ORIGINAL PAPER Interpretation of exact solution for fuzzy fractional non-homogeneous differential equation under the Riemann–Liouville sense and its application on the inventory management control problem Mostafijur Rahaman 1 • Sankar Prasad Mondal 2 • Shariful Alam 1 • Najeeb Alam Khan 3 • Amiya Biswas 4 Received: 28 July 2020 / Accepted: 23 September 2020 Ó Springer Nature Switzerland AG 2020 Abstract This article is presented for manifestation of the non-homogeneous linear fractional differential equation under fuzzy uncertainty. Taking the initial values and the coefficients of the fractional differential equations to be fuzzy numbers, the solutions are categorized into different problems and sub-problems. Based on the three different combinations of the fuzzy initial values and fuzzy coefficients, the theoretical foundation in this article is primarily divided into three major problems. Then, each problem again contains two different sub-problems according to the sign of the coefficients. Finally, the solutions for each of the sub-problems are given in the sense of two different cases of RL ½ i ðÞ a and RL ½ ii ð Þ a differentiability. The strong and the weak solution criteria have been discussed here. Also, the existence and uniqueness criterion for solution of the initial-valued fuzzy fractional differential equation has been established in this article. Finally, an EPQ model of deteriorated items is discussed for its production phase only as a proper validation of proposed theory. The greatness of the consideration of fuzzy fractional differential equation over crisp integer differential equation, fuzzy integer differential equation and crisp fractional differential equation to describe the EPQ model is established in this current article. In this context, a new defuzzification technique is developed to compare the fuzzy and crisp phenomena related to the EPQ model. Keywords Riemann–Liouville differentiability  Fractional differential equation  Laplace transformation  Deterioration  H differentiability  EPQ model 1 Introduction Fractional calculus (FC) is a topic of much enthusiasm in the current era of mathematical research having an origin in the distant past. The comprehensive knowledge of FC amazingly helps to analyze the dynamical behavior of the physical problems in real world, which had been effectively deployed in mathematical modeling to handle many complex problems in the fields of Science and Technology (see Diethelm et al. 2012; Agila et al. 2016; Podlubny 1999). The theory of the FC has been enriched by many cutting edge concepts and techniques with the needs of the time. Fractional differential equation (FDE) is one such con- cept where the integer orders of the differential equation are replaced by a fractional number. Miller and Ross (1993), Kilbas et al. (2006) and Magin (2006) discussed various features and aspects of the FDE and projected the advantage and smartness of this tool to tackle many complicated issues involved in several prob- lems in the domain of Morden Science and Technology. The notion of the fractional derivative seems to be a global one, whereas that of integer order derivative is a local one. The fractional order revelation is proved to be efficient to symbolize complex structures although computational & Shariful Alam salam50in@yahoo.co.in 1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, India 2 Department of Applied Science, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, India 3 Department of Mathematics, University of Karachi, Karachi, Pakistan 4 Department of Mathematics, Durgapur Government College, Durgapur, India 123 Granular Computing https://doi.org/10.1007/s41066-020-00241-3