The Cauchy problem for a high-order wave equation with a loaded convolution type.
https://doi.org/10.17586/2220-8054-2024-15-4-448-456
Abstract
The present paper is devoted to the problem for one of the loaded wave integro-differential equations, which is equivalent to the nonlocal problem for a higher-order wave equation. The study aims at nonlocal problems and constructs a representation of the solution to the problem for an equation of hyperbolic type. Also, the paper provides examples of some cases where it will be possible to construct solutions to the problem explicitly and in the graphs.
About the Authors
P. Agarwal
Anand International College of Engineerin; Ajman University
India
India
Praveen Agarwal – Department of Mathematic Jaipur-30301
Non-linear Dynamics Research Center (NDRC) Ajman
U. Baltaeva
Khorezm Mamun Academy; Urgench State University
Uzbekistan
Uzbekistan
Umida Baltaeva – Department of Exact science; Department of Applied Mathematics and Mathematical Physic
U. Madrakhimov
Urgench State University
Uzbekistan
Uzbekistan
Umrbek Madrakhimov – Department of Applied Mathematics and Mathematical Physic
J. I. Baltaev
RANCH University of Technology
Uzbekistan
Uzbekistan
Jamol I. Baltaev – Department of Technology
References
1. Bitsadze A.V. Some classes of partial differential equations. M., Nauka, 1981, 448 p. [in Russian].
2. Agarwal R.P. Boundary value problems for higher order differential equations, World Scientific, Singapore, 1986.
3. Obolashvili E. Higher Order Partial Differential Equations in Clifford Analysis. Progress in Mathematical Physics, eBook ISBN 978-1-4612-0015-4.
4. Weaver W., Timoshenko S.P., Young D.H. Vibration Problems in Engineering. New York, John Wiley and Sons, 1990.
5. Manolis G.D., Dineva P.S., Rangelov T., Sfyris D. Mechanical models and numerical simulations in nanomechanics: A review across the scales. Engineering Analysis with Boundary Elements, 2021, 128, P. 149–170.
6. Perelmuter M.N. Interface cracks bridged by nanofibers. Nanosystems: Phys. Chem. Math., 2022, 13(4), P. 356–364.
7. Baltaeva U., Alikulov Y., Baltaeva I.I., Ashirova A. Analog of the Darboux problem for a loaded integro-differential equation involving the Caputo fractional derivative. Nanosystems: Physics, Chemistry, Mathematics, 2021, 12(4), P. 418–424.
8. Syed Tauseef Mohyud-Din, Muhammad Aslam Noor, Solving higher-order Integro-differential equations using he’s polynomials. J. KSIAM, 2009, 13(2), P. 109–121.
9. Toaldo B. Convolution-type derivatives, hitting-times of subordinators and time-changed C 0-semigroups. Potential Anal., 2015, 42, P. 115–140.
10. Khan M. A new algorithm for higher order integro-differential equations. Afr. Mat., 2015, 26, P. 247–255.
11. Urinov A., Karimov Sh. Solution of the analogue of the Cauchy problem for the iterated multidimensional Klein-Gordon-Fock equation with the Bessel operator. arXiv preprint arXiv:1711.00093, 2017-arxiv.org.
12. Karimov S.T. Method of solving the Cauchy problem for one-dimensional polywave equation with singular Bessel operator, Russ Math., 2017, 61, P. 22–35.
13. Dai Z., Li H. and Li Q. Inequalities for the fractional convolution operator on differential forms. J Inequal Appl., 2018, 176.
14. Yuldashev T.K., Shabadikov K.K. Initial-value problem for a higher-order quasilinear partial differential equation. J. Math Sci., 2021, 254, P. 811– 822.
15. Marcello D’Abbicco. Asymptotics of higher order hyperbolic equations with one or two dissipative lower order terms arXiv preprint arXiv:2109.14067, 2021-arxiv.org.
16. Lorenzi A., Sinestrari E. An inverse problem in theory of materials with memory. Nonlinear Anal. TMA, 1988, 12, P. 411–423.
17. Yelmaz H., Kehmayer M., Chia Wei Hsu, Rotter S., Hui Cao. Customizing the Angular Memory Effect for Scattering Media. Phys. Rev. X, 2021, 11, P. 031010.
18. Durdiev D.K. An inverse problem for a three-dimensional wave equation in the medium with memory Math. Anal. and Disc. math., Novosibirsk, NGU, 1989, P. 19–26.
19. Grasselli M. An identification problem for an abstract linear hyperbolic integro-differential equation with applications Journal of Mathematical Analysis and Applications, 1992, 171(1), P. 27–60.
20. Durdiev D.K. Global solvability of an inverse problem for an integro-differential equation of electrodynamics Diff. Equ., 2008, 44(4), P. 893–899.
21. Kasemets K., Janno J. Inverse problems for a parabolic integro-differential equation in convolutional weak form. Abstract and Applied Analysis, 2013, Article ID 297104.
22. P.Podio-Guidugli. A virtual power format for hydromechanics,Continuum Mech. Thermodyn., 2009, 20, P. 479–487.
23. Safarov Zh.Sh., Durdiev D.K. Inverse problem for an integro-differential equation of acoustics, Diff. Equ., 2018, 54(1), P. 134–142.
24. Totieva Zh.D., Durdiev D.K. The problem of finding the one-dimensional kernel of the thermoviscoelasticity equation, Math. Notes, 2018, 103(1– 2), P. 118–132.
25. Durdiev D.K., Nuriddinov Zh.Z. Determination of a multidimensional kernel in some parabolic integro-differential equation Journal of siberian federal university. mathematics and physics 2021, 14(1), P. 117–127.
26. Nakhushev A.M. Equations of mathematical biology, Vishaya shkola, Moscow, 1995, p. 302.
27. Dzhenaliev M.T., Ramazanov M.I. On the boundary value problem for the spectrally loaded heat conduction operator. Sib. Math. J., 2006, 47, P. 433–451.
28. Islomov B., Baltaeva U.I. Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients, Electronic Journal of Differential Equations, 2015, 221, P. 1–10.
29. Sadarangani K.B., Abdullaev O.K. About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives. Int. J. Differential Equ., 2016, P. 1–6.
30. Assanova A.T., Kadirbayeva Z.M. Periodic problem for an impulsive system of the loaded hyperbolic equations. Electronic Journal of Differential Equations, 2018, 2018(72), P. 1–8.
31. Beshtokov M.K. Boundary-value problems for loaded pseudo parabolic equations of fractional order and difference methods of their solving, Russ Math., 2019, 63, P. 1–10.
32. Agarwal P., Baltaeva U., Alikulov Y. Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain. Chaos Solitons Fractals, 2020, 140, P. 110108.
33. Kozhanov A.I. Shipina T.N. Loaded differential equations and linear inverse problems for elliptic equations. Complex Variables and Elliptic Equations, 2021, 66, P. 910–928.
34. Baltaeva U., Baltaeva I., Agarwal P. Cauchy problem for a high-order loaded integro-differential equation. Math Meth Appl Sci., 2022, P. 1-10.
35. Khubiev K.U. Boundary-value problem for a loaded hyperbolic-parabolic equation with degeneration of order. J. Math. Sci., 2022, 260, P. 387– 391.
36. Karimov Sh. Erd‘lyi-Kober operators and their application to partial differential equations. Dissertation Abstract of Doctoral Dissertation (Dsc) on physical and Mathematical Sciences. Tashkent, 2019.
37. Courant R. Partial Differential Equations [in Russian]. Mir, Moscow, 1964
Review
For citations:
Agarwal P., Baltaeva U., Madrakhimov U., Baltaev J.I. The Cauchy problem for a high-order wave equation with a loaded convolution type. Nanosystems: Physics, Chemistry, Mathematics. 2024;15(4):448-456. https://doi.org/10.17586/2220-8054-2024-15-4-448-456
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