On a problem for the parabolic-hyperbolic type equation of~fractional order with non-linear loaded term

O. Kh. Abdullayev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 25 (2021) no. 1, pp. 7-20.

Résumé

We study the existence and uniqueness of solution of the non-local problem for the parabolic-hyperbolic type equation with non linear loaded term involving Caputo derivative $$ f(x) = \begin{cases} {u_{xx}}-_CD_{0y}^\alpha u+a_1(x)u^{p_1}(x,0), y > 0, \\ {u_{xx}}-{u_{yy}}+a_2(x)u^{p_2}(x,0), y0, \\ \end{cases} $$ where $$ {}_CD_{0y}^{\alpha }f(y) = \frac{1}{{\Gamma (1-\alpha )}} \int_0^y {(y - t)}^{-\alpha}f'(t)\,dt, \quad 0 \alpha 1, $$ $a_i(x)$ are given functions, $p_i$, $\alpha=\mathrm{const}$, besides $ p_i>0$ $(i=1,2)$, $0 \alpha 1$ in the domain $\Omega$ bounded with segments: $$ A_1 A_2 = \{ (x,y): x = 1, 0 y h\}, \quad B_1 B_2 = \{ (x,y): x = 0, 0 y h\},$$ $$B_2 A_2 = \{ (x,y): y = h, 0 x 1\} $$ at the $y > 0$, and characteristics: $$ A_1C: x - y = 1,\quad B_1C: x + y = 0$$ of the considered equation at $y 0$, where $A_1 (1, 0)$, $A_2 (1, h)$, $B_1( 0, 0)$, $B_2( 0, h)$, and $C(1/2, -1/2)$. Uniqueness of solution of the investigated problem was proved by an integral of energy. The existence of solution of the problem was proved by the method of integral equations. The theory of the second kind Fredholm type integral equations and the successive approximations method were widely used. We notice, that boundary value problems for the mixed type equations of fractional order with non linear loaded term have not been investigated.

Détail

Keywords: loaded equation, parabolic-hyperbolic type, Caputo fractional derivative, nonlinear integral equation, integral gluing condition, existence and uniqueness of solution.

Bibliographie

[1] Nakhushev A. M., Drobnoe ischislenie i ego primenenie [Fractional Calculus and Its Applications], Fizmatlit, Moscow, 2003, 272 pp. (In Russian) | Zbl

[2] Nakhushev A. M., Nagruzhennye uravneniia i ikh primeneniia [Loaded Equations and Their Applications], Nauka, Moscow, 2012, 232 pp. (In Russian)

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

[4] Melisheva E. P., “Dirichlet problem for loaded equation of Lavrentiev–Bizadze”, Vestn. Samar. Gos. Univ., Estestvennonauchn. Ser., 2010, no. 6(80), 39–47 (In Russian)

[5] Abdullayev O. Kh., “A non-local problem for a loaded mixed-type equation with a integral operator”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:2 (2016), 220–240 (In Russian) | DOI | Zbl

[6] Pskhu A. V., “The fundamental solution of a diffusion-wave equation of fractional order”, Izv. Math., 73:2 (2009), 351–392 | DOI | DOI | MR | Zbl

[7] Kilbas A. A. Repin O. A, “An analog of the Tricomi problem for a mixed type equation with a partial fractional derivative”, Fract. Calc. Appl. Anal., 13:1 (2010), 69–84 | Zbl

[8] Kadirkulov B. J., “Boundary problems for mixed parabolic-hyperbolic equations with two lines of changing type and fractional derivative”, EJDE, 2014:57 (2014), 1–7

[9] Salakhitdinov M. S. Karimov E. T., “On a nonlocal problem with gluing condition of integral form for parabolic-hyperbolic equation with Caputo operator”, Dokl. Akad. Nauk Resp. Uzbekistan, 2014, no. 4, 6–9 (In Russian)

[10] Berdyshev A. S., Cabada A., Karimov E. T., “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann–Liouville fractional differential operator”, Nonlinear Anal. Theory, Methods and Appl., 75:6 (2012), 3268–3273 | DOI | Zbl

[11] Sadarangani K., Abdullaev O. K., “A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative”, Adv. Differ. Equ., 2016 (2016), 241 | DOI | Zbl

[12] Abdullaev O. Kh., “Analog of the Gellerstedt problem for the mixed type equation with integral-differential operators of fractional order”, Uzbek. Math. J., 2019, no. 3, 4–18 | DOI | Zbl

[13] Abdullaev O. K., “On the problem for a mixed-type degenerate equation with Caputo and Erdélyi–Kober pperators of fractional order”, Ukr. Math. J., 71:6 (2019), 825–842 | DOI | Zbl

[14] Pskhu A. V., Uravneniia v chastnykh proizvodnykh drobnogo poriadka [Fractional Partial Differential Equations], Nauka, Moscow, 2005, 200 pp. (In Russian)