A numerical algorithm for solving index-1 weakly singular integral-algebraic equations with non-smooth solutions

References

• Brunner H. Volterra integral equations: an introduction to theory and applications. New York: Cambridge University Press; 2017.
   Google Scholar
• Wazwaz AM. Linear and nonlinear integral equations. Berlin: Springer; 2011.
   Google Scholar
• Kant K, Nelakanti G. Approximation methods for second kind weakly singular Volterra integral equations. J Comput Appl Math. 2020;368:Article ID 112531.
   PubMed Web of Science ®Google Scholar
• Zhong XC, Wei HM, Long XY. Numerical solution of a singular integral equation arising in a cruciform crack problem. Appl Anal. 2017;96(10):1767–1783.
   Web of Science ®Google Scholar
• Assari P. Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations. Appl Anal. 2019;98(11):2064–2084.
   Web of Science ®Google Scholar
• Yang Y, Deng G, Tohidi E. High accurate convergent spectral Galerkin methods for nonlinear weakly singular Volterra integro-differential equations. Comput Appl Math. 2021;40(4):1–32.
   Google Scholar
• Zaky MA, Ameen IG. A novel Jacob spectral method for multi-dimensional weakly singular nonlinear Volterra integral equations with nonsmooth solutions. Eng Comput. 2021;37(4):2623–2631.
   Web of Science ®Google Scholar
• Zhou Y, Stynes M. Block boundary value methods for linear weakly singular Volterra integro-differential equations. BIT Numer Math. 2021;61(2):691–720.
   Web of Science ®Google Scholar
• Dixon J. On the order of the error in discretization methods for weakly singular second kind non-smooth solutions. BIT Numer Math. 1985;25(4):623–634.
   Google Scholar
• Sajjadi SA, Pishbin S. Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels. Numer Algorithms. 2021;86(1):25–54.
   Web of Science ®Google Scholar
• Brunner H. Collocation methods for volterra integral and related functional equations. New York: Cambridge University Press; 2004.
   Google Scholar
• Zolfaghari R, Nedialkov N. Structural analysis of linear integral-algebraic equations. J Comput Appl Math. 2019;353:243–252.
   Web of Science ®Google Scholar
• Kafarov VV, Mayorga B, Dallos C. Mathematical method for analysis of dynamic processes in chemical reactors. Chem Eng Sci. 1999;54(20):4669–4678.
   Web of Science ®Google Scholar
• Jumarhon B, Lamb W, McKee S, et al. A Volterra integral type method for solving a class of nonlinear initial-boundary value problems. Numer Methods Partial Differ Equ. 1996;12:265–281.
   Google Scholar
• Chistyakova EV, Solovarova LS, Son DT. A numerical method for solving singular integral algebraic equations with weakly singular kernels. Bull South Ural State Univ Ser Comput Math Softw Eng Russ. 2021;10(3):5–15.
   Google Scholar
• Janno J, von Wolfersdorf L. Identification of weakly singular memory kernels in heat conduction. Z Angew Math Mech. 1997;77:243–257.
   Web of Science ®Google Scholar
• Campbell SL, Gear CW. The index of general nonlinear DAEs. Numer Math. 1995;72:173–196.
   Web of Science ®Google Scholar
• Chistyakov VF, Chistyakova EV. Linear differential-algebraic equations perturbed by Volterra integral operators. Differ Equ. 2017;53(10):1274–1287.
   Web of Science ®Google Scholar
• Hanke M, März R. Convergence analysis of least-squares collocation methods for nonlinear higher-index differential-algebraic equations. J Comput Appl Math. 2021;387:Article ID 112514.
   Web of Science ®Google Scholar
• Hanke M, März R. Towards a reliable implementation of least-squares collocation for higher index differential-algebraic equations–part 1: basics and ansatz function choices. Numer Algorithms. 2022;89(3):931–963.
   Web of Science ®Google Scholar
• Kunkel P, Mehrmann V. Differential-algebraic equations: analysis and numerical solution. Zürich: European Mathematical Society (EMS); 2006. (EMS textbooks in mathematics).
   Google Scholar
• Lamour R, März R, Tischendorf C. Differential-algebraic equations: a projector based analysis. Heidelberg: Springer; 2013.
   Google Scholar
• März R. The index of linear differential-algebraic equations with properly stated leading terms. Results Math. 2002;42:308–338.
   Google Scholar
• Rabier PJ, Rheinboldt WC. Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet PG, Lions JL, editors. Handbook of numerical analysis. Vol. VIII. Amsterdam: North-Holland; 2002. p. 183–540.
   Google Scholar
• Shiri B, Baleanu D. System of fractional differential algebraic equations with applications. Chaos Solitons Fractals. 2019;120:203–212.
   Web of Science ®Google Scholar
• Liang H, Brunner H. Integral-algebraic equations: theory of collocation methods II. SIAM J Numer Anal. 2016;54(4):2640–2663.
   Web of Science ®Google Scholar
• Liang H, Brunner H. Integral-algebraic equations: theory of collocation methods I. SIAM J Numer Anal. 2013;51(4):2238–2259.
   Web of Science ®Google Scholar
• Pishbin S, Ghoreishi F, Hadizadeh M. The semi-explicit Volterra integral algebraic equations with weakly singular kernels: the numerical treatments. J Comput Appl Math. 2013;245:121–132.
   Web of Science ®Google Scholar
• Zhao J, Cao Y, Xu Y. Sinc collocation solutions for the integral algebraic equation of index-1. Adv Appl Math Mech. 2016;8(5):757–771.
   Web of Science ®Google Scholar
• Balakumar V, Murugesan K. Numerical solution of Volterra integral-algebraic equations using block pulse functions. Appl Math Comput. 2015;263:165–170.
   Web of Science ®Google Scholar
• Botoroeva MN, Budnikova OS, Bulatov MV, et al. Numerical solution of integral-algebraic equations with a weak boundary singularity by k-step methods. Comput Math Math Phys. 2021;61:1787–1799.
   Web of Science ®Google Scholar
• Budnikova OS, Bulatov MV. Numerical solution of integral algebraic equations for multistep methods. Comput Math Math Phys. 2012;52:691–701.
   Web of Science ®Google Scholar
• Bulatov MV. Regularization of singular system of Volterra integral equation. Comput Math Math Phys. 2002;42:315–320.
   Google Scholar
• Bulatov MV, Lima PM. Two-dimensional integral-algebraic systems: analysis and computational methods. J Comput Appl Math. 2011;236:132–140.
   Web of Science ®Google Scholar
• Chistyakova EV, Chistyakov VF. Solution of differential algebraic equations with the Fredholm operator by the least squares method. Appl Numer Math. 2020;149:43–51.
   Web of Science ®Google Scholar
• Farahani MS, Hadizadeh M. Direct regularization for system of integral-algebraic equations of index-1. Inverse Probl Sci Eng. 2018;26(5):728–743.
   Web of Science ®Google Scholar
• Hadizadeh M, Ghoreishi F, Pishbin S. Jacobi spectral solution for integral-algebraic equations of index-2. Appl Numer Math. 2011;61:131–148.
   Web of Science ®Google Scholar
• Liang H, Brunner H. Collocation methods for integro-differential algebraic equations with index 1. IMA J Numer Anal. 2020;40(2):850–885.
   Web of Science ®Google Scholar
• Pishbin S, Ghoreishi F, Hadizadeh M. A posteriori error estimation for the Legendre collocation method applied to integral-algebraic Volterra equations. Electron Trans Numer Anal. 2011;38:327–346.
   Web of Science ®Google Scholar
• Pishbin S. Numerical solution and structural analysis of two-dimensional integral-algebraic equations. Numer Algorithms. 2016;73:305–322.
   Web of Science ®Google Scholar
• Pishbin S. Operational Tau method for singular system of Volterra integro-differential equations. J Comput Appl Math. 2017;311:205–214.
   Web of Science ®Google Scholar
• Pishbin S. Optimal convergence results of piecewise polynomial collocation solutions for integral-algebraic equations of index-3. J Comput Appl Math. 2015;279:209–224.
   Web of Science ®Google Scholar
• Pishbin S. The numerical solution of the semi-explicit IDAEs by discontinuous piecewise polynomial approximation. Appl Math Comput. 2018;339:93–104.
   Web of Science ®Google Scholar
• Zhang T, Liang H, Zhang S. On the convergence of multistep collocation methods for integral-algebraic equations of index 1. Comput Appl Math. 2020;39(4):1–15.
   Google Scholar
• Pomp A. The boundary-domain integral method for elliptic systems with application to shells. Berlin, Heidelberg: Springer Science and Business Media; 1998.
   Google Scholar
• Linz P. Analytical and numerical methods for Volterra equations. Philadelphia: SIAM; 1985.
   Google Scholar
• Atkinson K, Han W. Theoretical numerical analysis: a functional analysis framework. 3rd ed. New York, NY: Springer; 2009.
   Google Scholar
• Mason JC, Handscomb DC. Chebyshev polynomials. New York: CRC Press; 2002.
   Google Scholar
• Hesthaven JS, Gottlieb S, Gottlieb D. Spectral methods for time-dependent problems. New York: Cambridge University Press; 2007.
   Google Scholar
• Atkinson KE. The numerical solution of integral equations of the second kind. New York: Cambridge University Press; 1996. (Cambridge monographs on applied and computational mathematics).
   Google Scholar
• Chen Y, Tang T. Convergence analysis of the Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel. Math Comput. 2010;79:147–167.
   Web of Science ®Google Scholar
• Xianjuan L, Tang T. Convergence analysis of Jacobi spectral collocation methods for Abel–Volterra integral equations of second kind. Front Math China. 2012;7(1):69–84.
   Web of Science ®Google Scholar
• Canuto C, Hussaini MY, Quarteroni A, et al. Spectral methods fundamentals in single domains. Berlin, Heidelberg: Springer-Verlag; 2006.
   Google Scholar