Homotopy Analysis Method for Solving Multi-Fractional Order Random Ordinary Differential Equations - Sahar Mohammed; Fadhel Fadhel; Kasim Hussain (مجلة جامعة الانبار للعلوم الصرفة, 2023)
The homotopy analysis method may be considered as one of the most important and efficient methods for solving several problems in mathematics with different operators, linear and nonlinear, ordinary or partial differential equations, integral equations, etc. In this paper, the main objective is to introduce random ordinary differential equations with multi fractional derivatives, in which the homotopy analysis method is used to find the approximate solution of such equations with different generations of the Weiner process or Brownian motion. In addition to that, the convergence analysis for such equations is studied and proved, as well as, stating and proving the existence and uniqueness theorem. Three examples are considered (for linear, multi-fractional order and nonlinear equations) in order to check the validity and applicability of the proposed approach. These examples are simulated using computer programs written in Mathcad 14 computer program and the results are sketch using Microsoft Excel. The results show that the examples solutions are vary with respect to the stochastic process generation which are nowhere differentiable, as it is expected.