Solving second-order fuzzy differential equations by the fuzzy Laplace transform method

Abstract

In this paper, we study the fuzzy Laplace transforms introduced by the authors in (Allahviranloo and Ahmadi in Soft Comput. 14:235-243, 2010) to solve only first-order fuzzy linear differential equations. We extend and use this method to solve second-order fuzzy linear differential equations under generalized Hukuhara differentiability.

On the Existence and Uniqueness for High Order Fuzzy Fractional Differential Equations with Uncertainty

“…In [32], Allahviranloo and Ahmadi introduced the fuzzy Laplace transform, which they used under the strongly generalized differentiability. Recently, ElJaoui et al [33] developed it further. The newly defined fuzzy Laplace transform [32] for high order fuzzy derivatives is one of the most useful methods as mentioned by Jafarian et al in [34]: ".…”

“…becomes ( ) ≤ ( (( /(1− )) +1) + (( /(1− )) +(1− [ ])/(1− )) ) (33). Advances in Fuzzy Systems 5 This leads to the following estimates on and , for ∈ [0, ]:( ) ≤ ( ( /(1− )) + ( /(1− )+ (1−[ ])/(1− )) ) , ( ) ≤ ( (( /(1− )) +1) + (( /(1− )) +(1− [ ])/(1− )) ) .…”

On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations

“…This notion was exploited by certain researchers to study fuzzy differential equations (FDEs) of first or second order utilizing fuzzy Laplace transform, namely, by Allahviranloo and Ahmadi in [2], then by Salahshour et al (see [3,4]), and by ElJaoui et al in [5].…”

Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

“…To achieve this goal, we first introduce the Aumann fuzzy improper integral concept, which we utilize instead of the Riemann fuzzy improper integral used in [10,[12][13][14].…”

“…Salahshour and Allahviranloo gave in [11] some applications of fuzzy Laplace transform and studied sufficient conditions ensuring its existence. Recently in [12], we extended and used the fuzzy Laplace transform method to solve second-order fuzzy linear differential equations under strongly generalized Hukuhara differentiability. Then we established in [13] some important results about continuity and strongly generalized Hukuhara differentiability of functions defined via improper fuzzy Riemann integrals, and we proved some properties of fuzzy Laplace transforms for two variables functions, which we applied to solve fuzzy linear partial differential equations of first order.…”