Solving Three Types Of Fractional Order And Variable Order Fractional Differential Equations With Proportional Delays Based Shifted Chebyshev Polynomials

Fractional and variable order fractional delay differential systems have been widely used in physics and engineering fields.Fractional and variable order fractional delay differ-ential equations are the most important mathematical models for describing delay differential systems.Thus,the research of theory and application is of great significance.At present,the study of numerical solutions of fractional and variable order fractional delay differential e-quations has become a hot research direction.Proportional delay differential equations,as an important type of delay differential equations,can more accurately describe some complex mathematical problems.Therefore,based on the shifted Chebyshev polynomial approxi-mation theory,combined with the idea of fractional or variable order fractional differential definition and operator matrix,the method for solving three kinds of fractional and vari-able order fractional proportional delay differential equations is studied.The paper mainly includes the following contents:Firstly,the paper will introduce the research background and significance of polynomial approximation theory and will expound the general situation of fractional and variable order fractional delay differential equations.This thesis will mainly solve the problems of numer-ical solutions about three types of fractional order and variable order fractional differential equations,which include generalized fractional pantograph equations with variable coeffi-cients,fractional partial differential equations with proportional delays and variable order fractional nonlinear equations with proportional delays.In addition,according to the defi-nition and properties of the Chebyshev polynomials,the analytical expression of the shifted Chebyshev polynomials is derived.Secondly,in Chapter 2,combining the definition of shifted Chebyshev polynomials and Caputo fractional derivative,the generalized pantograph operational matrix of fractional derivative and operational matrix of product are derived.We can express each part of the generalized fractional pantograph equations in a form of matrix products.And the numerical solutions of the original equations are obtained by discreting the variables.Meanwhile,the error analysis of our method is made.The effectiveness and feasibility of the algorithm are verified by numerical examples.In Chapter 3,polynomial approximation theory is extended to solve the two-dimensional fractional proportional delays partial differential equations,and the higher-order proportional delays operator matrix is derived.In order to effectively solve such equations with low-order polynomials,the error correction method is introduced.The effectiveness of the method and the feasibility of the error correction method are verified by numerical examples.Finally,in Chapter 4,the application of function approximation theory to variable or-der fractional differential equations with proportional delays is studied.The variable order fractional differential operator matrix of shifted Chebyshev polynomials is derived and the processing method about solving the nonlinear term in the equations is introduced.In addi-tion,by constructing differential equations with respect to the error function,approximated error function is obtained,and the numerical solutions are corrected.