Payame Noor University (PNU)Control and Optimization in Applied Mathematics2383-31309120240501Hesitant Fuzzy Equation67791036010.30473/coam.2023.68208.1240ENFatemehBabakordiDepartment of Mathematics, Faculty of Science, Gonbad Kavous University, Gonbad Kavous, Iran.0000-0003-0745-1595Nemat AllahTaghi-NezhadDepartment of Mathematics, Faculty of Science, Gonbad Kavous University, Gonbad Kavous, Iran.0000-0002-4568-0527Journal Article20230616This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation. Additionally, an efficient method is proposed to solve these equations by defining four solution categories: Controllable, Tolerable Solution Set (TSS), Controllable Solution Set (CSS), and Algebraic Solution Set (ASS). Furthermore, the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions. The practicality of the proposed method is demonstrated through numerical examples.https://mathco.journals.pnu.ac.ir/article_10360_40062fefd890ab189664babdc44e862c.pdf