Construction Of The Fifth Kind Chebyshev Wavelets And Its Application In Solving Fractional Differential Equations

Wavelets play an extremely important role in many fields such as signal,biochemistry,and solving differential equation.In this paper,from the perspective of extended wavelet types,a new wavelet function is constructed,which is the fifth kind Chebyshev wavelets.Theoretically,the convergence of the wavelet series expansion and the truncation error estimate of its approximating function are proved,and the fractional integration formula and the fractional integral operator matrix of the wavelets are derived.In the numerical calculation,the fifth kind Chebyshev wavelets are used to solve a class of linear and nonlinear fractional differential equations,and its high-precision numerical solution is obtained.The first chapter summarizes the history of wavelets and fractional calculus,and the research status of wavelets in fractional order equations.The second chapter introduces the first to fourth kind Chebyshev wavelets,and the fifth kind Chebyshev polynomials and Block Pulse function.In the third chapter,taking the fifth Chebyshev polynomials as the wavelet generating function,the fifth kind Chebyshev wavelets with orthogonality are constructed by stretching and translating the fifth Chebyshev polynomials,and coefficient orthogonalization.Then,the recursive formula of the wavelet function group is obtained.The fourth chapter finds the trigonometric expression of the polynomial odd term by using the recursive formula of the fifth kind Chebyshev polynomials.Then,combining with the integration by parts method and Cauchy Schwartz inequality,this paper proves the uniform convergence of the fifth kind Chebyshev wavelet series with odd and even terms in L~2space.Because wavelet series take finite terms in practical numerical applications,truncation error estimate of the fifth kind Chebyshev wavelets approximating functions is given.In the fifth chapter,combining with the definition of fractional order integral and the range of independent variable in the fifth kind Chebyshev wavelets expression,the fractional order integral formula of the wavelet is derived.The fractional order integral formula is applied to solve a class of linear fractional order differential equation.The corresponding error charts are given by numerical experiment,which shows the high precision of this method.The sixth chapter derives the fractional integral operator matrix of the fifth kind Chebyshev wavelets by Block Pulse function.Based on the properties of Block Pulse function and Taylor's formula,the transformation formula of the fifth kind Chebyshev wavelets integral operator matrix in composite function is derived.The transformation formula is applied to solving a class of linear and nonlinear fractional differential equations.And numerical experiments show the effectiveness of the method.Through the same numerical example,the error of approximating the original function by the two methods of the fifth kind Chebyshev wavelets fractional integral formula and integral operator matrix is analyzed.Theoretically,it is demonstrated that the accuracy of using the wavelet fractional integral formula is higher than its fractional integral operator matrix.However,the fractional integral operator matrix is more practical.The seventh chapter summarizes the work of this paper and makes an outlook for related issues that need to be studied in the future.