SOLVING A BOUNDARY VALUE PROBLEM FOR THE FIRST ORDER DIFFERENTIAL EQUATION INVOLVING THE PRABHAKAR OPERATOR
In this study, a first order partial differential equation involving the Prabhakar fractional derivative
is examined within a rectangular domain. A boundary value problem associated with this equation
is investigated, and the existence and uniqueness of its solution are established. To construct the
solution, the Riemann method is employed. An auxiliary problem is formulated in terms of the
Riemann function. By applying the Laplace transform, the auxiliary problem is reduced to a Cauchy
problem for an ordinary differential equation. Subsequently, the inverse Laplace transform is utilized
to derive an explicit expression for the solution of the auxiliary problem, which corresponds to the
Riemann function of the original equation. The solution to the initial boundary value problem is
then obtained using the Riemann method. Furthermore, sufficient conditions are derived for the
given functions to ensure that the obtained solution satisfies the problem’s constraints.
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