Abstract:
This paper is devoted to the study of the problem of unique solvability of a nonlocal problem associated with fractional derivatives in the context of boundary conditions for an equation of mixed elliptic-hyperbolic type. An important feature of the equation under consideration is that its order degenerates along the line of type change. On the elliptic part of the boundary of the domain, we establish the Dirichlet condition, while on the characteristic part of the boundary, the domain specifies a condition that connects the Riemann–Liouville fractional derivatives pointwise with the values of the solution on the characteristics. The order of the derivative depends on the order of degeneration of the equation, as well as on the values of the solution and its derivative on the degeneration line located inside the domain. To prove the uniqueness of the solution to this problem, we apply the principles of extremum. The question of the existence of a solution is reduced to the study of the solvability of a singular integral equation with a Cauchy kernel. The paper also presents a condition that guarantees the existence of a regularizer that allows transforming a singular equation into a Fredholm equation of the second kind. Given the possibility of reducing the problem to an equivalent Fredholm integral equation of the second kind, as well as the proven uniqueness of the desired solution, we can conclude that there is a solution to the problem in the required class of functions.
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