Fuzzy Homotopy Analysis Method For Solving Fuzzy Riccati Differential Equation

Abstract

In this work, we have used fuzzy homotopy analysis method to find the fuzzy series solution (fuzzy semi-analytical solution) of the first order fuzzy Riccati differential equation. The fuzzy approximate-analytical solutions that we obtained during this paper are accurate solutions and very close to the fuzzy exact-analytical solutions. Some numerical results are given to illustrate the method. The obtained numerical results are compared with the exact solutions.

Applying the fuzzy homotopy analysis method to solve fuzzy initial value problems with variable coefficients

“…The basic definitions in the fuzzy theory, which are: Fuzzy set, α − level set, fuzzy number, fuzzy function, fuzzy derivative, etc., are found in detail in Wang and Guo (2011), Citil (2019), Sabr et al (2021), and Suhhiem and Khwayyit (2022). In this section, we will touch on definitions that are directly related to our work.…”

Semi Analytical Solution for Fuzzy Autonomous Differential Equations

“…A fuzzy ordinary differential equation is called autonomous if it is independent of its independent crisp variable x. This implies that the nth order fuzzy autonomous differential equation is of the form [11]: 𝑢 (𝑛) (x) = f ( 𝑢 (x) , 𝑢 ′ (𝑥) , 𝑢 ′′ (𝑥) , … . , 𝑢 (𝑛−1) (𝑥)) , 𝑥 ∈ [𝑥 0 , ℎ] where: 𝑢 is a fuzzy function of the crisp variable 𝑥, f (𝑢 (x) , 𝑢 ′ (𝑥) , 𝑢 ′′ (𝑥) , … .…”

“…In this work, we will need many basic concepts in the fuzzy theory. These concepts can be found in detail in [5,8,11].…”

Approximate Series Solution For Two-Point Fuzzy Boundary Value Problems

“…during this work, we need many fundamental concepts in the fuzzy set theory, such as fuzzy number, fuzzy function and fuzzy derivative. These concepts can be found in detail in [6,9,12].…”