A generalized barycentric rational interpolation method for generalized Abel integral equations. (English) Zbl 1460.65159
Summary: The paper is devoted to the numerical solution of generalized Abel integral equation. First, the generalized barycentric rational interpolants have been introduced and their properties investigated thoroughly. Then, a numerical method based on these barycentric rational interpolations and the Legendre-Gauss quadrature rule is developed for solving the generalized Abel integral equation. The main advantages of the presented method is that it provides an infinitely smooth approximate solution with no real poles for the generalized Abel integral equation.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45D05 | Volterra integral equations |
| 41A20 | Approximation by rational functions |
| 65D32 | Numerical quadrature and cubature formulas |
Keywords:
generalized Abel integral equation; generalized barycentric rational interpolation; Legendre-Gauss quadrature; error analysisReferences:
| [1] | Gorenflo, R.; Vessella, S., Abel Integral Equation (1991), Berlin: Springer, Berlin · Zbl 0717.45002 |
| [2] | Sadri, K.; Amini, A.; Cheng, C., A new operational method to solve Abel’s and generalized Abel’s integral equations, Appl. Math. Comput., 317, 49-67 (2018) · Zbl 1426.65218 |
| [3] | Mohammadi, F.; Machado, JT, A comparative study of integer and non-integer order wavelets for fractional nonlinear Fredholm integro-differential equations, J. Comput. Nonlinear Dyn. (2018) |
| [4] | Mohammadi, F., An efficient fractional-order wavelet method for fractional Volterra integro-differential equations, Int. J. Comput. Math., 95, 12, 2396-2418 (2017) · Zbl 1499.65285 |
| [5] | Singh, I.; Kumar, S., Haar wavelet method for some nonlinear Volterra integral equations of the first kind, J. Comput. Appl. Math., 292, 541-552 (2016) · Zbl 1327.65284 |
| [6] | Heydari, MH, Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations, Appl. Math. Comput., 144, 190-203 (2019) · Zbl 1433.65352 |
| [7] | Avazzadeh, Z.; Heydari, MH; Cattani, C., Legendre wavelets for fractional partial integro-differential viscoelastic equations with weakly singular kernels, Eur. Phys. J. Plus, 134, 7, 368 (2019) |
| [8] | Muftahov, I.; Tynda, A.; Sidorov, D., Numeric solution of Volterra integral equations of the first kind with discontinuous kernels, J. Comput. Appl. Math., 313, 119-128 (2017) · Zbl 1353.65137 |
| [9] | Dezhbord, A.; Lotfi, T.; Mahdiani, K., A new efficient method for cases of the singular integral equation of the first kind, J. Comput. Appl. Math., 296, 156-169 (2016) · Zbl 1342.65240 |
| [10] | Assari, P.; Dehghan, M., The approximate solution of nonlinear Volterra integral equations of the second kind using radial basis functions, Appl. Numer. Math., 131, 140-157 (2018) · Zbl 1446.65205 |
| [11] | 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) · Zbl 1391.45002 |
| [12] | Assari, P.; Dehghan, M., Solving a class of nonlinear boundary integral equations based on the meshless local discrete Galerkin (MLDG) method, Appl. Numer. Math., 123, 137-158 (2018) · Zbl 1377.65158 |
| [13] | Zhang, T.; Liang, H., Multistep collocation approximations to solutions of first-kind Volterra integral equations, Appl. Numer. Math., 130, 171-183 (2018) · Zbl 1387.65131 |
| [14] | Pourgholi, R.; Tahmasebi, A.; Azimi, R., Tau approximate solution of weakly singular Volterra integral equations with Legendre wavelet basis, Int. J. Comput. Math., 94, 7, 1337-1348 (2017) · Zbl 1369.65174 |
| [15] | Piessen, R., Computing integral transforms and solving integral equation using Chebyshev polynomial approximations, J. Comput. Appl. Math., 121, 113-124 (2000) · Zbl 0966.65103 |
| [16] | Huang, L.; Huang, Y.; Li, X., Approximate solution of Abel integral equation, Math. Comput. Appl., 56, 1748-1757 (2008) · Zbl 1152.45307 |
| [17] | Yousefi, SA, B-polynomial multiwavelets approach for the solution of Abel’s integral equation, Int. J. Comput. Math., 87, 2, 310316 (2010) · Zbl 1182.65206 |
| [18] | Yousefi, SA, Numerical solution of Abel’s integral by using Legendre wavelets, Appl. Math. Comput., 175, 574-580 (2006) · Zbl 1088.65124 |
| [19] | Gulsu, M.; Ozturk, Y.; Sezer, M., On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials, Appl. Math. Comput., 217, 4827-4833 (2011) · Zbl 1207.65098 |
| [20] | Jahanshahi, S.; Babolian, E.; Torres, D.; Vahidi, A., Solving Abel integral equation of first kind via fractional calculus, J. King Saud Univ. Sci., 27, 161-167 (2015) |
| [21] | Yang, C., An efficient numerical method for solving Abel integral equation, Appl. Math. Comput., 227, 656-661 (2014) · Zbl 1364.65301 |
| [22] | Vogeli, U.; Nedaiasl, K.; Sauter, SA, A fully discrete Galerkin method for Abel-type integral equations, Adv. Comput. Math., 44, 1-26 (2016) |
| [23] | Floater, MS; Hormann, K., Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107, 315-331 (2007) · Zbl 1221.41002 |
| [24] | Berrut, JP; Mittelmann, HD, Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval, Comput. Math. Appl., 33, 6, 77-86 (1997) · Zbl 0893.41009 |
| [25] | Berrut, JP; Trefethen, LN, Barycentric lagrange interpolation, SIAM Rev., 46, 3, 501-517 (2004) · Zbl 1061.65006 |
| [26] | Berrut, JP, Rational functions for guaranteed and experimentally well-conditioned global interpolation, Comput. Math. Appl., 15, 1, 1-16 (1988) · Zbl 0646.65006 |
| [27] | Cirillo, E.; Hormann, K.; Sidon, J., Convergence rates of derivatives of Floater-Hormann interpolants for well-spaced nodes, Appl. Numer. Math., 116, 108-118 (2017) · Zbl 1372.65031 |
| [28] | Berrut, JP; Floater, MS; Klein, G., Convergence rates of derivatives of a family of Barycentric rational interpolants, Appl. Numer. Math., 61, 9, 989-1000 (2011) · Zbl 1222.41011 |
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