Numerical Simulations for Fractional Differential Equations of Higher Order and a Wright-Type Transformation

TL;DR

This work develops a Wright-type transformation that ties higher-order Caputo fractional derivatives to expected values of a random-time process, enabling a probabilistic interpretation of fractional differential equations. Theoretical core is Lemma 2 and Theorem 1, which express solutions as $\mathbb{E}[z(\mathcal{T}_{\beta}(t))]$ and extend solvability to higher-order FODEs under broad initial data; these solutions can be interpreted via the random time $\mathcal{T}_{\beta}(t)$ with density $g_{\beta}$. The authors present two numerical schemes: a Monte Carlo approach combining Runge-Kutta on randomized timelines and a deep feedforward neural network trained on transformed data, both achieving high accuracy. Applications span the fractional embedded beam, fractional electric circuits with Mittag-Leffler and related external sources, and a fractional d'Alembert formula for wave equations, illustrating practical impact for engineering and applied science problems.

Abstract

In this work, a new relationship is established between the solutions of higher fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, and derive dAlemberts formula for the fractional wave equation. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In the first approach, we use the Runge-Kutta method of hybrid orders 4 and 5 to solve ordinary differential equations combined with the Monte Carlo integration to conduct the Wrighttype transformation. The second method uses a feedforward neural network to simulate the fractional differential equation.

Figures (9)

• Figure 1: Monte Carlo simulation of the solution of $EI\,D_{c}^{\beta }D_{c}^{\beta }D_{c}^{\beta }D_{c}^{\beta }\phi (t)=w_{0}$ with $L=1$ and $EI=w_0$.
• Figure 2: A Monte Carlo simulation of the solution of $D_{c}^{\beta }y_{\beta }(t)+ay_{\beta }(t)=E_{\beta }(t^{\beta }),$ with $a=1$.
• Figure 3: A Monte Carlo simulation of the solution of $D_{c}^{\beta }D_{c}^{\beta }y_{\beta }(t)+\omega ^{2}y_{\beta }(t)=\frac{2t^{2\beta }}{\Gamma (2\beta +1)},0<\beta <1$, and $c_1=c_2=1$.
• Figure 4: A Monte Carlo simulation of the solution of $D_{c}^{\beta }D_{c}^{\beta }$$y(t)+\omega ^{2}y(t)=t^{\beta }E_{2\beta ,\beta +1}(-t^{2\beta })=sin_{\beta }(t),$$0<\beta <1$ with $\omega=2$, and $c_1=c_2=1$.
• Figure 5: A Monte Carlo simulation of the solution of $D_{C}^{\beta }y(t)+y(t)=\sin _{\beta }{(t)}$ with $y(0)=0$ evaluated at $\beta =0.5$ (left) and $\beta =0.9$ (right).

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Lemma 1
• Lemma 2
• Theorem 1
• Corollary 1
• Example 1: RC circuit
• Example 2: LC Circuits