Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives

Ahmed M. Rajab; Saeed Pishbin; Javad Shokri; Ahmed M. Rajab; Saeed Pishbin; Javad Shokri

doi:10.3934/math.20241127

AIMS Mathematics

2024, Volume 9, Issue 9: 23182-23196. doi: 10.3934/math.20241127

Previous Article Next Article

Research article

Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives

Department of Mathematics, Faculty of Science, Urmia University, P.O. Box 165, Urmia, Iran

Received: 06 May 2024 Revised: 15 June 2024 Accepted: 24 June 2024 Published: 30 July 2024
MSC : 35R09, 45A05, 65N35

In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.
- integro-differential equations,
- weakly singular integral equations,
- Volterra–Fredholm integral equations,
- spectral method,
- convergence analysis
Citation: Ahmed M. Rajab, Saeed Pishbin, Javad Shokri. Analyzing the structure of solutions for weakly singular integro-differential equations with partial derivatives[J]. AIMS Mathematics, 2024, 9(9): 23182-23196. doi: 10.3934/math.20241127

Related Papers:

Abstract

In this work, we analyze the approximate solution of a specific partial integro-differential equation (PIDE) with a weakly singular kernel using the spectral Tau method. It present a numerical solution procedure for this PIDE, which is transferred into a Volterra–Fredholm integral equation (VFIE), and the spectral method is performed on VFIE. In some illustrated examples, we show that the VFIE problem has high numerical stability with respect to the original form of the PIDE problem. For this aim, we apply the spectral Tau method in two cases, first for the problem in the form of VFIE and then also for the problem in the form of PIDE. The remarkable numerical results obtained from the VFIE problem form compared to those gained from the PIDE problem form show the efficiency of the proposal method. Also, we prove the convergence theorem of the numerical solution of the Tau method for the VFIE problem, and then it is generalized to the PIDE problem.

References

[1]	C. V. Pao, Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics, J. Math. Anal. Appl., 48 (1974), 470–492. https://doi.org/10.1016/0022-247X(74)90171-1 doi: 10.1016/0022-247X(74)90171-1
[2]	C. V. Pao, L. Payne, H. Amann, Bifurcation analysis of a nonlinear diffusion system in reactor dynamics, Appl. Anal., 9 (2007), 107–119. https://doi.org/10.1080/00036817908839258 doi: 10.1080/00036817908839258
[3]	H. Engler, On some parabolic integro-differential equations: Existznce and asymptotics of solutions, In: Lecture notes in mathematics, Berlin, Heidelberg: Springer, 1017 (1983), 161–167. https://doi.org/10.1007/BFb0103248
[4]	E. G. Yanik, G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differetial equations, Nonlinear Anal., 12 (1988), 785–809. https://doi.org/10.1016/0362-546X(88)90039-9 doi: 10.1016/0362-546X(88)90039-9
[5]	G. J. Habetler, R. L. Schiffman, A finite difference method for analyzing the compression of poro-viscoelastic media, Computing, 6 (1970), 342–348. https://doi.org/10.1007/BF02238819 doi: 10.1007/BF02238819
[6]	R. M. Christensen, Theory of viscoelasticity, 2 Eds., Academic Press, 1982. https://doi.org/10.1016/B978-0-12-174252-2.X5001-7
[7]	R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl., 66 (1978), 313–332. https://doi.org/10.1016/0022-247X(78)90234-2 doi: 10.1016/0022-247X(78)90234-2
[8]	W. Gao, H. Rezazadeh, Z. Pinar, H. M. Baskonus, S. Sarwar, G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Quant. Electron., 52 (2020), 52. https://doi.org/10.1007/s11082-019-2162-8 doi: 10.1007/s11082-019-2162-8
[9]	F. Ghoreishi, M. Hadizadeh, Numerical computation of the Tau approximation for the Volterra–Hammerstein integral equations, Numer. Algor., 52 (2009), 541–559. https://doi.org/10.1007/s11075-009-9297-9 doi: 10.1007/s11075-009-9297-9
[10]	M. E. Gurtin, A. C. Pipkin, A general theory of heat conduction with finite wave speed, Arch. Rational Mech. Anal., 31 (1968), 113–126. https://doi.org/10.1007/BF00281373 doi: 10.1007/BF00281373
[11]	M. Renardy, Mathematical analysis of viscoelastic flows, Annu. Rev. Fluid Mech., 21 (1989), 21–34. https://doi.org/10.1146/annurev.fl.21.010189.000321 doi: 10.1146/annurev.fl.21.010189.000321
[12]	H. Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511543234
[13]	H. O. Al-Humedi1, Z. A. Jameel, Cubic B-spline least-square method combine with a quadratic weight function for solving integro-differential equations, Earthline J. Math. Sci., 4 (2020), 99–113. https://doi.org/10.34198/ejms.4120.99113 doi: 10.34198/ejms.4120.99113
[14]	H. O. Al-Humedi, Z. A. Jameel, Combining cubic B-spline Galerkin method with quadratic weight function for solving partial integro-differential equations, J. Al-Qadisiyah Comput. Sci. Math., 12 (2020), 9–20. https://doi.org/10.29304/jqcm.2020.12.1.660 doi: 10.29304/jqcm.2020.12.1.660
[15]	A. K. Hussain, Numerical solution of partial integro-diffrential equation using Legendre multi wavelets, J. Southwest jiaotong Univ., 55 (2020). https://doi.org/10.35741/issn.0258-2724.55.2.24 doi: 10.35741/issn.0258-2724.55.2.24
[16]	H. Zhang, X. Han, X. Yang, Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel, Appl. Math. Comput., 219 (2013), 6565–6575. https://doi.org/10.1016/j.amc.2013.01.012 doi: 10.1016/j.amc.2013.01.012
[17]	A. G. Atta, Y. H. Youssri, Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel, Comput. Appl. Math., 41 (2022), 381. https://doi.org/10.1007/s40314-022-02096-7 doi: 10.1007/s40314-022-02096-7
[18]	S. M. Sayed, A. S. Mohamed, E. M. Abo El-Dahab, Y. H. Youssri, Alleviated shifted Gegenbauer spectral method for ordinary and fractional differential equations, Contemp. Math., 5 (2024), 1344–1370. https://doi.org/10.37256/cm.5220244559 doi: 10.37256/cm.5220244559
[19]	D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Inequalities involving functions and their integrals and drivatives, In: Mathematics and its applications, Dordrecht: Springer, 53 (1991). https://doi.org/10.1007/978-94-011-3562-7
[20]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral methods: Fundamentals in single domains, In: Scientific computation, Heidelberg: Springer Berlin, 2006. https://doi.org/10.1007/978-3-540-30726-6
[21]	X. Li, T. Tang, Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind, Front. Math. China, 7 (2012), 69–84. https://doi.org/10.1007/s11464-012-0170-0 doi: 10.1007/s11464-012-0170-0

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)