Mixed Boussinesq-type differential equation

T. K. Yuldashev

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 2 (2016), pp. 13-26.

Résumé

Mathematical modeling of many processes occurring in the real world leads to the study of direct and inverse problems for equations of mathematical physics. Mixed and boundary value problems for partial differential and integro-differential equations by virtue of their importance in the application are one of the most important parts of the theory of differential equations. In the case, when the boundary of the flow of physical process not applicable for measurements, as an additional information can be used at nonlocal conditions in the integral form. We propose a method of studying the one-value solvability of the nonlocal problem for a mixed Boussinesq-type differential equation. Such type of differential equations models many natural phenomena and appears in many fields of sciences. For this reason, a great importance in the works of many researchers was given to this type of equations. We use the spectral method based on Fourier method of separation of variables. Application of this method of separation of variables can improve the quality of formulation of the considered problem and facilitate the processing procedure. So in this article we consider in the rectangular area $D=\{(x,t)| 0$ a nonlocal mixed problem for a mixed Boussinesq-type differential equation \begin{equation*} \Im U = \begin{cases} U_t - U_{txx} - U_{xx} + {\nu}^2 U = 0, t>0,\\ U_{tt} - U_{ttxx} - U_{xx} + {\nu}^2 U = 0, t0. \end{cases} \tag{1} \end{equation*} where $\nu, \alpha$ and $\beta$ are real positive numbers. We study the problem: Find in the domain $D$ a function $U(x, t)$ satisfying the following conditions \begin{equation*} U(x,t) \in C (\overline{\Omega}) \cap C^1 (\Omega \cup \{x=0\} \cup \{x=1\}) \cap C^2 (\Omega_{\_}) \cap C^{2,1}_{x,t} (\Omega_{+} \cup \{t=\beta\}), \tag{2} \end{equation*} \begin{equation*} \Im U(x,t) \equiv 0, (x,t) \in \Omega_{\_} \cup \Omega_{+} \cup \{t=\beta\}, \tag{3} \end{equation*} \begin{equation*} U(0,t)=U(1,t), \quad -\alpha \le t \le \beta, \tag{4}, \end{equation*} \begin{equation*} U_x(0,t)=U_x(1,t), \quad -\alpha \le t \le \beta, \tag{5}, \end{equation*} \begin{equation*} \int_{-\alpha}^0 U(x,t) dt = \psi (x), \quad 0 \le x \le 1, \tag{6} \end{equation*} where $\psi (x)$ is given a sufficiently smooth function, $\psi (0) = \psi (1)$, $\psi ' (0) = \psi ' (1)$. First we prove that, if there exists a solution $U(x, t)$ of the problem (2)–(6), then it is unique only when the following condition is fulfilled \begin{equation*} \Delta_n (\alpha) = 1 - \cos \lambda_n \alpha + \frac {1} {\lambda_n} \sin \lambda_n \alpha \ne 0, \tag{7} \end{equation*} where $\lambda_n = \sqrt{\frac{\nu^2 + \mu_n^2}{1 + \mu_n^2}}$, $\mu_n = 2 \pi n, n = 1,2,\ldots$ We consider the case, when the condition (7) is violated. We suppose that $\Delta_n(\alpha)=0$ for some $\alpha$ and $n = m$, then homogeneous problem (2)–(6) in $\psi (x) \equiv 0$ has nontrivial solution $$ U_m (x,t) = X_m (x) \cdot T_m(t), $$ where $X_m(x): \{1, \cos \mu_m x, \sin \mu_m x\}$, \begin{equation*} T_m(t) = \begin{cases} e^{- \lambda_m^2 t}, t>0,\\ \cos \lambda_m t -\lambda_m \sin \lambda_m t, t 0. \end{cases} \end{equation*} The condition $\Delta_n (\alpha) = 0$ is equivalent to the equality $\lambda_n = \lambda_n \cos \lambda_n \alpha - \sin \lambda_n \alpha$. Hence we see that the equality $\Delta_n (\alpha) = 0$ is possible only when $$ \alpha = \frac{2 \pi k}{\lambda_n}, \quad k \in N. $$ For other values of $\alpha$ the condition (7) holds. We note that there is a constant $C_0 > 0$ such that for sufficiently large $n$ holds the estimate \begin{equation*} \inf_n |\Delta_n (\alpha)| \ge C_0 > 0. \tag{8} \end{equation*} If the conditions (7) and (8) are fulfilled, then the solution of problem (2)–(6) exists and it can be presented as the sum of series $$ U(x,t) = \frac{\vartheta_0 (t)}{2}+\sum_{n=1}^{\infty} \vartheta_n (t) \cos \mu_n x + \sum_{n=1}^{\infty} u_n (t) \sin \mu_n n x, $$ where \begin{equation*} u_n(t) = \begin{cases} A_n \psi_n e^{- \lambda_n^2 t}, t>0,\\ A_n (\cos \lambda_n t - \lambda_n \sin \lambda_n t) \widetilde{\psi}_n, t0, \end{cases} \end{equation*} \begin{equation*} \vartheta_n(t) = \begin{cases} A_n \widetilde{\psi}_n e^{- \lambda_k^2 t}, t>0,\\ A_n (\cos \lambda_n t - \lambda_n \sin \lambda_n t) \widetilde{\psi}_n, t0, \end{cases} \end{equation*} $$ A_n = \frac{1}{\Delta_n (\alpha)}. $$ According to the problem: $\psi(x) \in C^3[0;1]$ and on the segment $[0; 1]$ it has piecewise continuous derivative of fourth order and $$ \psi (0) = \psi (1), \psi '(0) = \psi '(1), \psi ''(0) = \psi ''(1), \psi '''(0) = \psi '''(1). $$ Further the theorem is held. We suppose $\Delta_n (\alpha) = 0$ 0 for some $\alpha$ and $n = k_1, \ldots, k_s$, where $1 \le k_1 k_2 \ldots k_s$, $s$ is fixed natural number and there are orthogonality conditions \begin{align*} \psi_n = 2 \int\limits_0^1 \psi (x) \sin \mu_n x dx = 0, n = k_1, \ldots, k_s,\\ {\widetilde{\psi}}_n = 2 \int\limits_0^1 \psi (x) \cos \mu_n x dx = 0, n = k_1, \ldots, k_s. \end{align*} Then the solution of the problem (2)–(6) exists, and it is defined as the sum of series \begin{align*} U(x,t) = \frac{\vartheta_0 (t)}{2} + \left( \sum_{n=1}^{k_1 - 1} + \sum_{n=k_1+1}^{k_2 - 1} + \ldots + \sum_{n=k_s+1}^{\infty} \right) u_n(t) \sin \mu_n x +\\ + \left( \sum_{n=1}^{k_1 - 1} + \sum_{n=k_1+1}^{k_2 - 1} + \ldots + \sum_{n=k_s+1}^{\infty} \right) \vartheta_n (t) \cos \mu_n x + \sum_m C_m U_m (x,t), \end{align*} where in the last sum $m$ takes the values $k, k_1, \ldots, k_s, C_m$ as arbitrary constants.

Détail

Keywords: mixed-value problem, mixed-type differential equation, spectral method, one-value solvability.
Mots-clés : Boussinesq-type equation, nonlocal condition

Bibliographie

[1] S.\;D. Algazin, I.\;A. Kiyko, Flutter of Plates and Shells, Nauka Publ., M., 2006, 248 pp.

[2] A.\;M. Akhtyamov, A.\;R. Ayupova, “On the Solution of the Problem of Diagnosing Defects in the Form of a Small Cavity in the Rod”, Zhurnal Srednevolzhskogo matematicheskogo obshestva, 12:3 (2010), 37–42

[3] A.\;D. Baev, S.\;A. Shabrov, Meach Mon, “On Uniqueness of Solution of the Mathematical Model of Forced String Oscillation Singularities”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2014, no. 1, 50–55

[4] M.\;Kh. Beshtokov, “A Numerical Method for Solving One Nonlocal Boundary-Value Problem for a Third-Order Hyperbolic Equation”, Zhurnal vychislitelnoy matematiki i matematicheskoy fiziki, 54:9 (2014), 1497–1514 | DOI | MR | Zbl

[5] I.\;M. Gelfand, “Some Questions of Analyses and Differential Equations”, Uspekhi matematicheskih nauk, 14:3 (1959), 3–19 | MR | Zbl

[6] D.\;G. Gordeziani, G.\;A. Avilishbili, “Solving the Nonlocal Problems for One-Dimensional Medium Oscillation”, Matematicheskoe modelirovanie, 12:1 (2000), 94–103 | MR | Zbl

[7] T.\;D. Dzhuraev, A. Sopuev, M. Mamazhanov, Boundary-Value Problems for the Equations of Parabolic-Hyperbolic Type, Fan Publ., Tashkent, 1986, 220 pp. | MR

[8] A.\;A. Zamyshlyaeva, “Sobolev-Type Mathematical Models of Higher Order”, Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Seriya: Matematicheskoe modelirovanie i programmirovanie, 7:2 (2014), 5–28 | Zbl

[9] O.\;S. Zikirov, “On Dirichlet Problem for Third-Order Hyperbolic Equations”, Izvestiya vuzov. Matematika, 2014, no. 7, 63–71 | MR | Zbl

[10] E.\;I. Moiseev, “Solvability of a Nonlocal Boundary Value Problem”, Differentsialnye uravneniya, 37:11 (2001), 1565–1567 | MR | Zbl

[11] L. S. Pulkina, “A Nonlocal Problem for a Hyperbolic Equation With Integral Conditions of the 1st Kind With Time-Dependent Kernels”, Izvestiya vuzov. Matematika, 2012, no. 10, 32–44 | MR | Zbl

[12] L.\;Kh. Rakhmanova, “Solution of a Nonlocal Problem for a Mixed-Type Parabolic-Hyperbolic Equation in a Rectangular Domain by the Spectral Method”, Izvestiya vuzov. Matematika, 2007, no. 11, 36–40 | MR | Zbl

[13] O.\;A. Repin, “An Analogue of the Nakhushev Problem for the Bitsadze–Lykov Equation”, Differentsialnye uravneniya, 38:10 (2002), 1412–1417 | MR | Zbl

[14] K.\;B. Sabitov, On the Theory of Mixed-Type Equations, Fizmatlit Publ., M., 2014, 301 pp.

[15] Yu.\;K. Sabitova, “Boundary-Value Problem With Nonlocal Integral Condition for Mixed-Type Equations With Degeneracy on the Transition Line”, Matematicheskie zametki, 98:3 (2015), 393–406 | DOI | MR | Zbl

[16] M.\;S. Salakhitdinov, A.\;K. Urinov, Boundary-Value Problems for the Mixed-Type Equations With Spectral Parameter, Fan Publ., Tashkent, 1997, 165 pp. | MR

[17] A. Sopuev, N.\;K. Arkabaev, “Conjugation Problems for Linear Pseudoparabolic Equations of Third Order”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika. Mekhanika, 21:1 (2013), 16–23 | MR

[18] M.\;V. Turbin, “Investigation of Initial-Boundary Value Problem for the Herschel–Bulkley Mathematical Fluid Model”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2013, no. 2, 246–257

[19] G. Whitham, Linear and Nonlinear Waves, Mir Publ., M., 1977, 622 pp. | MR

[20] E.\;A. Utkina, “On Third Order Equations With Pseudoparabolic Operator and With Shift of Arguments of Initial Function”, Izvestiya vuzov. Matematika, 2015, no. 5, 62–68 | Zbl

[21] Ya.\;S. Uflyand, “On the Question of the Distribution of Fluctuations in the Composite Electrical Lines”, Inzhenerno-fizicheskiy zhurnal, 7:1 (1964), 89–92

[22] F.\;I. Frankl, Selected Works in Gas Dynamics, Nauka Publ., M., 1973, 711 pp. | MR

[23] S.\;A. Shabrov, “On Estimates of the Impact of a Mathematical Function of the Fourth-Order Model”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, no. 2, 168–179

[24] M.\;Kh. Shkhanukov, “Some Boundary-Value Problems for a Third-Order Equation Arising in the Simulation of Fluid Flow in Porous Media”, Differentsialnye uravneniya, 18:4 (1982), 689–699 | MR | Zbl

[25] T.\;K. Yuldashev, “Nonlinear Integro-Differential Equation of Pseudoparabolic Type With Nonlocal Integral Condition”, Science Journal of Volgograd State University. Mathematics. Physics, 2016, no. 1 (32), 11–23

[26] T.\;K. Yuldashev, “On an Inverse Problem for a Partial Linear Fredholm Integro-Differential Equation”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika, 2015, no. 2, 180–189

[27] T.\;K. Yuldashev, “On a Certain Fredholm Partial Integro-Differential Equation of the Third Order”, Izvestiya vuzov. Matematika, 2015, no. 9, 74–79 | MR | Zbl

[28] T.\;K. Yuldashev, “Inverse Problem for Nonlinear Fredholm Integro-Differential Equation of Fourth Order With Degenerate Kernel”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seria «Fiziko-matematicheskie nauki», 19:4 (2015), 736–749 | DOI

[29] T.\;K. Yuldashev, “Inverse Problem for a Partial Fredholm Integro-Differential Equation of Third Order”, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seria «Fiziko-matematicheskie nauki», 34:1 (2014), 56–65 | DOI

[30] D.\;J. Benney, J.\;C. Luke, “Interactions of permanent waves of finite amplitude”, Journ. Math. Phys., 43 (1964), 309–313 | DOI | MR | Zbl