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.

翻译：本文建立了高阶分数阶微分方程解与Wright型变换之间的新关系。解可解释为随机时间过程中函数的期望值。作为应用，我们求解了分数阶梁方程、以特殊函数为外部源的分数阶电路方程，并推导了分数阶波动方程的达朗贝尔公式。基于该关系，我们提出了两种模拟分数阶微分方程解的方法。这两种方法均利用Caputo导数作为函数高阶导数的Wright型变换的诠释。第一种方法采用混合阶4/5龙格-库塔法求解常微分方程，并结合蒙特卡洛积分实现Wright型变换。第二种方法则使用前馈神经网络模拟分数阶微分方程。