Geodesic
Bi-variationality, symmetries and approximate solutions
Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 596-608.

Voir la notice de l'article provenant de la source Math-Net.Ru

By a bi-variational system we mean any system of equations generated by two different Hamiltonian actions. A connection between their variational symmetries is established. The effective use of the nonclassical Hamiltonian actions for the construction of approximate solutions with the high accuracy for the given dissipative problem is demonstrated. We also investigate the potentiality of the given operator equation with the second-order time derivative, construct the corresponding functional and find necessary and sufficient conditions for the operator $S$ to be a generator of symmetry of the constructed functional. Theoretical results are illustrated by some examples. 
@article{CMFD_2021_67_3_a12,
     author = {V. M. Filippov and V. M. Savchin and S. A. Budochkina},
     title = {Bi-variationality, symmetries and approximate solutions},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {596--608},
     publisher = {mathdoc},
     volume = {67},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a12/}
}
TY  - JOUR
AU  - V. M. Filippov
AU  - V. M. Savchin
AU  - S. A. Budochkina
TI  - Bi-variationality, symmetries and approximate solutions
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2021
SP  - 596
EP  - 608
VL  - 67
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a12/
LA  - ru
ID  - CMFD_2021_67_3_a12
ER  - 
%0 Journal Article
%A V. M. Filippov
%A V. M. Savchin
%A S. A. Budochkina
%T Bi-variationality, symmetries and approximate solutions
%J Contemporary Mathematics. Fundamental Directions
%D 2021
%P 596-608
%V 67
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a12/
%G ru
%F CMFD_2021_67_3_a12
V. M. Filippov; V. M. Savchin; S. A. Budochkina. Bi-variationality, symmetries and approximate solutions. Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 596-608. http://geodesic.mathdoc.fr/item/CMFD_2021_67_3_a12/

[1] Budochkina S. A., Savchin V. M., “Variatsionnye simmetrii eilerovykh i neeilerovykh funktsionalov”, Diff. uravn., 47:6 (2011), 811–818 | Zbl

[2] Kozlov V. V., Simmetrii, topologiya i rezonansy v gamiltonovoi mekhanike, Izd-vo Udmurtskogo gos. un-ta, Izhevsk, 1995

[3] Olver P., Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989

[4] Savchin V. M., Matematicheskie metody mekhaniki beskonechnomernykh nepotentsialnykh sistem, RUDN, M., 1991

[5] Savchin V. M., Budochkina S. A., “O suschestvovanii variatsionnogo printsipa dlya operatornogo uravneniya so vtoroi proizvodnoi po «vremeni»”, Mat. zametki, 80:1 (2006), 87–94 | Zbl

[6] Savchin V. M., Budochkina S. A., “Simmetrii i pervye integraly v mekhanike beskonechnomernykh sistem”, Dokl. RAN, 425:2 (2009), 169–171 | Zbl

[7] Filippov V. M., Variatsionnye printsipy dlya nepotentsialnykh operatorov, RUDN, M., 1985

[8] Filippov V. M., “O variatsionnom printsipe dlya gipoellipticheskikh uravnenii s postoyannymi koeffitsientami”, Diff. uravn., 22:2 (1986), 338–343 | Zbl

[9] Filippov V. M., “O poluogranichennykh resheniyakh obratnykh zadach variatsionnogo ischisleniya”, Diff. uravn., 23:9 (1987), 1599–1607 | Zbl

[10] Budochkina S. A., “Symmetries and first integrals of a second order evolutionary operator equation”, Eurasian Math. J., 3:1 (2012), 18–28 | Zbl

[11] Budochkina S. A., “On connection between variational symmetries and algebraic structures”, Ufa Math. J., 13:1 (2021), 46–55 | DOI | Zbl

[12] Filippov V. M., Savchin V. M., Budochkina S. A., “On the existence of variational principles for differential-difference evolution equations”, Proc. Steklov Inst. Math., 283 (2013), 20–34 | DOI | Zbl

[13] Filippov V. M., Savchin V. M., Shorokhov S. G., “Variational principles for nonpotential operators”, J. Math. Sci. (N.Y.), 68:3 (1994), 275–398 | DOI

[14] Marchuk G. I., “Construction of adjoint operators in non-linear problems of mathematical physics”, Sb. Math., 189:10 (1998), 1505–1516 | DOI | Zbl

[15] Mikhlin S. G., Numerical performance of variational methods, Wolters-Noordhoff Publ, Groningen, 1965

[16] Popov A. M., “Potentiality conditions for differential-difference equations”, Differ. Equ., 34:3 (1998), 423–426 | Zbl

[17] Popov A. M., “Inverse problem of the calculus of variations for systems of differential-difference equations of second order”, Math. Notes, 72:5 (2002), 687–691 | DOI | Zbl

[18] Savchin V. M., Budochkina S. A., “Invariance of functionals and related Euler–Lagrange equations”, Russ. Math., 61:2 (2017), 49–54 | DOI | Zbl

[19] Tleubergenov M. I., Azhymbaev D. T., “On the solvability of stochastic Helmholtz problem”, J. Math. Sci., 253 (2021), 297–305 | DOI | Zbl

[20] Tleubergenov M. I., Ibraeva G. T., “On inverse problem of closure of differential systems with degenerate diffusion”, Eurasian Math. J., 10:2 (2019), 93–102 | DOI | Zbl

[21] Tleubergenov M. I., Ibraeva G. T., “On the solvability of the main inverse problem for stochastic differential systems”, Ukr. Math. J., 71:1 (2019), 157–165 | DOI | Zbl

[22] Tonti E., “On the variational formulation for linear initial value problems”, Ann. Mat. Pura Appl., 95 (1973), 331–359 | DOI | Zbl

[23] Tonti E., “Variational formulation for every nonlinear problem”, Int. J. Eng. Sci., 22:1 (1984), 1–12 | DOI