87
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Using 2D and 1D block-pulse functions simultaneously for solving the Barbashin integro-differential equations

ORCID Icon &
Pages 1957-1970 | Received 03 May 2022, Accepted 13 Mar 2023, Published online: 06 Aug 2023

References

  • Mirzaee, F., Hadadiyan, E. A new numerical method for solving two-dimensional Volterra-Fredholm integral equations. J. Appl. Math. Comput. 52, 489–513 (2015). DOI:10.1007/s12190-015-0951-1
  • N. Aghazadeh and A.A. Khajehnasiri, Solving nonlinear two-dimensional Volterra integrodifferential equations by block-pulse functions, Math. Sci. 7(2013), pp. 3–6.
  • K. Al-Khaled and F. Allan, Decomposition method for solving nonlinear integro-differential equation, J. Appl. Math. Comput. 19(1-2) (2005), pp. 415–425.
  • V.M. Aleksandrov and E.V. Kovalenko, Problems in Continuous Mechanics with Mixed Boundary Conditions, Nauka, Moscow, 1986. (Russian).
  • J. Appell, S. Kalitvin, and P.P. Zabrejko, Partial Integral Equations and Integro-differential Equations, Marcel Dekker, New York–Basel, 2000. 560 pp.
  • E. Babolian, Z. Masouri, and S. Hatamzadeh-Varmazyar, Numerical solution of nonlinear Volterra– Fredholm integro-differential equations via direct method using triangular functions, Comput. Math. Appl. 58(2) 2009), pp. 239–247.
  • E. Babolian, K. Maleknejad, M. Roodaki, and H. Almasieh, Two-dimensional triangular functions and their applications to nonlinear 2D Volterra–Fredholm integral equations, Comput. Math. Appl. 60 (2010), pp. 1711–1722.
  • E.A. Barbashin, On conditions for the conservation of stability of solutions to integrodifferential equations [in Russian], Izv. VUZov Mat. 1 (1957), pp. 25–34.
  • H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 2004.
  • K.M. Case and P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA, 1967.
  • M.C. Cercignani, Mathematical Methods in Kinetic Theory, Macmillian, New York, NY, 1969.
  • M.B. Dhakne and S.D. Kendre, On abstrct Nonlinear Mixed Volterra–Fredhom Integrodifferential equations, Commun. Appl. Nonlinear Anal. 13(4) (2006), pp. 101–112.
  • L. Hacia, On approximate solution for integral equations of mixed type, ZAMM Z. Angew. Math. Mech 76(1996), pp. 415–416.
  • Z.H. Jiang and W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Schaufelberger, Springer–Verlag, 1992.
  • Z.H. Jiang and W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Published by Springer-Verlag, 1992.
  • A.S. Kalitvin, On two problems for the Barbashin integro-differential equations, J. Math. Sci. 126(6) (2005), pp. 1600–1606.
  • A.S. Kalitvin, Generalized solutions to the caushy for the Barbashin integro-differential equations, J. Math. Sci. 208(1) (2015), pp. 1–7.
  • A.S. Kalitvin and V.A. Kalitvin, Volterra and Volterra–Fredholm Integral Equations with Partial Integrals, LGPU, Lipetsk, 2006.
  • A.S. Kalitvin, Some aspects of the theory of integro-differential Barbashin equations in function spaces [in Russian], Probl. Mat. Anal. 67, 61–68 (2012); J. Math. Sci., New York 188, No. 3, 241–249 (2013).
  • H.G. Kaper, C.G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear TransportTheory, Birkhauser, Basel, 1982.
  • A.L. Khoteev, Necessary multi-point conditions for singular controls for integrodifferential equations of Barbashin type, Vestn. Akad. Nauk. SSSR. 2 (1984), pp. 20–25. (Russian).
  • A.L. Khoteev, An optimal control problem for integrodifferential equations of Barbashin type, in Probl. Upravl. Optim. Minsk, pp. 74–87, (1976).
  • K. Maleknejad, B. Basirat, and E. Hashemizadeh, Hybrid legendre polynomials and blockPulse functions approach for nonlinear Volterra–Fredholm integro-differential equations, Comput. Math. Appl. 61(2011), pp. 2821–2828.
  • K. Maleknejad and M. Hadizadeh, New computational method for Volterra–Fredholm integral equations, Comput. Math. Appl. 37(1999), pp. 1–8. mixed.
  • K. Maleknejad and K. Mahdiani, Solving nonlinear mixed Volterra–Fredholm integral equations with two dimensional block-pulse functions using direct method, Commun. Nonlinear Sci. Numer Simulat.16(2011), pp. 35123519.
  • K. Maleknejad and K. Mahdiani, Solution and error analysis of two dimensional Fredholm–Volterra integral equations using piecewise constant functions, Am. J. CAM 2(1) (2012), pp. 53–57.
  • K. Maleknejad and K. Mahdiani, Solution and error analysis of two dimensional Fredholm–Volterra integral equations using piecewise constant functions, Am. J. Comput. Appl. Math. 2(1) (2012), pp. 53–57.
  • K. Maleknejad and Y. Mahmoudi, Taylor polynomial solution of high–order nonlinear Volterra–Fredholm integro-differential equations, Appl. Math. Comput. 145(2003), pp. 641–653.
  • G.I. Marchuk, The Methods of Calculation for Nuclear Reactors, Atomizdat, Moscow, 1961. (Russian).
  • G. Michael, On stability of linear Barbashin type integrodifferential equations, Math. Probl. Eng.2015, pp.5. Article ID 962565.
  • B.G. Pachpatte, On mixed Volterra-Fredholm type integral equations, Indian J. Pure Appl. Math. 17 (1986), pp. 488–496.
  • B.G. Pachpattey, On a general partial integral equation of Barbashin type, Tamsui. Oxf. J. Inf. Math. Sci. 27(1) (2011), pp. 99–115.
  • E. Poorfattah and A. Shaerlar Jafari, Direct method for solving nonlinear two-dimensional Volterra– Fredholm integro-differential equations by block pulse functions, Int. J. Inf. Security Syst. Management4(1) (2015), pp. 418–423.
  • L. Rahmani, B. Rahimi, and M. Mordad, Numerical solution of Volterra–Fredholm integro-differential equation by block pulse functions and operational matrices, Gen. Math. Notes 4(2) (2011), pp. 37–48june.
  • A. Roohollahi, B. Ghazanfari, and S. Akhavan, Numerical solution of the mixed Volterra–Fredholm integro-differential multi-term equations of fractional order, J. Comput. Appl. Math. 376(2020), pp. 112828.
  • H.L. Tidke, Existence of global solution to nonlinear mixed Volterra–Fredholm integro-differential equations with nonlocal conditions, Electron. J. Differ. Equ. 2009( 55 )17.
  • W. Wang, An algorithm for solving the high-order nonlinear Volterra–Fredholm integro-differential equation with mechanization, AMC 172(2006), pp. 1–23.
  • L.M. Wurzburg, Partial Integral Operators and Integro-differential Equations, Chapman and Hall/CRC, 2000. February 29.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.