A dynamical process of several variables

V.S. Mokeichev^∗, A.M. Sidorov^∗∗

Kazan Federal University, Kazan, 420008 Russia

E-mail: ^∗Valery.Mokeychev@kpfu.ru, ^∗∗Anatoly.Sidorov@kpfu.ru

Received March 24, 2018

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Abstract

In the space of ϕ-distributions with values belonging to a Banach space, the process described by the problem of partial differential equation has been considered. Conditions under which the process is dynamic have been given. The notion of ϕ-distributions and ϕ-solutions has been introduced by V.S. Mokeichev as a tool for studying the solvability of some partial differential equations and mathematical models. Thus, it is possible to solve certain problems without any generalized solution (Schwartz distribution). Furthermore, an opportunity to explain the theory of solvability without assumptions on the type of the investigated partial differential equation (elliptic, parabolic, hyperbolic) and on whether the equation is scalar. One of principal advantages of the space of ϕ-distributions is that its elements and only they expand in the series by a given system of elements ϕ.

Keywords: partial differential equation, ϕ-distribution, ϕ-solution

Acknowledgments. The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (project no. 1.13556.2019/13.1).

References

1. Mokeichev V.S., Mokeichev A.V. A new approach to the theory of linear problems for the systems of partial differential equations. I. Izv. Vyssh. Uchebn. Zaved., Mat., 1999, no. 1, pp. 25–35. (In Russian)
2. Mokeichev V.S., Mokeichev A.V. A new approach to the theory of linear problems for the systems of partial differential equations. II. Izv. Vyssh. Uchebn. Zaved., Mat., 1999, no. 7, pp. 30–41. (In Russian)
3. Mokeichev V.S., Mokeichev A.V. A new approach to the theory of linear problems for the systems of partial differential equations. III. Izv. Vyssh. Uchebn. Zaved., Mat., 1999, no. 11, pp. 50–59. (In Russian)
4. Egorov Yu.V. Lineinye differentsial’ye uravneniya glavnogo tipa [Linear Differential Equations of Principal Type]. Moscow, Nauka, 1984. 360 p. (In Russian)
5. Mokeichev V.S., Sidorov A.M. On an expansion in the series by given system of elements. Issled. Prikl. Mat. Inf., 2004, no. 25, pp. 163–167. (In Russian)
6. Mokeichev V.S., Sidorov A.M. A pseudodifferential operator on a torus. Materialy 18-i mezhdunar. Sarat. zimnei shk. “Sovremennye problemy theorii fuktsii i ikh prilozhenia” [Proc. 18th Sarat. Int. Winter Math. Sch. “Modern Problems of the Theory of Functions and Their Applications”]. Saratov, Nauchn. Kniga, 2016, p. 193. (In Russian)
7. Mokeichev V.S., Sidorov A.M. Correctly solvable problems in linear partial differential equations. Materialy 13-i mezhdunar. Kazan. letnei nauch. shk.-konf. Theoriya funktsii, eye prilozhenia i smezhnye voprosy [Proc. 13th Int. Kazan. Summer Sch.–Conf. “The Theory of Functions, Its Applications and Related Questions” (Kazan, August 21–27, 2017)]. Kazan, Izd. Kazan. Mat. O-va., Izd. Akad. Nauk RT, 2017, pp. 264–265. (In Russian)
8. Mokeichev V.S., Sidorov A.M. On the notion of ϕ solutions of linear problems (using the example of string oscillations). Tezisy dokl. 10-i Saratovskoi zimnei shk. “Sovremennye problemy theorii funktsii i ikh prilozhenia” [Proc. 10th Sarat. Winter Math. Sch. “Modern Problems of the Theory of Functions and Their Applications”]. Saratov, Izd. Sarat. Univ., 2000, pp. 94–95. (In Russian)
9. Mokeichev V.S. Boundary-value problems for partial differential equations. Izv. Vyssh. Uchebn. Zaved., Mat., 1975, no. 4, pp. 103–107. (In Russian)
10. Mokeichev V.S. On expansion into series by a given system of elements. Issled. Prikl. Mat. Inf., Kazan, 2011, no. 7, pp. 144–152. (In Russian)
11. Mokeichev V.S. A space with the only elements characterized by Fourier-series expansion by a given system of elements. Evrraz. Nauchn. Ob”edinenie, 2016, vol. 1, no. 10, pp. 24–31. (In Russian)

For citation: Mokeichev V.S., Sidorov A.M. A dynamical process of several variables. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 4, pp. 762–770. (In Russian)

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