Boundary value problem for mixed-compound equation with fractional derivative, functional delay and advance

A. N. Zarubin ; E. V. Chaplygina

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 1, pp. 20-36

Résumé

We study the Tricomi problem for the functional-differential mixed-compound equation $LQu(x,y)=0$ in the class of twice continuously differentiable solutions. Here $L$ is a differential-difference operator of mixed parabolic-elliptic type with Riemann–Liouville fractional derivative and linear shift by $y$. The $Q$ operator includes multiple functional delays and advances $a_1(x)$ and $a_2(x)$ by $x$. The functional shifts $a_1(x)$ and $a_2(x)$ are the orientation preserving mutually inverse diffeomorphisms. The integration domain is $D=D^+\cup D^-\cup I$. The “parabolicity” domain $D^+$ is the set of $(x,y)$ such that $x_0$, $y>0$. The ellipticity domain is $D^-=D_0^-\cup D_1^-\cup D_2^-$, where $D_k^-$ is the set of $(x,y)$ such that $x_k$, $-\rho_k(x)$, and $\rho_k=\sqrt{a_1^k(x)(x_1-a_1^k(x))}$, $\rho_k(x)=\rho_0(a_1^k(x))$, $k=0, 1, 2$. A general solution to this Tricomi problem is found. The uniqueness and existence theorems are proved.

Détail

Keywords: mixed-compound equation, fractional derivative, difference operator, Tricomi problem.