Задача Коши для уравнения четвертого порядка с дробной производной в смысле Капуто

В данной статье исследуется задача Коши в полуплоскости для неоднородного уравнения четвертого порядка с дробной производной в смысле Капуто. Единственность решения показывается с помощью преобразования Лапласа. При построении решения сначала находятся частные решения, выраженные через функции Райта. Затем с помощью этих частных решений строятся функции Грина. Решение строится в явном виде с использованием функции Грина. Также получен явный вид фундаментального решения.

1. Tarasov V.E. Review of some promising fractional physical models. International Journal of Modern Physics B, 2013, 27(9), P. 1330005.

2. Basharov A.M. The Stochastic foundation of the nanoparticle kinetic description by differential equations with fractional derivatives. Nanosystems: Physics, Chemistry, Mathematics, 2012, 3(6), P. 47–63. (In Russ.)

3. Sabitov K.B. Cauchy problem for the beam vibration equation. Differential Equations, 2017, 53(5), P. 658–664.

4. Selvamani R., Prabhakaran T., Yaylaci M. et al. Fractional nonlocal couple stress waves in magnetoelastic nanobeam using homotopy perturbation technique. Acta Mechanica, 2025, 236, P. 5477-–5494.

5. Abouelregal A.E., Marin M., Foul A. et al. Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative. Scientific Reports, 2024, 14(9122), P. 1–19.

6. Lewandowski R. and Wielentejczyk P. Nonlinear vibration of viscoelastic beams described using fractional order derivatives. Journal of Sound and Vibration, 2017, 399, P. 228–243.

7. Eyebe G.J, Betchewe G., Mohamadou A., Kofane T.C. Nonlinear vibration of a nonlocal nanobeam resting on fractional-order viscoelastic Pasternak foundations. Fractal and Fractional, 2018, 2(3), P. 21.

8. Chong, N., Wang, L., Lei, D. et al. Nonlinear vibration of fractional viscoelastic piezoelectric nanobeams based on nonlocal theory. Arch Appl Mech, 2025, 95(150), P. 1–18.

9. Baltaeva U., Agarwal P., Khasanov B., Hayitbayev H., Hubert F. Inverse analysis of a loaded heat conduction equation. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16(6), P. 727–736.

10. Kochubei A.N. Fractional-order diffusion. Differential Equations, 1990, 26(4), P. 485—-492.

11. Mainardi F., Rionero S., Ruggeri T. On the initial value problem for the fractional diffusion-wave equation. Waves and stability in continuous media, 1994, 7, P. 246–251.

12. Mainardi F. The time fractional diffusion-wave equation. Radiophys Quantum Electron, 1995, 38(1-2), P. 13-–24.

13. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett., 1996, 9(6), P. 23-–28.

14. Voroshilov A.A., Kilbas A.A. The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative. Differ. Equ., 2006, 42(5), P. 638-–649.

15. Pskhu A.V. The fundamental solution of a diffusion-wave equation of fractional order. Izvestiya: Mathematics, 2009, 73(2), P. 351-–392.

16. Pskhu A.V. Fundamental solutions and Cauchy problems for an odd-order partial differential equation with. Electronic Journal of Differential Equations, 2019, 2019(21), P. 1—13.

17. Karasheva L.L. Cauchy problem for high even order parabolic equation with time fractional derivative. Sibirskie Elektronnye Matematicheskie Izvestiya Siberian Electronic Mathematical Reports., 2018, 15, P. 696–706. In Russian..

18. 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: Phys. Chem. Math., 2021, 12(4), P. 418–424.

19. Karimov Sh., Mamajonova R. The Cauchy problem for a beam vibration equation involving the Caputo time-fractional operator. Bull. Inst. Math., 2025, 8(6), P. 154–160.

20. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier, Amsterdam, 2006.