Research Article

Year 2024,
Volume: 4 Issue: 2, 149 - 164, 30.06.2024
### Abstract

### References

In this manuscript, we present and test a numerical scheme with an algorithm to solve Volterra and Abel's integral equations utilizing generalized sampling operators. Illustrative computational examples are included to indicate the validity and practicability of the proposed technique. All of the computational examples in this research have been computed on a personal computer implementing some program coded in MATLAB.

- [1] Archibald, T. and Tazzioli, R. Integral equations between theory and practice: the cases of Italy and France to 1920. Archive for History of Exact Sciences, 68, 547-597, (2014).
- [2] Abel, N.H. Aufloesung einer mechanischen Aufgabe. Journal für die Reine und Angewandte Mathematik (Crelle), 1, 153-157, (1826).
- [3] Gorenflo, R. and Mainardi, F. Fractional Calculus: Integral and Differential Equations of Fractional Order. Springer: Vienna, (1997).
- [4] Moiseiwitsch, B.L. Integral Equations. Longman: London and New York, (1977).
- [5] Babolian, E. and Davari, A. Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind. Applied Mathematics and Computation, 165, 223-227, (2005).
- [6] Izadian, J., Salahshour, S. and Soheil, S. A numerical method for solving Volterra and Fredholm integral equations using homotopy analysis method. AWER Procedia Information Technology & Computer Science, 1, 406-411, (2012).
- [7] Rashidinia, J. and Zarebnia, M. Solution of a Volterra integral equation by the Sinc-collocation method. Journal of Computational and Applied Mathematics, 206(2), 801-813, (2007).
- [8] Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind (Vol. 4). Cambridge University Press: England, (1997).
- [9] Maleknejad, K. and Aghazadeh, N. Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method. Applied Mathematics and Computation, 161(3), 915–922, (2005).
- [10] Maleknejad, K., Tavassoli Kajani, M. and Mahmoudi, Y. Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets. Kybernetes, 32(9/10), 1530–1539, (2003).
- [11] Maleknejad, K., Mollapourasl, R. and Alizadeh, M. Numerical solution of Volterra type integral equation of the first kind with wavelet basis. Applied Mathematics and Computation, 194(2), 400-405, (2007).
- [12] Shoukralla, E.S. and Markos, M.A. The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind. Asian-European Journal of Mathematics, 13(01), 2050030, (2020).
- [13] Wang, W. A mechanical algorithm for solving the Volterra integral equation. Applied Mathematics and Computation, 172(2), 1323-1341, (2006).
- [14] Maleknejad, K., Hashemizadeh, E. and Ezzati, R. A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Communications in Nonlinear Science and Numerical Simulation, 16(2), 647–655, (2011).
- [15] Usta, F., Ilkhan, M. and Kara, E.E. Numerical solution of Volterra integral equations via Szász-Mirakyan approximation method. Mathematical Methods in the Applied Sciences, 44(9), 7491-7500, (2021).
- [16] Usta, F. Numerical analysis of fractional Volterra integral equations via Bernstein approximation method. Journal of Computational and Applied Mathematics, 384, 113198, (2021).
- [17] Usta, F. Bernstein Approximation technique for numerical solution of Volterra integral equations of the third kind. Computational and Applied Mathematics, 40, 161, (2021).
- [18] Acar, T., Cappelletti Montano, M., Garrancho, P. and Leonessa, V. On sequence of J. P. King type operators. Journal of Function Spaces, 2019, 2329060, (2019).
- [19] Butzer, P.L. and Stens, R.L. Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Review, 34(1), 40-53, (1992).
- [20] Butzer, P.L., Fischer, A. and Stens, R.L. Generalized sampling approximation of multivariate signals: general theory. Atti Sem. Mat. Fis. Univ. Modena, 41(1), 17-37, (1993).
- [21] Butzer, P.L., Ries, S. and Stens, R.L. Approximation of continuous and discontinuous functions by generalized sampling series. Journal of Approximation Theory, 50, 25-39, (1987).
- [22] Butzer, P.L. and Stens, R.L. Prediction of non-bandlimited signals from past samples in terms of splines of low degree.Mathematische Nachrichten, 132(1), 115-130, (1987).
- [23] Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press: Cambridge, (1997).
- [24] Paez-Rueda, C.I. and Bustamante-Miller, R. Novel computational approach to solve convolutional integral equations: method of sampling for one dimension, Pontificia Universidad Javeriana: Engineering for Development, 23(2), 1-32, (2019).
- [25] Wazwaz, A.M. A First Course in Integral Equations, World Scientific: Singapore, (1997).
- [26] Ansari, K.J., Sessa, S. and Alam, A. A class of relational functional contractions with applications to nonlinear integral equations. Mathematics, 11(15), 3408, (2023).
- [27] Say, F. Asymptotics of singularly perturbed Volterra type integro-differential equation. Konuralp Journal of Mathematics, 8(2), 365-369, (2020).
- [28] Wang, Q. and Zhou, H. Two-grid iterative method for a class of Fredholm functional integral equations based on the radial basis function interpolation. Fundamental Journal of Mathematics and Applications, 2(2), 117-122, (2019).
- [29] Zhou, H., Wang, Q. The Nyström method and convergence analysis for system of Fredholm integral equations. Fundamental Journal of Mathematics and Applications, 2(1), 28-32, (2019).
- [30] Shestopalov, Y.V. and Smirnov, Y.G. Integral Equations. Karlstad University, 2002.
- [31] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Amsterdam: Gordon and Breach Science Publishers, 1993.
- [32] P.L. Butzer, W. Splettstößer, R.L. Stens, The sampling theorem and linear prediction, Jahresberichte Dt. Math.-Verein, 90 (1988), 1-70.
- [33] Bardaro, C. and Mantellini, I. Asymptotic expansion of generalized Durrmeyer sampling type series. Jaen Journal on Approximation, 6(2), 143-165, (2014).
- [34] Bardaro, C. and Mantellini, I. On linear combinations of multivariate generalized sampling type series. Mediterranean Journal of Mathematics, 10, 1833-1852, (2013).
- [35] Bardaro, C., Butzer, P.L., Stens, R.L. and Vinti, G. Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Transactions on Information Theory, 56(1), 614-633, (2010).
- [36] Bardaro, C., Vinti, G., Butzer, P.L. and Stens, R. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampling Theory in Signal and Image Processing, 6, 29-52, (2007).
- [37] Costarelli, D. and Vinti, G. Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Bollettino dell’Unione Matematica Italiana, 4(3), 445-468, (2011).
- [38] Costarelli, D. and Vinti, G. Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization, 34(8), 819-844, (2013).
- [39] Costarelli, D. and Vinti, G. Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some function spaces. Numerical Functional Analysis and Optimization, 36(8), 964- 990, (2015).
- [40] Bardaro, C. and Mantellini, I. Asymptotic formulae for linear combinations of generalized sampling operators. Zeitschrift für Analysis und ihre Anwendunge, 32, 279-298, (2013).
- [41] Bardaro, C. and Mantellini, I. A quantitative Voronvoskaja formula for generalized sampling operators. East Journal on Approximations, 15(4), 459-471, (2009).
- [42] Bardaro, C. and Vinti, G. Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator. Results in Mathematics, 34, 224-240, (1998).

Year 2024,
Volume: 4 Issue: 2, 149 - 164, 30.06.2024
### Abstract

### References

- [1] Archibald, T. and Tazzioli, R. Integral equations between theory and practice: the cases of Italy and France to 1920. Archive for History of Exact Sciences, 68, 547-597, (2014).
- [2] Abel, N.H. Aufloesung einer mechanischen Aufgabe. Journal für die Reine und Angewandte Mathematik (Crelle), 1, 153-157, (1826).
- [3] Gorenflo, R. and Mainardi, F. Fractional Calculus: Integral and Differential Equations of Fractional Order. Springer: Vienna, (1997).
- [4] Moiseiwitsch, B.L. Integral Equations. Longman: London and New York, (1977).
- [5] Babolian, E. and Davari, A. Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind. Applied Mathematics and Computation, 165, 223-227, (2005).
- [6] Izadian, J., Salahshour, S. and Soheil, S. A numerical method for solving Volterra and Fredholm integral equations using homotopy analysis method. AWER Procedia Information Technology & Computer Science, 1, 406-411, (2012).
- [7] Rashidinia, J. and Zarebnia, M. Solution of a Volterra integral equation by the Sinc-collocation method. Journal of Computational and Applied Mathematics, 206(2), 801-813, (2007).
- [8] Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind (Vol. 4). Cambridge University Press: England, (1997).
- [9] Maleknejad, K. and Aghazadeh, N. Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method. Applied Mathematics and Computation, 161(3), 915–922, (2005).
- [10] Maleknejad, K., Tavassoli Kajani, M. and Mahmoudi, Y. Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelets. Kybernetes, 32(9/10), 1530–1539, (2003).
- [11] Maleknejad, K., Mollapourasl, R. and Alizadeh, M. Numerical solution of Volterra type integral equation of the first kind with wavelet basis. Applied Mathematics and Computation, 194(2), 400-405, (2007).
- [12] Shoukralla, E.S. and Markos, M.A. The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind. Asian-European Journal of Mathematics, 13(01), 2050030, (2020).
- [13] Wang, W. A mechanical algorithm for solving the Volterra integral equation. Applied Mathematics and Computation, 172(2), 1323-1341, (2006).
- [14] Maleknejad, K., Hashemizadeh, E. and Ezzati, R. A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Communications in Nonlinear Science and Numerical Simulation, 16(2), 647–655, (2011).
- [15] Usta, F., Ilkhan, M. and Kara, E.E. Numerical solution of Volterra integral equations via Szász-Mirakyan approximation method. Mathematical Methods in the Applied Sciences, 44(9), 7491-7500, (2021).
- [16] Usta, F. Numerical analysis of fractional Volterra integral equations via Bernstein approximation method. Journal of Computational and Applied Mathematics, 384, 113198, (2021).
- [17] Usta, F. Bernstein Approximation technique for numerical solution of Volterra integral equations of the third kind. Computational and Applied Mathematics, 40, 161, (2021).
- [18] Acar, T., Cappelletti Montano, M., Garrancho, P. and Leonessa, V. On sequence of J. P. King type operators. Journal of Function Spaces, 2019, 2329060, (2019).
- [19] Butzer, P.L. and Stens, R.L. Sampling theory for not necessarily band-limited functions: a historical overview. SIAM Review, 34(1), 40-53, (1992).
- [20] Butzer, P.L., Fischer, A. and Stens, R.L. Generalized sampling approximation of multivariate signals: general theory. Atti Sem. Mat. Fis. Univ. Modena, 41(1), 17-37, (1993).
- [21] Butzer, P.L., Ries, S. and Stens, R.L. Approximation of continuous and discontinuous functions by generalized sampling series. Journal of Approximation Theory, 50, 25-39, (1987).
- [22] Butzer, P.L. and Stens, R.L. Prediction of non-bandlimited signals from past samples in terms of splines of low degree.Mathematische Nachrichten, 132(1), 115-130, (1987).
- [23] Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press: Cambridge, (1997).
- [24] Paez-Rueda, C.I. and Bustamante-Miller, R. Novel computational approach to solve convolutional integral equations: method of sampling for one dimension, Pontificia Universidad Javeriana: Engineering for Development, 23(2), 1-32, (2019).
- [25] Wazwaz, A.M. A First Course in Integral Equations, World Scientific: Singapore, (1997).
- [26] Ansari, K.J., Sessa, S. and Alam, A. A class of relational functional contractions with applications to nonlinear integral equations. Mathematics, 11(15), 3408, (2023).
- [27] Say, F. Asymptotics of singularly perturbed Volterra type integro-differential equation. Konuralp Journal of Mathematics, 8(2), 365-369, (2020).
- [28] Wang, Q. and Zhou, H. Two-grid iterative method for a class of Fredholm functional integral equations based on the radial basis function interpolation. Fundamental Journal of Mathematics and Applications, 2(2), 117-122, (2019).
- [29] Zhou, H., Wang, Q. The Nyström method and convergence analysis for system of Fredholm integral equations. Fundamental Journal of Mathematics and Applications, 2(1), 28-32, (2019).
- [30] Shestopalov, Y.V. and Smirnov, Y.G. Integral Equations. Karlstad University, 2002.
- [31] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Amsterdam: Gordon and Breach Science Publishers, 1993.
- [32] P.L. Butzer, W. Splettstößer, R.L. Stens, The sampling theorem and linear prediction, Jahresberichte Dt. Math.-Verein, 90 (1988), 1-70.
- [33] Bardaro, C. and Mantellini, I. Asymptotic expansion of generalized Durrmeyer sampling type series. Jaen Journal on Approximation, 6(2), 143-165, (2014).
- [34] Bardaro, C. and Mantellini, I. On linear combinations of multivariate generalized sampling type series. Mediterranean Journal of Mathematics, 10, 1833-1852, (2013).
- [35] Bardaro, C., Butzer, P.L., Stens, R.L. and Vinti, G. Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Transactions on Information Theory, 56(1), 614-633, (2010).
- [36] Bardaro, C., Vinti, G., Butzer, P.L. and Stens, R. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampling Theory in Signal and Image Processing, 6, 29-52, (2007).
- [37] Costarelli, D. and Vinti, G. Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Bollettino dell’Unione Matematica Italiana, 4(3), 445-468, (2011).
- [38] Costarelli, D. and Vinti, G. Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization, 34(8), 819-844, (2013).
- [39] Costarelli, D. and Vinti, G. Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some function spaces. Numerical Functional Analysis and Optimization, 36(8), 964- 990, (2015).
- [40] Bardaro, C. and Mantellini, I. Asymptotic formulae for linear combinations of generalized sampling operators. Zeitschrift für Analysis und ihre Anwendunge, 32, 279-298, (2013).
- [41] Bardaro, C. and Mantellini, I. A quantitative Voronvoskaja formula for generalized sampling operators. East Journal on Approximations, 15(4), 459-471, (2009).
- [42] Bardaro, C. and Vinti, G. Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator. Results in Mathematics, 34, 224-240, (1998).

There are 42 citations in total.

Primary Language | English |
---|---|

Subjects | Approximation Theory and Asymptotic Methods |

Journal Section | Research Articles |

Authors | |

Early Pub Date | June 30, 2024 |

Publication Date | June 30, 2024 |

Submission Date | May 22, 2024 |

Acceptance Date | June 29, 2024 |

Published in Issue | Year 2024 Volume: 4 Issue: 2 |

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