Основні поняття
Solutions of higher-order fractional differential equations can be interpreted as expected values of functions in a random time process.
Анотація
The content presents a new relationship between solutions of higher-order fractional differential equations and a Wright-type transformation. Key highlights:
Lemma 2 establishes a connection between fractional derivatives of higher order and expected values of derivatives, improving on previous work.
Theorem 1 allows solving higher-order fractional differential equations with certain initial conditions, where the solutions can be interpreted as expected values of functions in a random time process.
Applications include solving the fractional beam equation, fractional electric circuits with special function sources, and deriving d'Alembert's formula for the fractional wave equation.
Two numerical approaches are presented: one using Monte Carlo integration with the Runge-Kutta method, and another using feedforward neural networks.
The content also includes a discussion on the properties of the Wright-type function gβ(x;t) and its applications in fractional calculus.
Статистика
Γ(k+1)/tkβ
Γ(kβ+1)
Eβ(-stβ)
sβ-1e-xsβ
Цитати
"Solutions could be interpreted as expected values of functions in a random time process."
"Lemma 2 provides a weaker condition" than previous work.
"Dr. Mark Meerschaert passed away recently, but he contributed tremendously to the theory of fractional calculus."