Solutions of random fractional order differential equations using aboodh and aboodh-adomian decomposition method

Many physical problems in nature can be described by mathematical models involving ordinary or partial differential equations. These mathematical models are a simplified description of physical reality expressed in mathematical terms. Therefore, the search for exact or approximate solutions to these physical problems helps us understand the tools of these mathematical models. Scientists have developed many methods to obtain exact solutions or near-exact solutions of ordinary or partial differential equations. Among the developed methods, they mostly benefited from integral transformations. In this study, we will talk about the Aboodh transformation, which is a new integral transformation. The Aboodh transform method was introduced by Khalid Aboodh in 2013 to simplify the process of solving linear and nonlinear fractional differential equations. Aboodh transform is one of the convenient methods to obtain exact solutions of fractional differential equations or systems of equations. Due to the difficulties caused by nonlinear terms, approximate solutions of nonlinear differential equations or systems of equations can be found by using the Adomian decomposition method together with the Aboodh transformation. In this study, the definition of the Aboodh integral transformation, the definition of the Caputo-Fabrizio fractional order derivative, some of the continuous probability distributions, and the definitions of the function that produces the expected value, variance and moment of the random variable are given. In addition, a summary of the literature on the Aboodh transform, the Aboodh-Adomian decomposition method and random differential equations is given, and the application of the method is explained. Fractional differential equations were made random by randomly selecting the coefficients and initial conditions of the differential equations. Additionally, the probability characteristics of the resulting random differential equations were examined by finding their solutions. Parameters were selected from different probability distributions and the expected value, variance and confidence interval of the solutions were calculated, and the solution behaviors were examined graphically using MATLAB and MAPLE13 programs.