Solution of random fractional order differential equations by variational iteration method

This study aims to find and analyze the solutions of random differential equations using the Variational Iteration Method (VIM). In this context, ordinary, partial, and fractional differential equations, as well as systems of equations, are considered and solved under specified probability distributions by introducing randomness. The expected value, variance, confidence intervals, and, in some cases, the coefficient of variation of the solutions are also calculated, and their graphs are generated using MATLAB. The thesis begins with a brief overview of relevant literature, followed by the necessary theoretical background. In the first section, random ordinary differential equations are addressed, and their solutions are obtained using the Variational Iteration Method (VIM), with an analysis of the solution behaviors. In the second section, the method is applied to random differential equations involving the Caputo fractional derivative, and the solutions are analyzed similarly. The third section focuses on random partial differential equations with the Atangana–Baleanu fractional derivative, where the classical VIM is used to obtain and analyze the solutions. Finally, in the fourth section, the Adomian–VIM approach is employed to solve random differential equations involving the Riemann–Liouville fractional derivative, with a corresponding analysis of solution behaviors. This study demonstrates that VIM is an effective method for solving various types of differential equations and enables the accurate computation of expected values and other statistical properties when randomness is introduced.