Linear and Non-Linear Mathematical Models with Imprecise Parameters having Uncertain Memberships

Abstract

Linear and non-linear mathematical equations are essential tools for understanding and describing a wide range of phenomena across scienti_c and engineering disciplines. These equations can be algebraic systems, di_erential equations, integral equations, or integro-di_erential equations, etc. Typically, parameters associated with these governing equations are regarded as deterministic entities. However, these parameters may contain imprecision or uncertainty due to observational or experimental errors or a lack of information, which challenges the idealised assumption of determinism. Uncertainty in parameters may be e_ciently managed by utilising probability theory, interval analysis, and fuzzy set theory. Unfortunately, probabilistic methods may not be able to deliver reliable results at the required precision without su_cient data. It may be due to the probability density functions involved in it. On the other hand, interval and fuzzy theory approaches emerge as valuable tools in such situations. In this work, we have focused on the fuzzy sets to handle the parameters of the problems undertaken. Fuzzy sets have assigned membership grades. Further, uncertainty in the membership grade of the fuzzy parameters may also be possible. In these scenarios, interval type-2, type-2 and interval type-3 fuzzy sets may be bene_cial. Accordingly, this thesis is dedicated to investigate di_erent linear and non-linear mathematical models under higher orders of fuzzy uncertainty, and we have used the type-2 and type-3 fuzzy sets to target impreciseness. Particularly, we focused on analysing mathematical equations, such as systems of linear equations, eigenvalue problems, di_erential equations (both integer and fractional order), and integral equations in type-2 or type-3 fuzzy environments. In this regard, new analytical and numerical methods are developed to solve the type-2 and type-3 fuzzy mathematical equations. Various analytical methods are proposed to address linear systems of equations, linear and non-linear eigenvalue problems, di_erential equations, and Fredholm integral equations under type-2 and type-3 fuzzy environments. Additionally, numerical and semi-analytical methods, viz., the Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Elzaki Transformed HPM (ETHPM), Fractional Reduced Di_erential Transform Method (FRDTM), Legendre Wavelet Method (LWM), and Generalised Modi_ed Euler Method (GMEM), are also developed in the type-2 fuzzy environment to solv the fractional order ordinary and partial di_erential equations. Numerical examples and application problems are solved to demonstrate the e_ciencies and capabilities of the developed methods. In this regard, various sample problems are solved _rst. Further application problems governed by a linear system of equations, such as beams, trusses, and rectangular sheets, are considered. Structural problems, viz., spring-mass systems, are studied under linear and non-linear eigenvalue problems. Application problems related to di_erential equations, viz., electric circuits and spring-mass systems, are also investigated. Fractional order di_erential models, viz., prey-predator model, HIV transmission model, SEIR model of measles, smoking giving up model, heat equation, modi_ed Camassa-Holm equation (MCHE) and modi_ed Degasperis-Procesi equation (MDPE) are analysed in this investigation. In special cases, comparisons are made with existing solutions to show the e_cacy and reliability of the present methods.