Source: http://journals.uran.ua/eejet/article/view/155148
Timestamp: 2019-04-25 04:04:03+00:00

Document:
Research and analysis of dynamic processes in oscillatory systems are closely connected to the establishment of exact or approximate analytical solutions to the problems of mathematical physics, which model such systems. The mathematical models of wave propagation in oscillatory systems under certain initial conditions at a fixed time are well known in the literature. However, wave processes in lengthy structures subject to an external force only and at the assigned states of the process at two points in time have been insufficiently studied. Such processes are modeled by a two-point time problem for the inhomogeneous wave equation in an unbounded domain t>0, x∈ℝs. The model takes into consideration the assignment of a linear combination with unknown amplitude of oscillations and the rate of its change at two points in time. A two-point problem, generally speaking, is the ill-posed boundary value problem, since the respective homogeneous problem has non-trivial solutions. A class of quasi-polynomials has been established as the class of the existence of a single solution to the problem. This class does not contain the non-trivial elements from the problem's kernel, which ensures the uniqueness of solution to the problem. We have proposed a precise method to build the solution in the specified class. The essence of the method is that the problem's solution is represented as the action of a differential expression, whose symbol is the right-hand side of the equation, on some function of parameters. The function is constructed in a special way using the equation and two-point conditions, and has special features associated with zeroes of the denominator – the characteristic determinant of the problem.
The method is illustrated by the description of oscillatory processes within an infinite string and a membrane.
Chaban, A. (2008). Matematychne modeliuvannia kolyvnykh protsesiv v elektromekhanichnykh systemakh. Lviv: Vydavnytstvo Tarasa Soroky, 328.
Pukach, P. Ya., Kuzio, I. V. (2015). Resonance phenomena in quasi-zero stiffness vibration isolation systems. Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversytetu, 3, 62–67.
Bondarenko, V. I., Samusya, V. I., Smolanov, S. N. (2005). Mobil'nye pod'emnye ustanovki dlya avariyno-spasatel'nyh rabot v shahtnyh stvolah. Gorniy zhurnal, 5, 99–100.
Magrab, E. B. (2014). An Engineer’s Guide to Mathematica. John Wiley and Sons, 452.
Friedman, A. (2011). Partial differential equation. New York: Dover Publications, 272.
Nytrebych, Z. M., Malanchuk, O. M., Il'kiv, V. S., Pukach, P. Ya. (2017). Homogeneous problem with two-point conditions in time for some equations of mathematical physics. Azerbaijan Journal of Mathematics, 7 (2), 180–196.
Ptashnyk, B. Y. (1967). Zadacha typu Valle-Pussena dlia hiperbolichnykh rivnian iz stalymy koefitsientamy. Dop. AN URSR. Ser. A, 10, 1254–1257.
Ptashnyk, B. Y. (1967). N-liniyna zadacha dlia hiperbolichnykh rivnian iz stalymy koefitsientamy. Visnyk Lvivskoho politekhnichnoho instytutu, 16, 80–87.
Ptashnik, B. I. (1984). Nekorrektnye granichnye zadachi dlya differenci-al'nyh uravneniy s chastnymi proizvodnymi. Kyiv: Naukova dumka, 264.
Il’kiv, V. S., Nytrebych, Z. M., Pukach, P. Y. (2016). Boundary-value problems with integral conditions for a system of Lamé equations in the space of almost periodic functions. Electronic Journal of Differential Equations, 304, 1–12.
Borok, V. M. (1968). Klassy edinstvennosti resheniya kraevoy zadachi v beskonechnom sloe. DAN SSSR, 183 (5), 995–998.
Borok, V. M., Perel'man, M. A. (1973). Unique solution classes for a multipoint boundary value problem in an infinite layer. Izv. vuzov. Matematika, 8, 29–34.
Vilents, I. L. (1974). Klasy yedynosti rozviazku zahalnoi kraiovoi zadachi v shari dlia system liniynykh dyferentsialnykh rivnian u chastynnykh pokhidnykh. Dop. AN URSR. Ser. А., 3, 195–197.
Vallee-Poussin, Ch. J. (1929). Sur l'equation differentielle lineaire du second ordre. Determination d'une integrale par deux valeurs assignees. Extension aux equations d'ordre n. Journ. Math. de pura et appl., 8, 125–144.
Picone, M. (1910). On the exceptional values of a parameter on which an ordinary linear differential equation of the second order depends. Pisa: Scuola Normale Superiore, 144.
Feynman, R. P., Leighton, R. B., Sands, M. (1963). The Feynman Lectures on Physics. Mainly electromagnetism and matter. New York: New millennium ed., 566.
Lamoureux, M. P. (2006). The mathematics of PDEs and the wave equation. Calgary: Seismic Imaging Summer School, 39.
Lie, K. A. (2005). The wave equation in 1D and 2D. Dep. of Informatics University of Oslo, 48.
Chalyi, A. V. (2017). Medical and biological physics. Vinnytsia: Nova knyga, 476.
Blagitko, B., Zayachuk, I., Pyrogov, O. (2006). The Mathematical Model of the Pulse Wave Propagation in Large Blood Vascular. Fizyko-matematychne modeliuvannia ta informatsiyni tekhnolohiyi, 4, 7–11.
Nahushev, A. M. (1995). Uravneniya matematicheskoy fiziki. Moscow: Vysshaya shkola, 304.
Nytrebych, Z., Malanchuk, O., Il'kiv, V., Pukach, P. (2017). On the solvability of two-point in time problem for PDE. Italian Journal of Pure and Applied Mathematics, 38, 715–726.
Bondarenko, B. A. (1987). Bazisnye sistemy polinomnyh i kvazipolinomnyh resheniy uravneniy v chastnyh proizvodnyh. Tashkent: Fan, 148.
Chaban A. Matematychne modeliuvannia kolyvnykh protsesiv v elektromekhanichnykh systemakh. Lviv: Vydavnytstvo Tarasa Soroky, 2008. 328 p.
Pukach P. Ya., Kuzio I. V. Resonance phenomena in quasi-zero stiffness vibration isolation systems // Naukovyi Visnyk Natsіonalnoho Hіrnychoho Unіversytetu. 2015. Issue 3. P. 62–67.
Bondarenko V. I., Samusya V. I., Smolanov S. N. Mobil'nye pod'emnye ustanovki dlya avariyno-spasatel'nyh rabot v shahtnyh stvolah // Gorniy zhurnal. 2005. Issue 5. P. 99–100.
Magrab E. B. An Engineer’s Guide to Mathematica. John Wiley and Sons, 2014. 452 p.
Friedman A. Partial differential equation. New York: Dover Publications, 2011. 272 p.
Homogeneous problem with two-point conditions in time for some equations of mathematical physics / Nytrebych Z. M., Malanchuk O. M., Il'kiv V. S., Pukach P. Ya. // Azerbaijan Journal of Mathematics. 2017. Vol. 7, Issue 2. P. 180–196.
Ptashnyk B. Y. Zadacha typu Valle-Pussena dlia hiperbolichnykh rivnian iz stalymy koefitsientamy // Dop. AN URSR. Ser. A. 1967. Issue 10. P. 1254–1257.
Ptashnyk B. Y. N-liniyna zadacha dlia hiperbolichnykh rivnian iz stalymy koefitsientamy // Visnyk Lvivskoho politekhnichnoho instytutu. 1967. Issue 16. P. 80–87.
Ptashnik B. I. Nekorrektnye granichnye zadachi dlya differenci-al'nyh uravneniy s chastnymi proizvodnymi. Kyiv: Naukova dumka, 1984. 264 p.
Il’kiv V. S., Nytrebych Z. M., Pukach P. Y. Boundary-value problems with integral conditions for a system of Lamé equations in the space of almost periodic functions // Electronic Journal of Differential Equations. 2016. Issue 304. P. 1–12.
Borok V. M. Klassy edinstvennosti resheniya kraevoy zadachi v beskonechnom sloe // DAN SSSR. 1968. Vol. 183, Issue 5. P. 995–998.
Borok V. M., Perel'man M. A. Unique solution classes for a multipoint boundary value problem in an infinite layer // Izv. vuzov. Matematika. 1973. Issue 8. P. 29–34.
Vilents I. L. Klasy yedynosti rozviazku zahalnoi kraiovoi zadachi v shari dlia system liniynykh dyferentsialnykh rivnian u chastynnykh pokhidnykh // Dop. AN URSR. Ser. А. 1974. Issue 3. P. 195–197.
Vallee-Poussin Ch. J. Sur l'equation differentielle lineaire du second ordre. Determination d'une integrale par deux valeurs assignees. Extension aux equations d'ordre n // Journ. Math. de pura et appl. 1929. Vol. 8. P. 125–144.
Picone M. On the exceptional values of a parameter on which an ordinary linear differential equation of the second order depends. Pisa: Scuola Normale Superiore, 1910. 144 p.
Feynman R. P., Leighton R. B., Sands M. The Feynman Lectures on Physics. Mainly electromagnetism and matter. New York: New millennium ed., 1963. 566 p.
Lamoureux M. P. The mathematics of PDEs and the wave equation. Calgary: Seismic Imaging Summer School, 2006. 39 p.
Lie K. A. The wave equation in 1D and 2D. Dep. of Informatics University of Oslo, 2005. 48 p.
Chalyi A. V. Medical and biological physics. Vinnytsia: Nova knyga, 2017. 476 p.
Blagitko B., Zayachuk I., Pyrogov O. The Mathematical Model of the Pulse Wave Propagation in Large Blood Vascular // Fizyko-matematychne modeliuvannia ta informatsiyni tekhnolohiyi. 2006. Issue 4. P. 7–11.
Nahushev A. M. Uravneniya matematicheskoy fiziki. Moscow: Vysshaya shkola, 1995. 304 p.
On the solvability of two-point in time problem for PDE / Nytrebych Z., Malanchuk O., Il'kiv V., Pukach P. // Italian Journal of Pure and Applied Mathematics. 2017. Issue 38. P. 715–726.
Bondarenko B. A. Bazisnye sistemy polinomnyh i kvazipolinomnyh resheniy uravneniy v chastnyh proizvodnyh. Tashkent: Fan, 1987. 148 p.

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