Qualitative Study of Fractional Differential Equations and Analytical Study of Fractional Order Mathematical Model by Various Methods

Abstract

In first chapter, we provides the development of the subject which contains newlinethe necessary mathematical machinery for investigating existence and uniqueness newlineof factional differential equations , factional Integro differential equations and newlineapplication. Also discuss some useful definitions and existing results which are newlinenecessary for our work. Moreover, we discuss result of stability analysis of models newlineand Analytical Methods. newlineIn second chapter, by utilising Shauder s fixed point technique and the Bihary newlineinequality, we shall explore the results on the existence and uniqueness of fractional newlinedifferential equations under the Caputo sense. Through the use of the Banach newlinefixed point theorem and the theory of the strongly continuous cosine family, we newlinewill also analyse the existence and uniqueness of a moderate solution to the initial newlinevalue use of fractional order subject to non-local circumstances under Caputo sense. newlineAdditionally, by utilising the Banach fixed point theorem and Semi group theory, we newlinewill explore results on the existence and uniqueness of non linear fractional mixed newlineIntegrodifferential equations with non local conditions under Caputo sense. newlineIn third chapter, we examine the following fractional order neutral stochastic newlinefunctional differential equations with random impulses, which are denoted as, newlineand#63729;and#63732;and#63732;and#63732;and#63730; newlineand#63732;and#63732;and#63732;and#63731; newlinec newline0Dand#945; newlineand#947; [and#977;and#945;(and#947;) and#8722; and#933;(and#947;, and#977;and#947;)] = [and#8501;and#977;(and#947;) + and#967;(and#947;, and#977;and#947;)]dand#947; + and#963;(and#947;, and#977;and#947;)dW(and#947;), and#947; gt and#947;0, and#947; and#824;= and#958;k, newlineand#977;(and#958;k) = bk(and#964;k)and#977;(and#958;and#8722; newlinek ), and#977;and#945;(and#958;k) = bk(and#964;k)and#977;and#945;(and#958;and#8722; newlinek ), k = 1, 2, ..., newlineand#977;and#947;0 = and#981;, and#977;and#945;(and#947;0) = and#968;, newline(0.0.1) newlineWith the use of the M¨o nch fixed point theorem and the non-compact measurement newlinetechnique, the current chapter aims to investigate the existence and uniqueness of newlinemild solutions to equations in the Caputo sense. newlineAdditionally, using the Banach fixed point newline