Solving Pythagorean fuzzy fractional differential equations using Laplace transform

In this research article, we discuss an important class of modern differential equations in the Pythagorean fuzzy environment, called the Pythagorean fuzzy fractional differential equations (PFFDE). Fuzzy fractional differential equations under the generalized Hukuhara Caputo fractional derivative are extended in a Pythagorean fuzzy context to extract their analytical solutions. To solve the PFFDE, we define the Riemann-Liouville fractional integral, the Riemann-Liouville (RL) fractional derivative, the Caputo fractional derivative, and the Laplace transform in a Pythagorean fuzzy fashion. Furthermore, we present the solution procedure for homogeneous and inhomogeneous PFFDEs in the form of theorems. We then extract the closed-form solution of the PFFDE using the Pythagorean fuzzy Laplace transform and the Mittag-Leffler function. We extract two possible solutions for PFFDE based on the type of gH-differentiability and the Pythagorean fuzzy initial conditions. Moreover, we discuss some applications of PFFDE and its graphical representation to ensure the effectiveness of the proposed method.