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### Solving linear Fredholm integro-differential equation by Nyström method

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Języki publikacji
EN
Abstrakty
EN
The study of the solution’s existence and uniqueness for the linear integro-differential Fredholm equation and the application of the Nyström method to approximate the solution is what we will present in this paper. We use the Neumann theorem to construct a sufficient condition that ensures the solution’s existence and uniqueness of our problem in the Banach space C1 [a,b]. We have applied the Nyström method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method’s convergence. The Nyström method discretizes the integro-differential equation into solving a linear system. Only with the existence and uniqueness condition, we show the solution’s existence and uniqueness of the linear system and the convergence of the numerical solution to the exact solution in infinite norm sense. We present two theorems to give a good estimate of the error. Also, to show the efficiency and accuracy of the Nyström method, some numerical examples will be provided at the end of this work.
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EN
PL
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53--64
Opis fizyczny
Bibliogr. 25 poz., rys., tab.
Twórcy
autor
• Laboratoire des Mathématiques Appliquées et Modélisation, Université 8 Mai 1945 Guelma 24000, Algeria
autor
• Laboratoire des Mathématiques Appliquées et Modélisation, Université 8 Mai 1945 Guelma 24000, Algeria
autor
• Laboratoire des Mathématiques Appliquées et Modélisation, Université 8 Mai 1945 Guelma 24000, Algeria
autor
• Laboratoire des Mathématiques Appliquées et Modélisation, Université 8 Mai 1945 Guelma 24000, Algeria
Bibliografia
• [1] Lakshmikantham, V. (1995). Theory of Integro-Differential Equations (Vol. 1). CRC Press.
• [2] Jerri, A. (1999). Introduction to Integral Equation with Application. John Wiley and Sons.
• [3] Singh, H., Dutta, H., & Cavalcanti, M.M. (2021). Topics in Integral and Integro-Differential Equations, Studies in Systems. Springer.
• [4] Zemyan, S.M. (2012). The Classical Theory of Integral Equations. Birkhauser Basel, Springer, New York.
• [5] Yusufoglu, E. (2007). An efficient algorithm for solving integro-differential equations system. Applied Mathematics and Computing, 192, 51-55.
• [6] Singh, C.S., Singh, H., Singh, V.K., & Singh, Om, P. (2016). Fractional order operational matrix methods for fractional singular integro-differential equation. Applied Mathematical Modelling, 40, 10705-10718.
• [7] Singh, C.S., Singh, H., Singh, S., & Kumar, D. (2018). An efficient computational method for solving system of nonlinear generalized Abel integral equations arising in astrophysics. Physica A, 525, 1440-1448.
• [8] Singh, H., Baleanu, D., Srivastava, H.M., Dutta, H., & Jha, N.K. (2020). Solution of multi-dimensional Fredholm equations using Legendre scaling functions. Applied Numerical Mathematics, 150, 313-324.
• [9] Tiwari, S., Pandey, R.K., Singh, H., & Singh, J. (2020). Embedded pseudo-Runge-Kutta methods for first and second order initial value problems. Science & Technology Asia, 25(1), 128-141.
• [10] Pandey, R.K., & Singh, H. (2020). An efficient numerical algorithm to solve volterra integral equation of second kind. Topics in Integral and Integro-Differential Equations. Studies in Systems, Decision and Control, 340, 215-228.
• [11] Shanga, X., & Hanb, D. (2010). Application of the variational iteration method for solving nth-order integro-differential equations. Journal of Computianal Applied Mathematics, 234, 1442-1447.
• [12] Mennouni, A. (2012). A projection method for solving Cauchy singular integro-differential equations. Applied Mathematics Letters, 25, 986-989.
• [13] Atkinson, K., & Han, W. (2001). Theoretical Numerical Analysis: A Functional Analysis Framework. Springer-Verlag, New York.
• [14] Guebbai, H., Aissaoui, M.Z., Debbar, I., & Khalla, B. (2014). Analytical and numerical study for an integro-differential nonlinear volterra equation. Applied Mathematics and Computation, 299, 376-373.
• [15] Guebbai, H., Lemita, S., Segni, S., & Merchela, W. (2020). Difference derivative for an integro-differential nonlinear volterra equation. Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp’uternye Nauki, 30(2), 176-188.
• [16] Zhou, H., & Wang, Q. (2019). The Nystrom method and convergence analysis for system of Fredholm integral Equations. Fundamental Journal of Applied Mathematics, 2(1), 28-32.
• [17] Segni, S., Ghiat, M., & Guebbai, H. (2019). New approximation method for Volterra nonlinear integro-differential aquation. Asian-European Journal of Mathematics, 12(1), 1950016.
• [18] Ghiat, M., Guebbai, H., Kurulay, M., & Segni, S. (2020). On the weakly singular integro differential nonlinear Volterra equation depending in acceleration term. Computational and Applied Mathematics, 39(2), 206.
• [19] Salah, S., Guebbai, H., Lemita, S., & Aissaoui, M.Z. (2019). Solution of an integro-differential nonlinear equation of Volterra arising of earthquake model. Boletim da Sociedade Paranaense de Matematica , 1-14.
• [20] Lemita, S., & Guebbai, H. (2019). New process to approach linear Fredholm integrale quations defined on large interval. Asian-European Journal of Mathematics, 12(1), 1950009.
• [21] Touati, S., Lemita, S., Ghiat, M., & Aissaoui M.Z. (2019). Solving a non-linear Volterra-Fredholm integro-differentail equation with weakly singular kernels. Fasciculi Mathematics, 62, 155-168.
• [22] Ghiat, M., & Guebbai, H. (2018). Analytical and numerical study for an integro-differential nonlinear volterra equation with weakly singular kernel. Computational and Applied Mathematics, 37(4), 4661-4974.
• [23] Davies, E.B. (2007). Linear Operators and their Spectra. Cambridge University Press, Cambridge.
• [24] Ahues, M., Largillier, A., & Limaye, B.V. (2001). Spectral Computations for Bounded Operators. Chapman and Hall/CRC, Boca Raton.
• [25] Quarteroni, A., Sacco, R., & Saleri, F. (2004). Méthodes Numériques: Algorithmes, analyse et applications. Springer.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia