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This dissertation mainly intends to elucidate the concept of fuzzy theory on differential calculus. Differential calculus being the study of derivatives of functions at chosen input value has wide ranging practical applications to nearly all quantitative disciplines. Its advance development for fractional order derivatives has increased its significance in every area of science and engineering. While modeling ordinary and fractional differential equations of physical phenomenon, issues of every uncertainty is coped out by means of various theories, among which fuzzy theory is most popular. Specifically, fuzzy theory was designed to mathematically represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. This theory is proposed to make the membership function to operate over the range of real numbers [0, 1]. In this connection, here we have considered linear, non-linear, integer and fractional order differential models with uncertainty. Illustratively, exercises are constructed to present numerical-analytical solutions of initial value problems of fuzzy differential equations (FDEs) and fuzzy fractional differential equations (FFDEs). These differential equations are considered under strongly generalized Hukuhara differentiability. We proposed improved fractional Euler’s method (IFEM) and modified homotopy perturbation method (MHPM) for FFDEs. Also utilized max-min improved fractional Euler’s method and average improved fractional Euler’s method. Additionally, a novel operator method is investigated for the solution of linear FFDEs. Furthermore, we also dealt with the extension of applications of the new integral transform, Sumudu transform on FDEs and FFDEs. These methods are illustrated by solving several examples. Efficiency and exactness of results worked out are examined from the tables and graphs. The exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained.
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