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.