Fuzzy Sumudu transform
Ordinary differential Equations (ODEs) have long been used to model real-life problems. As times goes on, researchers realized that ODEs are no longer the best modelling tool, since they are inaccurate, especially when dealing with stuff that are surrounded by uncertainties. To deal with this, resea...
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| Format: | Thesis |
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| Language: | English |
| Subjects: | |
| Online Access: | http://dspace.unimap.edu.my:80/xmlui/bitstream/123456789/77980/1/Page%201-24.pdf http://dspace.unimap.edu.my:80/xmlui/bitstream/123456789/77980/2/Full%20text.pdf http://dspace.unimap.edu.my:80/xmlui/bitstream/123456789/77980/4/Norazrizal.pdf |
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| Summary: | Ordinary differential Equations (ODEs) have long been used to model real-life problems. As times goes on, researchers realized that ODEs are no longer the best modelling tool, since they are inaccurate, especially when dealing with stuff that are surrounded by uncertainties. To deal with this, researchers utilized the fuzzy set theory to propose fuzzy differential equations (FDEs). Unlike ODEs which could not cope with uncertainties at the initial values, FDEs handle this efficiently. However, the methods that can be
used to deal with FDEs are very limited, and many of them are still at development stages. In this research, a new analytical method is proposed for solving a class of FDEs, including FDEs of integer order, fuzzy fractional differential equations (FFDEs), fuzzy partial differential equations (FPDEs) and system of linear fuzzy differential equations (SLFDEs). For this purpose, a novel integral transform called fuzzy Sumudu transform (FST) is proposed. This is done by integrating the fuzzy set theory and the classical Sumudu transform (CST). New properties and fundamental results concerning FST are proposed and proved mathematically. These new properties comprised results on duality with fuzzy Laplace transform (FLT), linearity, preserving, shifting, convolution, as well as theorems for fuzzy derivatives. The proposed results are then used to construct
detailed procedures for solving FDEs, FFDEs, FPDEs and SLFDEs where the class of FDEs are interpreted under the strongly generalized differentiability concept. From there, the procedures are demonstrated on some examples, in order to illustrate FST
applicability. Some analyses are performed where the behaviour of the solutions obtained are described in particulars. Furthermore, a comparison between the solutions obtained for the class of FDEs are compared with the solutions obtained using ODEs, where the ODEs include integer order ODEs, fractional differential equations (FrDEs), partial differential equations (PDEs) and system of linear differential equations (SLDEs). The main advantage of FST highlighted in this thesis is that it possessed the scale preserving property. This means that the transformed function, with a new domain, is a similitude to the original function. Additionally, this research provides an alternative for researchers when dealing with the class of FDEs. |
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