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    Multi-step gH-difference-based methods for fuzzy differential equations
    (Springer Nature, 2023) Safikhani, Leila; Vahidi, Alireza; Allahviranloo, Tofigh; Afshar Kermani, Mozhdeh
    The main purpose of this paper is to introduce fuzzy Adams–Bashforth (A–B) and fuzzy Adams–Moulton (A–M) methods based on the generalized Hukuhara (gH)-differentiability and employ them as the predictor and corrector, respectively. The local truncation error, stability and convergence of these methods are discussed in the sequel. Finally, some fuzzy linear and nonlinear initial value problems (IVPs) are solved. The numerical results obtained here show that our methods provide a suitable approximation for the exact solution. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
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    Z+-laplace transforms and Z+-differential equations of the arbitrary-order, theory and applications
    (Elsevier Inc., 2022) Ardeshiri Lordejani, Maryam; Afshar Kermani, Mozhdeh; Allahviranloo, Tofigh
    This paper provides a framework to study a class of arbitrary-order uncertain differential equations known as arbitrary-order Z+-differential equations. For this purpose, we first present the parametric form of Z+-numbers. Then, we introduce the basic algebraic operations on Z+-numbers, including addition, scalar multiplication, and Hukuhara difference. Definitely, these operations lead us to define the Z+-valued function. Afterward, the limit and continuity concepts of a Z+-valued function are provided under the definition of a metric on the space of Z+-numbers. Furthermore, the concepts of Z+-differentiability, Z+-integral, and Z+-Laplace transform with the convergence theorem for the Z+-valued function and its nth-order derivatives are introduced in detail. Considering all these concepts, a Z+-differential equation (Z+DE) can be expressed in the form of a bimodal differential equation combining a fuzzy initial value problem (FIVP) and a random differential equation (RDE). To this end, we use a combination of “FIVP under strongly generalized Hukuhara differentiability (SGH-differentiability)” and “random differential equation under mean-square differentiability (ms-differentiability)” to define the nth-order differential equations with Z+-number initial values. Further, the existence and uniqueness of the Z+-differential equations are examined by presenting several theorems. Finally, the effectiveness of the approaches is illustrated by solving two examples.