Construction of Singular Particular Solutions Expressed via Hypergeometric Functions for an Equation with Multiple Characteristics

Obtaining a representation of the solution to the Cauchy problem for equation high-order fractional derivative, by the method of finding self-similar solutions

“…Note that self-similar solutions of an equation with the usual derivative of the form ∂ p u (x, y) ∂x p − ∂ q u (x, y) ∂y q = 0, p < q, were constructed in [1].…”

“…In [2], using the methods of special operators, self-similar solutions of equation (1) were found in the cases β 2 ≤ α < β ≤ 2, n − 1 < β ≤ n ∈ N . In this paper, we construct self-similar solutions by a method that does not require knowledge of special operators.…”

Solution of the problem with initial conditions for the equation high-order fractional Hilfer derivative

“…In [2], using the methods of special operators, self-similar solutions of an equation with constant coefficients containing fractional derivatives in the sense of Riemann-Liouville were found. For an equation with an ordinary derivative of the form ∂ p u (x, y) ∂x p − ∂ q u (x, y) ∂y q = 0, p < q, self-similar solutions were constructed in [3]. Note that various initial-boundary value problems for an equation with a fractional Hilfer derivative were studied in [4] - [6] and others.…”

Building singular solutions for degenerate high order equations

“…In our works [1], [2], particular solutions were constructed with singularities for high-order equations with multiple characteristics with constant coefficients. In this paper, we will construct self-similar solutions for equations of the form…”