GENERALIZATIONS TO SOME INTEGRO-DIFFERENTIAL EQUATIONS EMBODYING POWERS OF A DIFFERENTIAL OPERATOR


Cite item

Abstract

The abstract equations containing the operators of the second, third and fourth degree are investigated in this work.
The necessary conditions for the solvability of the abstract equations, containing the operators of the second and fourth degree, are proved without using linear independence of the vectors included in these equations. Previous authors have essentially used the linear independence of the vectors to prove the necessary
solvability condition.
The present paper also gives the correctness criterion for the abstract equation, containing the operators of the third degree with arbitrary vectors, and its exact solution in terms of these vectors in a Banach space.
The theory presented here, can be useful for investigation of Fredholm integro-differential equations embodying powers of an ordinary differential operator or a partial differential operator.

About the authors

M. M. Baiburin

L.N. Gumilyov Eurasian National University, 2, Satpayev street, Nur-Sultan, 010008, Republic of Kazakhstan.

Author for correspondence.
Email: morenov.sv@ssau.ru

Candidate of Physical and Mathematical Sciences, associate professor of the Department of Fundamental Mathematics

Kazakhstan

References

  1. Apreutesei N., Ducrot A., Volpert V. Travelling waves for integro-differential equations in population dynamics. Discrete and Continuous Dynamical Systems, 2009, Ser. B, vol. 11, no. 3, pp. 541–561. doi: 10.3934/dcdsb.2009.11.541 .
  2. Baiburin M.M., Providas E. Exact Solution to Systems of Linear First-Order Integro-Differential Equations with Multipoint and Integral Conditions. In: Rassias T., Pardalos P. (eds) Mathematical Analysis and Applications. Springer Optimization and Its Applications book series, volume 154, 2019, pp. 1-16. doi: 10.1007/978-3-030-31339-5_1 .
  3. Bloom F. Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory. SIAM Studies in Applied Mathematics, Philadelphia, 1981, 231 P. ISBN: 0-89871-171-1 Available at: http://bookre.org/reader?file=725637 .
  4. Cushing J.M. Integrodifferential equations and delay models in population dynamics. Springer-Verlag Berlin Heidelberg, 1977. DOI: https://doi.org/10.1002/bimj.4710210608 .
  5. Medlock J., Kot M. Spreading disease: integro-differential equations old and new. Mathematical Biosciences, August 2003, vol. 184, pp. 201–222. DOI: https://doi.org/10.1016/S0025-5564(03)00041-5 .
  6. Oinarov R.O., Parasidi I.N. Correct extensions of operators with finite defect in Banach spases. Izvestiya Akademii Nauk Kazakhskoi SSR, 1988, vol. 5, pp. 42-46 .
  7. Parasidis I.N., Providas E. Integro-differential equations embodying powers of a differential operator. Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya , 2019, vol. 25, no. 3, pp. 13–21. DOI: https://doi.org/10.18287/2541-7525-2019-25-3-12-21 .
  8. Parasidis I.N., Providas E. On the Exact Solution of Nonlinear Integro-Differential Equations. In: Applications of Nonlinear Analysis, 2018, pp. 591–609. doi: 10.1007/978-3-319-89815-5_21 .
  9. Parasidis I.N., Tsekrekos P.C., Lokkas Th.G. Correct and self-adjoint problems for biquadratic operators. Journal of Mathematical Sciences, 2010, vol. 166, issue 2, pp. 420–427. DOI: https://doi.org/10.1007/s10958-011-0621-2 .
  10. Parasidis I.N., Providas E. Extension Operator Method for the Exact Solution of Integro-Differential Equations. In: Pardalos P., Rassias T. (eds) Contributions in Mathematics and Engineering. Springer, Cham, 2016, pp. 473–496. DOI: https://doi.org/10.1007/978-3-319-31317-7_23 .
  11. Polyanin A.D., Zhurov A.I. Exact solutions to some classes of nonlinear integral, integro-functional, and integro-differential equations. Doklady Mathematics, 2008, issue 77, pp. 315–319. DOI: https://doi.org/10.1134/S1064562408020403 .
  12. Sachs E.W., Strauss A.K. Efficient solution of a partial integro-differential equation in finance. Applied Numerical Mathematics, 2008, issue 58, pp. 1687–1703. DOI: https://doi.org/10.1016/j.apnum.2007.11.002 .
  13. Shishkin G.A. Linear Fredholm integro-differential equations. Ulan-Ude, Buryat State University, 2007. .
  14. Shivanian E. Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Engineering Analysis with Boundary Elements, 2003, vol. 37, pp. 1693–1702. DOI: https://doi.org/10.1016/j.enganabound.2013.10.002 .
  15. Vassiliev N.N., Parasidis I.N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method. Information and Control Systems, 2018, issue 6, pp. 14–23. DOI: https://doi.org/10.31799/1684-8853-2018-6-14-23 .
  16. Vassiliev N.N., Parasidis I.N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators. Information and Control Systems, 2019, issue 2, pp. 2–9. DOI: https://doi.org/10.31799/1684-8853-2019-2-2-9 .
  17. Wazwaz A.M. Linear and nonlinear integral equations, methods and applications. Berlin, Heidelberg: Springer, 2011. DOI: https://doi.org/10.1007/978-3-642-21449-3 .

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