Solutions of ordinary and partial differantial equations of random fractional with adomian decomposition method

Application of nonlinear differential equations in physics, engineering, etc. There are applications in fields, such equations often do not have analytical solutions. Many problems, such as physics and engineering, can be modeled mathematically. Since most of these models are expressed with the help of nonlinear ordinary and partial differential equations, there are many methods developed in the literature to obtain approximate analytical solutions of such equations. One of the most frequently used methods is the Adomian Decomposition Method. The Adomian Decomposition Method was first introduced by George Adomian in the early 1980s. Adomian is an American mathematician who developed this method for ordinary, partial, linear and nonlinear differential equations. Adomian applied this method to find approximate solutions of deterministic, stochastic, linear and nonlinear problems with boundary and initial conditions. The method is created by decomposition nonlinear terms. Nonlinear terms are defined as the sum of Adomian Polynomials. For every Adomian Polynomial n>0 it depends on the y arguments. The formulas to obtain these polynomials were developed by Adomian. In this study, some definitions of fractional derivative and important functions used in the calculation of fractional derivative, some continuous probability distributions, expected value, variance and moment generating function definitions of random variables are given. A summary of the literature on the Adomian Decomposition Method is given and the application of the method is explained. The coefficients and initial conditions of the differential equations were chosen randomly, and the fractional differential equations were randomized. In addition, the solutions of the obtained random differential equations were found and the probabilistic characters of the solutions were examined. The expected values ​​and variances of the solutions were calculated by choosing the parameters from different probability distributions, and the solution behaviors were analyzed graphically in Maple and Matlab. In addition to the Adomian Decomposition Method, information about the Residual Power Series Method has been given, the Laplace-Adomian Decomposition Method has been explained and its application to the differential equations has been made.