Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions
https://doi.org/10.17586/2220-8054-2024-15-2-160-169
Abstract
In this paper, we consider a linear parabolic type partial differential equation in the space of generalized functions as the equation of neutron diffusion in the presence of neutron absorption by the atomic nucleus with nonlinear impulsive effects. Spectral equation is obtained from the Dirichlet boundary value conditions and this spectral problem is studied. The Fourier method of variables separation is used. Countable system of nonlinear functional integral equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of the countable system of functional integral equations is proved. The method of successive approximations is used in combination with the method of contracting mapping. Criteria of uniqueness and existence of generalized solution of the impulsive mixed problem is obtained. Solution of the mixed problem is derived in the form of the Fourier series. It is shown that the Fourier series converges uniformly.
About the Authors
T. K. Yuldashev
Tashkent State University of Economics
Uzbekistan
Uzbekistan
Tursun K. Yuldashev.
Karimov street, 49, TSUE, Tashkent, 100066
A. K. Fayziyev
Tashkent State University of Economics
Uzbekistan
Uzbekistan
Aziz K. Fayziyev.
Karimov street, 49, TSUE, Tashkent, 100066
References
1. Blinova I.V., Grishanov E.N., Popov A.I., Popov I.Y., Smolkina M.O. On spin flip for electron scattering by several delta-potentials for 1D Hamiltonian with spin-orbit interaction. Nanosystems: Phys. Chem. Math., 2023, 14(4), P. 413–417.
2. Deka H., Sarma J. A numerical investigation of modified Burgers’ equation in dusty plasmas with non-thermal ions and trapped electrons. Nanosystems: Phys. Chem. Math., 2023, 14(1), P. 5–12.
3. Dweik J., Farhat H., Younis J. The Space Charge Model. A new analytical approximation solution of Poisson-Boltzmann equation: the extended homogeneous approximation. Nanosystems: Phys. Chem. Math., 2023, 14(4), P. 428–437.
4. Fedorov E.G., Popov I.Yu. Analysis of the limiting behavior of a biological neurons system with delay. J. Phys.: Conf. Ser., 2021, 2086, P. 012109.
5. Fedorov E.G., Popov I.Yu. Hopf bifurcations in a network of Fitzhigh-Nagumo biological neurons. International Journ. Nonlinear Sciences and Numerical Simulation, 2021.
6. Fedorov E.G. Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Math., 2021, 12(5), P. 553–562.
7. Irgashev B.Yu. Boundary value problem for a degenerate equation with a Riemann-Liouville operator. Nanosystems: Phys. Chem. Math., 2023, 14(5), P. 511–517.
8. Kuljanov U.N. On the spectrum of the two-particle Schro¨dinger operator with point potential: one dimensional case. Nanosystems: Phys. Chem. Math., 2023, 14(5), P. 505–510.
9. Parkash C, Parke W.C., Singh P. Exact irregular solutions to radial Schro¨dinger equation for the case of hydrogen-like atoms. Nanosystems: Phys. Chem. Math., 2023, 14(1), P. 28–43.
10. Popov I.Y. A model of charged particle on the flat Mo¨bius strip in a magnetic field. Nanosystems: Phys. Chem. Math., 2023, 14(4), P. 418–420.
11. Sibatov R.T., Svetukhin V.V. Subdiffusion kinetics of nanoprecipitate growth and destruction in solid solutions. Theor. Math. Phys., 2015, 183, P. 846–859.
12. Vatutin A.D., Miroshnichenko G.P., Trifanov A.I. Master equation for correlators of normalordered field mode operators. Nanosystems: Phys. Chem. Math., 2022, 13(6), P. 628–631.
13. Uchaikin V.V., Sibatov R.T. Fractional kinetics in solids: Anomalous charge transport in semiconductors, dielectrics and nanosystems. CRC Press, Boca Raton, FL, 2013.
14. Matrasulov J., Yusupov J.R., Saidov A.A. Fast forward evolution in heat equation: Tunable heat transport in adiabatic regime. Nanosystems: Phys. Chem. Math., 2023, 14(4), P. 421–427.
15. Galaktionov V.A., Mitidieri E., Pohozaev S. Global sign-changing solutions of a higher order semilinear heat equation in the subcritical Fujita range. Advanced Nonlinear Studies, 2012, 12(3), P. 569–596.
16. Galaktionov V.A., Mitidieri E., Pohozaev S.I. Classification of global and blow-up sign-changing solutions of a semilinear heat equation in the subcritical Fujita range: second-order diffusion. Advanced Nonlinear Studies, 2014, 14(1), P. 1–29.
17. Denk R., Kaip M. Application to parabolic differential equations. In: General Parabolic Mixed Order Systems in Lp and Applications. Operator Theory: Advances and Applications, 239. Birkha¨user, Cham., 2013.
18. Van Dorsselaer H., Lubich C. Inertial manifolds of parabolic differential equations under high-order discretizations. Journ. of Numer. Analysis, 1099, 19(3), P. 455–471.
19. Ivanchov N.I. Boundary value problems for a parabolic equation with integral conditions. Differen. Equat., 2004, 40(4), P. 591–609.
20. Mulla M., Gaweash A., Bakur H. Numerical solution of parabolic in partial differential equations (PDEs) in one and two space variable. Journ. Applied Math. and Phys., 2022, 10(2), P. 311–321.
21. Nguyen H., Reynen J. A space-time least-square finite element scheme for advection-diffusion equations. Computer Methods in Applied Mech. and Engin., 1984, 42(3), P. 331–342.
22. Pinkas G. Reasoning, nonmonotonicity and learning in connectionist networks that capture propositional knowledge. Artificial Intelligence, 1995, 77(2), P. 203–247.
23. Pohozaev S.I. On the dependence of the critical exponent of the nonlinear heat equation on the initial function. Differ. Equat., 2011, 47(7), P. 955–962.
24. Pokhozhaev S.I. Critical nonlinearities in partial differential equations. Russ. J. Math. Phys., 2013, 20 (4), P. 476–491.
25. Yuldashev T.K. Mixed value problem for a nonlinear differential equation of fourth order with small parameter on the parabolic operator. Comput. Math. Math. Phys., 2011, 51 (9), P. 1596–1604.
26. Yuldashev T.K. Mixed value problem for nonlinear integro-differential equation with parabolic operator of higher power. Comput. Math. Math. Phys., 2012, 52(1), P. 105–116.
27. Yuldashev T.K. Nonlinear optimal control of thermal processes in a nonlinear Inverse problem. Lobachevskii Journ. Math., 2020, 41(1), P. 124–136.
28. Zonga Y., Heb Q., Tartakovsky A.M. Physics-informed neural network method for parabolic differential equations with sharply perturbed initial conditions. arXiv:2208.08635[math.NA], 2022, P. 1–50.
29. Agarwal R.P., O’Regan D., Triple solutions to boundary value problems on time scales. Applied Mathematics Letters, 2000, 13(4), P. 7–11.
30. Agur Z., Cojocaru L., Mazaur G., Anderson R.M., Danon Y.L. Pulse mass measles vaccination across age cohorts. Proceedings of the National Academy of Sciences of the USA, 1993, 90, P. 11698–11702.
31. Benchohra M., Henderson J., Ntouyas S.K. Impulsive differential equations and inclusions. Contemporary mathematics and its application. Hindawi Publishing Corporation, New York, 2006.
32. Catlla J., Schaeffer D.G., Witelski Th.P., Monson E.E., Lin A.L. On spiking models for synaptic activity and impulsive differential equations. SIAM Review, 2008, 50(3), P. 553–569.
33. Halanay A., Veksler D. Qualitative theory of impulsive systems. Mir, Moscow, 1971, 309 p. (in Russian).
34. Lakshmikantham V., Bainov D.D., Simeonov P.S. Theory of impulsive differential equations. World Scientific, Singapore, 1989, 434 p.
35. Milman V.D., Myshkis A.D. On the stability of motion in the presence of impulses. Sibirskiy Matemat. Zhurnal, 1960, 1(2), P. 233–237.
36. Perestyk N.A., Plotnikov V.A., Samoilenko A.M., Skripnik N.V. Differential equations with impulse effect: multivalued right-hand sides with discontinuities. DeGruyter Stud. 40, Math. Walter de Gruter Co., Berlin, 2011.
37. Samoilenko A.M., Perestyk N.A. Impulsive differential equations. World Sci., Singapore, 1995.
38. Stamova I., Stamov G. Impulsive biological models. In: Applied impulsive mathematical models. CMS Books in Mathematics. Springer, Cham., 2016.
39. Anguraj A., Arjunan M.M. Existence and uniqueness of mild and classical solutions of impulsive evolution equations. Elect. J. of Differen. Equat., 2005, 2005(111), P. 1–8.
40. Antunes D., Hespanha J., Silvestre C. Stability of networked control systems with asynchronous renewal links: An impulsive systems approach. Automatica, 2013, 49(2), P. 402–413.
41. Bai Ch., Yang D. Existence of solutions for second-order nonlinear impulsive differential equations with periodic boundary value conditions. Boundary Value Problems (Hindawi Publishing Corporation), 2007, 2007(41589), P. 1–13.
42. Bin L., Xinzhi L., Xiaoxin L. Robust global exponential stability of uncertain impulsive systems. Acta Mathematica Scientia, 2005, 25(1), P. 161–169.
43. Benchohra M., Salimani B.A. Existence and uniqueness of solutions to impulsive fractional differential equations. Elect. Journ. Differen. Equat., 2009, 2009(10), P. 1–11.
44. Chen J., Tisdell Ch.C., Yuan R. On the solvability of periodic boundary value problems with impulse. J. of Math. Anal. and Appl., 2007, 331, P. 902–912.
45. Fecken M., Zhong Y., Wang J. On the concept and existence of solutions for impulsive fractional differential equations. Communications in Non-Linear Science and numerical Simulation, 2012, 17(7), P. 3050–3060.
46. Mardanov M.J., Sharifov Ya.A., Habib M.H. Existence and uniqueness of solutions for first-order nonlinear differential equations with two-point and integral boundary conditions. Electr. J. of Differen. Equat., 2014, 2014(259), P. 1–8.
47. Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima. Nanosystems: Phys. Chem. Math., 2022, 13(1), P. 36–44.
48. Yuldashev T.K., Fayziev A.K. Integral condition with nonlinear kernel for an impulsive system of differential equations with maxima and redefinition vector. Lobachevskii Journ. Math., 2022, 43(8), P. 2332–2340.
49. Yuldashev T.K., Fayziyev A.K. Inverse problem for a second order impulsive system of integro-differential equations with two redefinition vectors and mixed maxima. Nanosystems: Phys. Chem. Math., 2023, 14(1), P. 13–21.
50. Gao Z., Yang L., Liu G. Existence and uniqueness of solutions to impulsive fractional integro-differential equations with nonlocal conditions. Applied Mathematics, 2013, 4(6), P. 859–863.
51. Cooke C.H., Kroll J. The existence of periodic solutions to certain impulsive differential equations. Computers and Mathematics with Applications, 2002, 44(5-6), P. 667–676.
52. Li X., Bohner M., Wang C.-K. Impulsive differential equations: Periodic solutions and applications. Automatica, 2015, 52, P. 173–178.
53. Yuldashev T.K. Periodic solutions for an impulsive system of nonlinear differential equations with maxima. Nanosystems: Phys. Chem. Math., 2022, 13(2), P. 135–141.
54. Yuldashev T.K. Periodic solutions for an impulsive system of integro-differential equations with maxima. Vestnik Sam. Gos. Universiteta. Seria: Fiz.-Mat. Nauki, 2022, 26(2), P. 368–379.
55. Yuldashev T.K., Sulaimonov F.U. Periodic solutions of second order impulsive system for an integro-differential equations with maxima. Lobachevskii Journ. Math., 2022, 43(12), P. 3674–3685.
56. Fayziyev A.K., Abdullozhonova A.N., Yuldashev T.K. Inverse problem for Whitham type multi-dimensional differential equation with impulse effects. Lobachevskii Journ. Math., 2023, 44(2), P. 570–579.
57. Yuldashev T.K., Fayziyev A.K. Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects. Nanosystems: Phys. Chem. Math., 2023, 14(3), P. 312–320.
58. Yuldashev T.K., Ergashev T.G., Fayziyev A.K. Coefficient inverse problem for Whitham type two-dimensional differential equation with impulse effects. Chelyab. Fiz-Mat. Zhurn., 2023, 8(2), P. 238–248.
Review
For citations:
Yuldashev T.K., Fayziyev A.K. Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions. Nanosystems: Physics, Chemistry, Mathematics. 2024;15(2):160-169. https://doi.org/10.17586/2220-8054-2024-15-2-160-169
Views:
12
Email this article 