DOI: https://doi.org/10.15587/1729-4061.2019.155148

Analytical method to study a mathematical model of wave processes under two­point time conditions

Zinovii Nytrebych, Volodymyr Ilkiv, Petro Pukach, Oksana Malanchuk, Ihor Kohut, Andriy Senyk

Abstract


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.

The main practical application of the constructed method is the possibility to adequately mathematically model the oscillatory systems, which takes into consideration a possibility to control the system's parameters. Such a control over parameters makes it possible to perform optimal synthesis and design of parameters for the relevant technical systems in order to analyze and account for special features in the dynamic modes of oscillations

Keywords


oscillatory systems; mathematical models of wave processes; differential-symbol method; two-point problem; wave equation

References


Pukach, P., Il’kiv, V., Nytrebych, Z., Vovk, M. (2017). On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model. Opuscula Mathematica, 37 (45), 735. doi: https://doi.org/10.7494/opmath.2017.37.5.735

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.

Samusya, V., Oksen, Y., Radiuk, M. (2013). Heat pumps for mine water waste heat recovery. Mining of Mineral Deposits, 153–157. doi: https://doi.org/10.1201/b16354-27

Bayat, M., Pakar, I., Bayat, M. (2014). Approximate analytical solution of nonlinear systems using homotopy perturbation method. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 230 (1), 10–17. doi: https://doi.org/10.1177/0954408914533104

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.

Pinchover, Y., Rubinstein, J. (2005). An Introduction to Partial Differential Equations. Cambridge University Press, 385. doi: https://doi.org/10.1017/cbo9780511801228

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.

Nytrebych, Z., Il’kiv, V., Pukach, P., Malanchuk, O. (2018). On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer. Kragujevac Journal of Mathematics, 42 (2), 193–207. doi: https://doi.org/10.5937/kgjmath1802193n

Malanchuk, O., Nytrebych, Z. (2017). Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables. Open Mathematics, 15 (1). doi: https://doi.org/10.1515/math-2017-0009

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.

Ptashnik, B. I., Salyga, B. O. (1984). An analog of the multipoint problem for partial differential equations with variable coefficients. Ukrainian Mathematical Journal, 35 (6), 628–635. doi: https://doi.org/10.1007/bf01056224

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.

Ptashnyk, B. I., Symotyuk, М. М. (2003). Multipoint Problem with Multiple Nodes for Partial Differential Equations. Ukrainian Mathematical Journal, 55 (3), 481–497. doi: https://doi.org/10.1023/a:1025881429063

Ptashnyk, B. I., Symotyuk, М. М. (2003). Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients. Ukrainian Mathematical Journal, 55 (2), 293–310. doi: https://doi.org/10.1023/a:1025468413500

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.

Borok, V. M. (1969). Uniqueness classes for the solution of a boundary problem with an infinite layer for systems of linear partial differential equations with constant coefficients. Mathematics of the USSR-Sbornik, 8 (2), 275–285. doi: https://doi.org/10.1070/sm1969v008n02abeh001120

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.

Agarwal, R. P., Kiguradze, I. (2004). On multi-point boundary value problems for linear ordinary differential equations with singularities. Journal of Mathematical Analysis and Applications, 297 (1), 131–151. doi: https://doi.org/10.1016/j.jmaa.2004.05.002

Akram, G., Tehseen, M., Siddiqi, S. (2013). Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM. British Journal of Mathematics & Computer Science, 3 (2), 180–194. doi: https://doi.org/10.9734/bjmcs/2013/2362

Graef, J. R., Kong, L., Kong, Q. (2010). Higher order multi-point boundary value problems. Mathematische Nachrichten, 284 (1), 39–52. doi: https://doi.org/10.1002/mana.200710179

Gupta, C. P., Trofimchuk, S. (1999). Solvability of a multi-point boundary value problem of Neumann type. Abstract and Applied Analysis, 4 (2), 71–81. doi: https://doi.org/10.1155/s1085337599000093

Olver, P. J. (2014). Introduction to partial differential equations. Springer. doi: https://doi.org/10.1007/978-3-319-02099-0

Friedman, A. (1961). The wave equation for differential forms. Pacific Journal of Mathematics, 11 (4), 1267–1279. doi: https://doi.org/10.2140/pjm.1961.11.1267

Feynman, R. P., Leighton, R. B., Sands, M. (1963). The Feynman Lectures on Physics. Mainly electromagnetism and matter. New York: New millennium ed., 566.

Li, B. (2008). Wave Equations for Regular and Irregular Water Wave Propagation. Journal of Waterway, Port, Coastal, and Ocean Engineering, 134 (2), 121–142. doi: https://doi.org/10.1061/(asce)0733-950x(2008)134:2(121)

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.

Hughes, T. J. R., Lubliner, J. (1973). On the one-dimensional theory of blood flow in the larger vessels. Mathematical Biosciences, 18 (1-2), 161–170. doi: https://doi.org/10.1016/0025-5564(73)90027-8

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.

Nitrebich, Z. M. (1996). An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations. Journal of Mathematical Sciences, 81 (6), 3034–3038. doi: https://doi.org/10.1007/bf02362589

Kalenyuk, P. I., Nytrebych, Z. M. (1999). On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables. Journal of Mathematical Sciences, 97 (1), 3879–3887. doi: https://doi.org/10.1007/bf02364928

Kalenyuk, P. I., Kohut, I. V., Nytrebych, Z. M. (2012). Problem with integral condition for a partial differential equation of the first order with respect to time. Journal of Mathematical Sciences, 181 (3), 293–304. doi: https://doi.org/10.1007/s10958-012-0685-7

Nitrebich, Z. M. (1996). A boundary-value problem in an unbounded strip. Journal of Mathematical Sciences, 79 (6), 1388–1392. doi: https://doi.org/10.1007/bf02362789

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.

Nytrebych, Z. М., Malanchuk, O. M. (2017). The differential-symbol method of solving the two-point problem with respect to time for a partial differential equation. Journal of Mathematical Sciences, 224 (4), 541–554. doi: https://doi.org/10.1007/s10958-017-3434-0

Nytrebych, Z. M., Malanchuk, O. M. (2017). The differential-symbol method of solving the problem two-point in time for a nonhomogeneous partial differential equation. Journal of Mathematical Sciences, 227 (1), 68–80. doi: https://doi.org/10.1007/s10958-017-3574-2

Bondarenko, B. A. (1987). Bazisnye sistemy polinomnyh i kvazipolinomnyh resheniy uravneniy v chastnyh proizvodnyh. Tashkent: Fan, 148.

Hayman, W. K., Shanidze, Z. G. (1999). Polynomial solutions of partial differential equations. Methods and Applications of Analysis, 6 (1), 97–108. doi: https://doi.org/10.4310/maa.1999.v6.n1.a7

Hile, G. N., Stanoyevitch, A. (2004). Heat polynomial analogues for equations with higher order time derivatives. Journal of Mathematical Analysis and Applications, 295 (2), 595–610. doi: https://doi.org/10.1016/j.jmaa.2004.03.039


GOST Style Citations


On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model / Pukach P., Il’kiv V., Nytrebych Z., Vovk M. // Opuscula Mathematica. 2017. Vol. 37, Issue 45. P. 735. doi: https://doi.org/10.7494/opmath.2017.37.5.735 

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.

Samusya V., Oksen Y., Radiuk M. Heat pumps for mine water waste heat recovery // Mining of Mineral Deposits. 2013. P. 153–157. doi: https://doi.org/10.1201/b16354-27 

Bayat M., Pakar I., Bayat M. Approximate analytical solution of nonlinear systems using homotopy perturbation method // Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. 2014. Vol. 230, Issue 1. P. 10–17. doi: https://doi.org/10.1177/0954408914533104 

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.

Pinchover Y., Rubinstein J. An Introduction to Partial Differential Equations. Cambridge University Press, 2005. 385 p. doi: https://doi.org/10.1017/cbo9780511801228 

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.

On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer / Nytrebych Z., Il’kiv V., Pukach P., Malanchuk O. // Kragujevac Journal of Mathematics. 2018. Vol. 42, Issue 2. P. 193–207. doi: https://doi.org/10.5937/kgjmath1802193n 

Malanchuk O., Nytrebych Z. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables // Open Mathematics. 2017. Vol. 15, Issue 1. doi: https://doi.org/10.1515/math-2017-0009 

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.

Ptashnik B. I., Salyga B. O. An analog of the multipoint problem for partial differential equations with variable coefficients // Ukrainian Mathematical Journal. 1984. Vol. 35, Issue 6. P. 628–635. doi: https://doi.org/10.1007/bf01056224 

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.

Ptashnyk B. I., Symotyuk М. М. Multipoint Problem with Multiple Nodes for Partial Differential Equations // Ukrainian Mathematical Journal. 2003. Vol. 55, Issue 3. P. 481–497. doi: https://doi.org/10.1023/a:1025881429063 

Ptashnyk B. I., Symotyuk М. М. Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients // Ukrainian Mathematical Journal. 2003. Vol. 55, Issue 2. P. 293–310. doi: https://doi.org/10.1023/a:1025468413500 

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.

Borok, V. M. Uniqueness classes for the solution of a boundary problem with an infinite layer for systems of linear partial differential equations with constant coefficients // Mathematics of the USSR-Sbornik. 1969. Vol. 8, Issue 2. P. 275–285. doi: https://doi.org/10.1070/sm1969v008n02abeh001120 

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.

Agarwal R. P., Kiguradze I. On multi-point boundary value problems for linear ordinary differential equations with singularities // Journal of Mathematical Analysis and Applications. 2004. Vol. 297, Issue 1. P. 131–151. doi: https://doi.org/10.1016/j.jmaa.2004.05.002 

Akram G., Tehseen M., Siddiqi S. Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM // British Journal of Mathematics & Computer Science. 2013. Vol. 3, Issue 2. P. 180–194. doi: https://doi.org/10.9734/bjmcs/2013/2362 

Graef J. R., Kong L., Kong Q. Higher order multi-point boundary value problems // Mathematische Nachrichten. 2010. Vol. 284, Issue 1. P. 39–52. doi: https://doi.org/10.1002/mana.200710179 

Gupta C. P., Trofimchuk S. Solvability of a multi-point boundary value problem of Neumann type // Abstract and Applied Analysis. 1999. Vol. 4, Issue 2. P. 71–81. doi: https://doi.org/10.1155/s1085337599000093 

Olver P. J. Introduction to partial differential equations. Springer, 2014. doi: https://doi.org/10.1007/978-3-319-02099-0 

Friedman A. The wave equation for differential forms // Pacific Journal of Mathematics. 1961. Vol. 11, Issue 4. P. 1267–1279. doi: https://doi.org/10.2140/pjm.1961.11.1267 

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.

Li B. Wave Equations for Regular and Irregular Water Wave Propagation // Journal of Waterway, Port, Coastal, and Ocean Engineering. 2008. Vol. 134, Issue 2. P. 121–142. doi: https://doi.org/10.1061/(asce)0733-950x(2008)134:2(121) 

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.

Hughes T. J. R., Lubliner J. On the one-dimensional theory of blood flow in the larger vessels // Mathematical Biosciences. 1973. Vol. 18, Issue 1-2. P. 161–170. doi: https://doi.org/10.1016/0025-5564(73)90027-8 

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.

Nitrebich Z. M. An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations // Journal of Mathematical Sciences. 1996. Vol. 81, Issue 6. P. 3034–3038. doi: https://doi.org/10.1007/bf02362589 

Kalenyuk P. I., Nytrebych Z. M. On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables // Journal of Mathematical Sciences. 1999. Vol. 97, Issue 1. P. 3879–3887. doi: https://doi.org/10.1007/bf02364928 

Kalenyuk P. I., Kohut I. V., Nytrebych Z. M. Problem with integral condition for a partial differential equation of the first order with respect to time // Journal of Mathematical Sciences. 2012. Vol. 181, Issue 3. P. 293–304. doi: https://doi.org/10.1007/s10958-012-0685-7 

Nitrebich Z. M. A boundary-value problem in an unbounded strip // Journal of Mathematical Sciences. 1996. Vol. 79, Issue 6. P. 1388–1392. doi: https://doi.org/10.1007/bf02362789 

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.

Nytrebych Z. М., Malanchuk O. M. The differential-symbol method of solving the two-point problem with respect to time for a partial differential equation // Journal of Mathematical Sciences. 2017. Vol. 224, Issue 4. P. 541–554. doi: https://doi.org/10.1007/s10958-017-3434-0 

Nytrebych Z. M., Malanchuk O. M. The differential-symbol method of solving the problem two-point in time for a nonhomogeneous partial differential equation // Journal of Mathematical Sciences. 2017. Vol. 227, Issue 1. P. 68–80. doi: https://doi.org/10.1007/s10958-017-3574-2 

Bondarenko B. A. Bazisnye sistemy polinomnyh i kvazipolinomnyh resheniy uravneniy v chastnyh proizvodnyh. Tashkent: Fan, 1987. 148 p.

Hayman W. K., Shanidze Z. G. Polynomial solutions of partial differential equations // Methods and Applications of Analysis. 1999. Vol. 6, Issue 1. P. 97–108. doi: https://doi.org/10.4310/maa.1999.v6.n1.a7 

Hile G. N., Stanoyevitch A. Heat polynomial analogues for equations with higher order time derivatives // Journal of Mathematical Analysis and Applications. 2004. Vol. 295, Issue 2. P. 595–610. doi: https://doi.org/10.1016/j.jmaa.2004.03.039 







Copyright (c) 2019 Zinovii Nytrebych, Volodymyr Ilkiv, Petro Pukach, Oksana Malanchuk, Ihor Kohut, Andriy Senyk

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ISSN (print) 1729-3774, ISSN (on-line) 1729-4061