:10.1023/A:1018994802659 |
| [6] | DOI: 10.1090/S0025-5718-97-00803-X · Zbl 0855.47006 · doi:10.1090/S0025-5718-97-00803-X |
| [7] | DOI: 10.1016/j.camwa.2014.03.016 · Zbl 1367.65147 · doi:10.1016/j.camwa.2014.03.016 |
| [8] | DOI: 10.1080/00207729608929258 · Zbl 0875.93116 · doi:10.1080/00207729608929258 |
| [9] | DOI: 10.1016/S0096-3003(02)00081-4 · Zbl 1027.65182 · doi:10.1016/S0096-3003(02)00081-4 |
| [10] | Jerri A., Introduction to Integral Equations with Applications (1999) · Zbl 0938.45001 |
| [11] | Karimi Vanani S., Math. Comput. 57 pp 494– (2013) |
| [12] | DOI: 10.1090/S0025-5718-1987-0878692-4 · doi:10.1090/S0025-5718-1987-0878692-4 |
| [13] | DOI: 10.1016/j.cam.2011.01.043 · Zbl 1216.65185 · doi:10.1016/j.cam.2011.01.043 |
| [14] | Lardy L.J., J. Int. Eqs. 3 pp 43– (1981) |
| [15] | DOI: 10.1007/s11464-012-0170-0 · Zbl 1260.65111 · doi:10.1007/s11464-012-0170-0 |
| [16] | Mandal B.N., Applied Singular Integral Equations (2011) · Zbl 1234.45002 |
| [17] | DOI: 10.1007/BF02243435 · Zbl 0449.65053 · doi:10.1007/BF02243435 |
| [18] | DOI: 10.1016/S0377-0427(00)00349-6 · Zbl 0966.65103 · doi:10.1016/S0377-0427(00)00349-6 |
| [19] | DOI: 10.1080/00207720120227 · doi:10.1080/00207720120227 |
| [20] | DOI: 10.1093/imanum/2.4.437 · Zbl 0501.65062 · doi:10.1093/imanum/2.4.437 |
| [21] | Saadatmandi A., Z. Naturforsch. 63 (10) pp 752– (2008) |
| [22] | DOI: 10.1002/num.20442 · Zbl 1186.65136 · doi:10.1002/num.20442 |
| [23] | Samko S.G., Fractional Integrals and Derivatives: Theory and Applications (1993) · Zbl 0818.26003 |
| [24] | DOI: 10.1007/978-3-540-71041-7 · Zbl 1227.65117 · doi:10.1007/978-3-540-71041-7 |
| [25] | Tikhonov A.N., Solution of Ill-Posed Problems (1977) |
| [26] | Zeilon N., Arkiv. Mat. Astr. Fysik. 18 pp 1– (1924) |
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.
