Two-point boundary value problem for a partial differential equation in spaces of periodic functions

  • V.S. Ilkiv Lviv Polytechnic National University, Lviv, Ukraine
  • Z.M. Nytrebych Lviv Polytechnic National University, Lviv, Ukraine
  • P.Y. Pukach Lviv Polytechnic National University, Lviv, Ukraine
  • M.I. Vovk Lviv Polytechnic National University, Lviv, Ukraine
Keywords: boundary value problem, partial differential equation, spatial variable, Fourier coefficients

Abstract

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$.

The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.
We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

Author Biographies

V.S. Ilkiv, Lviv Polytechnic National University, Lviv, Ukraine

Lviv Polytechnic National University, Lviv, Ukraine

Z.M. Nytrebych, Lviv Polytechnic National University, Lviv, Ukraine

Lviv Polytechnic National University, Lviv, Ukraine

P.Y. Pukach, Lviv Polytechnic National University, Lviv, Ukraine

Lviv Polytechnic National University, Lviv, Ukraine

M.I. Vovk, Lviv Polytechnic National University, Lviv, Ukraine

Lviv Polytechnic National University, Lviv, Ukraine

References

R.P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinary differential equations with singularities, Journ. Math. Anal. Appl., 297 (2004), 131-151. doi:10.1016/j.jmaa.2004.05.002

G. Akram, M. Tehseen, S. Siddiqi, Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM, British Journ. of Math. & Comp. Science, 3 (2013), 180-194. doi:10.9734/BJMCS/2013/2362.

I.I. Antypko, M.A. Perelman, On the uniqueness classes of the nonlocal multi-point boundary problem solutionsin the in nite layer, The function theory, funct. analysis and appl., 16 (1972), 98-109. (in Russian)

A.T. Assanova, On a solvability of a family of a multi-point boundary value problems for system of differential equations and the their application to the nonlocal boundary value problems, Math. Journ., Almaty, 13 (2013) no.3, 58-73.

A.T. Assanova, A.E. Imanchiev, On conditions of the solvability of nonlocal multi-point boundary value problems for quasi-linear systems of hyperbolic equations, Eurasian Math. Journ., 6 (2015), no.4, 19-28.

Banichuk NV, Ivanova SYu, Sharanyuk AV. Dynamics of Structures, Analysis and Optimization. Moscow: Nauka, 1989 (in Russian).

P. Bassanini, Iterative methods for quasilinear hyperbolic systems in the rst canonic form, Appl. Anal., 12,(1981), no.2, 105-117.

V.M. Borok, Uniqueness classes for the solution of a boundary problem with an in finite layer for systems of linear partial differential equations with constant coefficients, Math. USSR-Sb., 8 (1969), no.2, 275-285.

V.M. Borok, M.A. Perel'man, On uniqueness classes for a solution of a boundary value problem in an infi nite layer, Izv. Vyssh. Uchebn. Zaved., Ser. Math., 8, (1973), 29-34. (in Russian)

P. Cesari, A boundary value problem for quasilinear hyperbolic systems, Riv. math. Univ. Parma, 3 (1974), no.2, 107-131.

A. Dhamacharoen, K. Chompuvised, An efficient method for solving multipoint equation boundary value problems, Intern. Journ. of Math. and Comp. Sciences, 7 (2013), no.3, 329-333.

Yu.A. Dubinskii, A method of solution of partial differential equations, Dokl. Akad. Nauk SSSR, 258 (1981), no.4, 780-784.

V.I. Erofeev, V.V. Kazhaev, N.P. Semerikova, Waves in the rods. Dispersion. Dissipation. Nonlinearity. Moscow: Fizmatlit, 2002 (in Russian with an abstract in English).

L.V. Fardigola, Well-posed problems in a layer with differential operators in a boundary condition, Ukr. Mat. Zh., 44 (1992), no.8, 1083-1090. (in Russian)

L.V. Fardigola, Nonlocal two-point boundary-value problems in a layer with differential operators in a boundary condition, Ukr. Mat. Zh., 47 (1995), no.8, 1122-1128. (in Russian)

Filippov A.P. Oscillations of the deformable systems. Moscow: Mashinostroenie, 1970 (in Russian).

V.I. Gorbachuk, M.L. Gorbachuk, Boundary value problems for operator differential equations, Netherlands: Kluwer Academic Publishers, 1990.

J.R. Graef, L.Kong, Q. Kong, Higher order multi-point boundary value problems, Mathematische Nachrichten, 284 (2011), no.1, 39-52. doi:/10.1002/mana.200710179

R.J. Gu, K.L. Kuttler, M. Shillor, Frictional wear of a thermoelastic beam Journ. Math. Anal. And Appl., 242 (2000), 212-236.

C.P. Gupta, S.I. Tro mchuk, Solvability of a multi-point boundary value problem of Neumann type, Abstr. and Appl. Analysis, 4 (1999), no.2, 71-81.

R.A. Horn, Ch.R. Johnson, Matrix analysis, NY: Cambridge University Press, 2012.

V.S. Il'kiv, A multipoint nonlocal problem for partial differential equations, Diff. Uravn., 23 (1987), no.3, 487-492. (in Russian)

V.S. Il'kiv, Incorrect nonlocal boundary value problem for partial differential equations, In: Kadets V, Zelazko W (editors). North-Holland Math. Studies. Elsevier, 197 (2004), 115-121.

V.S. Il'kiv, Nytrebych Z.M., Estimate of the measure of level set for the solutions of differential equations with constant coefficients, Journ. of Math. Sciences. 217 (2016), no.2, 166-175. doi:10.1007/s10958-016-2964-1

V.S. Il'kiv, Nytrebych Z.M., Pukach P.Y. Boundary-value problems with integral conditions for a system of Lame equations in the space of almost periodic functions, Electron. Journ. of Diff. Eq., 304, (2016), 1-12.

V.S. Il'kiv, Nytrebych Z.M., Pukach P.Y., Nonlocal problem with moment conditions for hyperbolic equations, Electr Journ. of Diff. Eq., 265 (2017), 1-9.

V.S. Il'kiv, I.Ya. Savka, Nonlocal two-point problem for partial differential equations with linearly dependent coefficients, Journ. of Math. Sciences, 167 (2010), no.1, 47-61. doi:10.1007/s10958-010-9901-5

V.S. Il'kiv, N.I. Strap, Solvability of the nonlocal boundary-value problem for a system of differential operator equations in the Sobolev scale of spaces and in a re ned scale, Ukrain. Math. Journ., 67 (2015), no.5, 690-710. doi:10.1007/s11253-015-1108-y

V.S. Il'kiv, N.I. Strap, Nonlocal boundary value problem for a differential-operator equation with nonlinear right part in a complex domain, Math. Stud., 45 (2016),no.2, 170-181. doi:10.15330/ms.45.2.170-181

V.S. Il'kiv, N.I. Strap, Nonlocal boundary-value problem for a differential operator equation with weak nonlinearity in the spaces of Dirichlet-Taylor series with xed spectrum, Journ. of Math. Sciences, 231 (2018), no.4, 572-585. doi:10.1007/s10958-018-3835-8

V.S. Il'kiv, N.I. Strap, Solvability of a nonlocal boundary-value problem for the operator-differential equation with weak nonlinearity in a re ned scale of Sobolev spaces, Journ. of Math. Sciences, 218 (2016), no.1, 1-15.

doi:10.1007/s10958-016-3006-8

T.S. Kagadiy, A.H. Shporta, The asymptotic method in problems of the linear and nonlinear elasticity theory, Naukovyi Visnyk Natsionalnoho Hirnychoho Univ., 3, (2015), 76-81.

P.I. Kalenyuk, Z.M. Nytrebych, On an operational method of solving initial-value problems for partial differential equations induced by generalized separation of variables, Journ. of Math. Sciences, 97 (1999), no.1, 3879-3887.

I. Kiguradze, Some optimal conditions for the solvability of two-point singular boundary value problems, Funct. Diff. Equations, 10, (2003), 259-281.

L. Kong, Q. Kong, M.K. Kwong, J.S.W. Wong, Linear Sturm-Liouville problems with multi-point boundary conditions, Math. Nachricht, 286 (2013), no.11-12, 1167-1179. doi:10.1002/mana.201200187

R. Ma, D. O'Regan, Solvability of singular second order m-point boundary value problems, Journ. of Math. Analysis and Appl., 301 (2005), no.1, 124-134. doi:10.1016/j.jmaa.2004.07.009

Magrab EB. Vibrations of Elastic Systems with Applications to MEMS and NEMS. NY: Springer, 2012.

A.A. Makarov, On the existence of a well-posed two-point boundary value problem in a layer for systems of pseudo-differential equations, Differ. Uravn., 30 (1994), no.1, 144-150. (in Russian)

Z.M. Nitrebich, An operator method of solving the Cauchy problem for a homogeneous system of partial differential equations, J. Math. Sci., 81 (1996), no.6, 3034-3038.

J. Niu, Y.Zh. Lin, Chi.P. Zhang, Approximate solution of nonlinear multi-point boundary value problem on the half-line, Math. Modelling and Analysis, 17 (2012), no.2, 190-202. doi:10.3846/13926292.2012.660889

Z. Nytrebych, V. Il'kiv, P. Pukach, O. Malanchuk, On nontrivial solutions of homogeneous Dirichlet problem for partial differential equations in a layer, Krag. J. of Math., 42 (2018), no.2, 193-207.

Z. Nytrebych, O. Malanchuk, V. Il'kiv, P. Pukach, Homogeneous problem with two-point in time conditions for some equations of mathematical physics, Azerb. Jorn. of Math., 7 (2017), no.2, 174-190.

Z. Nytrebych, O. Malanchuk, V. Il'kiv, P. Pukach, On the solvability of two-point in time problem for PDE, Italian J. of Pure and Appl. Math., 38 (2017), 715-726.

M. Picone, Sui valori eccezionali di un parametro do cui dipende un equazione differentiale lineare ordinaria del secondo ordine, Pisa, 1909.

B.Yo. Ptashnyk, Problem of Vallee-Poussin type for hyperbolic equations with constant coefficients, Dopovidi Academii Nauk URSR, 10, (1966), 1254-1257. (in Ukrainian)

B.Yo. Ptashnyk, Problem of Vallee-Poussin type for linear hyperbolic equations with variable coefficients, Dopovidi Academii Nauk URSR, 2 (1967), 127-130. (in Ukrainian)

B.Yo. Ptashnyk, V.S. Il'kiv, I.Ya. Kmit', V.M. Polishchuk, Nonlocal boundary-value problems for partial differential equations. Kyiv: Naukova Dumka, 2002. (in Ukrainian)

B.Yo. Ptashnyk, M.M. Symotyuk, Multipoint problem with multiple nodes for partial differential equations, Ukr. Math. Journ., 55 (2003), no.3, 481-497.

P. Pucci, Problemi ai limiti per sistemi di equazioni iperboliche, Boll. Unione Mat. Ital. B, bf16 (1979), no.5, 187-99.

M.M. Symotyuk, I.R. Tymkiv, Problem with two-point conditions for parabolic equation of second order on time, Carpathian Math. Publ., 6 (2014), no.2, 351{359. (in Ukrainian) doi:10.15330/cmp.6.2.351-359

J.D. Tamarkin, On some general problems in the theory of ordinary linear differential equations and on the expansion in series of arbitrary functions, Petrograd, 1917 (in Russian).

M. Tatari, M. Dehgha, An efficient method for solving multi-point boundary value problems and applications in physics, Journ. of Vibr. and Contr., 18 (2011), no.8, 1116{1124. doi:10.1177/1077546311408467

Ch.J. Vallee-Poussin 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., 9 (1929), no.8, 125-144.

Published
2020-10-06
How to Cite
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Ilkiv V, Nytrebych Z, Pukach P, Vovk M. Two-point boundary value problem for a partial differential equation in spaces of periodic functions. Mat. Stud. [Internet]. 2020Oct.6 [cited 2020Oct.27];54(1):79-0. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/4
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