A non-local problem for a loaded mixed-type equation with a integral operator

O. Kh. Abdullayev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 220-240.

Résumé

We study the existence and uniqueness of the solution of non-local boundary value problem for the loaded elliptic-hyperbolic equation $$ u_{xx} + \mathop{\mathrm{sgn}} (y) u_{yy} + \frac{1 - \mathop{\mathrm{sgn}} (y)}{2} \sum\limits_{k = 1}^n {R_k}(x, u(x, 0)) = 0 $$ with integral operator $$ {R_k}\bigl(x, u(x, 0)\bigr) = \left\{ \begin{array}{lc} {p_k}(x)D_{x\,\,1}^{ - {\alpha _k}}u(x, 0), q \le x \le 1,\\[2mm] {r_k}(x)D_{ - 1\,x}^{ - {\beta _k}}u(x, 0), - 1 \le x \le - q, \end{array} \right. $$ where $$ \begin{array}{l} \displaystyle D_{ax}^{ - {\alpha _k}}f(x) = \frac{1}{{\Gamma ({\alpha _k})}} \int _a^x \frac{f(t)}{(x - t)^{1-{\alpha _k} }}dt, \\ \displaystyle D_{xb}^{ - {\beta _k}}f(x) = \frac{1}{{\Gamma ({\beta _k})}} \int _x^b \frac{f(t)}{(t - x)^{1-{\beta _k}}}dt , \end{array} $$ in double-connected domain $\Omega $, bounded with two lines: $$ \sigma _1:~x^2 + y^2 = 1,\quad \sigma _2:~ x^2 + y^2 = q^2 \quad \text{at $y > 0$,}$$ and characteristics: $$ A_j C_1:~ x + ( - 1)^j y = ( - 1)^{j + 1},\quad B_j C_2:~x + ( - 1)^j y = ( - 1)^{j + 1} \cdot q$$ of the considered equation at $y 0$, where $0 q 1$, $j = 1, 2$; $A_1 ( 1; 0)$, $A_2( - 1; 0)$, $B_1(q; 0)$, $B_2( - q; 0)$, $C_1(0; - 1)$, $C_2(0; - q)$, $\beta _k$, $\alpha _k > 0$. Uniqueness of the solution of investigated problem was proved by an extremum principle for the mixed type equations. Thus we need to prove that, the loaded part of the equation is identically equal to zero if considerate problem is homogeneous. Existence of the solution of the problem was proved by a method of the integral equations, thus the theory of the singular integral equations and Fredholm integral equations of the second kind were widely used.

Détail

Keywords: loaded equation, integral operator, elliptic-hyperbolic type equations, double-connected domain, existence and uniqueness of solution, extremum principle, integral equations.

Bibliographie

[1] Nahushev A. M., “The Darboux problem for a certain degenerate second order loaded integrodifferential equation”, Differ. Uravn., 12:1 (1976), 103–108 (In Russian) | MR | Zbl

[2] Nakhushev A. M., “Loaded equations and their applications”, Differ. Equations, 19:1 (1983), 74–81 | MR | Zbl

[3] Nakhushev A. M., Nagruzhennye uravneniia i ikh primeneniia [Loaded equations and their applications], Nauka, Moscow, 2012, 233 pp. (In Russian)

[4] Eleev V. A., “Some boundary value problems for mixed loaded equations of second and third order”, Differ. Equations, 30:2 (1994), 210–217 | MR | Zbl

[5] Dzarakhohov A. V., Eleev V. A., “On a nonlocal boundary value problem for a third-order loaded equation”, Vladikavkaz. Mat. Zh., 6:3 (2004), 36–46 (In Russian) | MR | Zbl

[6] Kaziev V. M., “Darboux's problem for a loaded second-order integrodifferential equation”, Differ. Equations, 14:1 (1978), 130–133 | MR | Zbl

[7] Kaziev V. M., “Goursat's problem for a loaded integrodifferential equation”, Differ. Equations, 17:2 (1981), 216–220 | MR | Zbl

[8] Lanin I. N., “Boundary-value problem for a loaded third-order hyperbolic-parabolic equation”, Differ. Equations, 17:1 (1981), 66–72 | MR | Zbl

[9] Islamov B. I., Kuryazov D. M., “On a boundary value problem for loaded equation of the second order”, Dokl. Akad. Nauk Resp. Uzb., 1996, no. 1–2, 3–6 (In Russian) | MR | Zbl

[10] Kuryazov D. M., “A boundary value problem for a loaded equation of mixed type with two lines of degeneracy”, Uzbekskii Matematicheskii Zhurnal, 1999, no. 5, 40–46 (In Russian) | MR

[11] Ramazanov M. I., “On a nonlocal problem for a loaded hyperbolic-elliptic equation in a rectangular domain”, Matematicheskii zhurnal. Almaty, 2:4 (2002), 75–81 (In Russian) http://www.math.kz/images/journal/2002-4/Ramazanov.pdf | MR | Zbl

[12] Khubiev K. U., “On one boundary value problem for a loaded mixed hyperbolic-parabolic type equation”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 7:2 (2005), 74–77 (In Russian)

[13] Sabitov K. B., Melisheva E. P., “The Dirichlet problem for a loaded mixed-type equation in a rectangular domain”, Russian Math. (Iz. VUZ), 57:7 (2013), 53–65 | DOI | MR | Zbl

[14] Sabitov K. B., “Initial-boundary problem for parabolic-hyperbolic equation with loaded summands”, Russian Math. (Iz. VUZ), 59:6 (2015), 23–33 | DOI | MR | Zbl

[15] Melisheva E. P., “Dirichlet problem for loaded equation of Lavrentiev–Bizadze”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2010, no. 6(80), 39–47 (In Russian)

[16] Abdullayev O. Kh., “About a method of research of the non-local problem for the loaded mixed type equation in double-connected domain”, Bulletin KRASEC. Phys. Math. Sci., 9:2 (2014), 3–12 | DOI

[17] Abdullayev O. Kh., “A boundary value problem for a loaded elliptic-hyperbolic type equation in a double-connected domain”, Vestnik KRAUNTs. Fiz.-mat. nauki [Bulletin KRASEC. Phys. Math. Sci.], 2014, no. 1(8), 33–48 (In Russian) | DOI | Zbl

[18] Islomov B. I., Abdullyaev O. Kh., “A Boundary value problem of Bitsadze type problem for a third-order elliptic-hyperbolic type equation in a double-connected domain”, Doklady Adygskoi (Cherkesskoi) Mezhdunarodnoi akademii nauk, 7:1 (2004), 42–46 (In Russian) | MR

[19] Bitsadze A. V., Boundary value problems for second order elliptic equations, North-Holland Series in Applied Mathematics and Mechanics, 5, North-Holland Publ., Amsterdam, 1968, 211 pp. | MR | Zbl | Zbl

[20] Muskhelishvili N. I., Singular integral equations, Wolters-Noordhoff Publ., Groningen, 1967, 447 pp. | MR | Zbl | Zbl