Numerical Solutions Of Several Kinds Of Differential Equations Based On The Third Kind Of Chebyshev Wavelet Collocation Method

Differential equations often appear in the fields of engineering and science,and are widely used.However,the exact solutions of most differential equations are not easy to solve.Therefore,it is very important to develop reliable and efficient numerical solution methods for differential equations.Wavelet analysis is an important technique of multiresolution analysis,which is often used as an important mathematical tool for numerical solution of integral and differential equations.The current wavelet methods for differential equations include CAS Picard method,Chebyshev wavelet method,Legendre wavelet method,Haar wavelet method,etc.The above methods are all two-scale Wavelets and three-scale wavelets are rarely mentioned.The concept of three-scale Haar wavelet was proposed in the paper of Indian scholar Ratesh Kumar.In order to study the influence of wavelet scale on numerical calculation results,this paper gives the definition of the three-scales third kind of Chebyshev wavelet,and solves the differential equation combined with Picard iteration.The main work of this paper is as follows:1.A third kind of Chebyshev wavelet collocation method or solving numerical solutions of higher order differential equations is proposed.Based on the shifted third kind of Chebyshev polynomial,the Riemann-Liouville fractional integral expression of the third kind of Chebyshev wavelet function is derived with the help of Laplace transform,and the error estimation under this method is given.Using the wavelet collocation method,the higher-order differential equations are transformed into algebraic equations and solved.The numerical results show the effectiveness of the method.2.A three-scale third kind of Chebyshev wavelet function is constructed,and the orthogonality of the wavelet function and the convergence and error estimation of the wavelet function expansion are proved.Based on the first part,the Riemann Liouville fractional integral expression of the three-scales third kind of Chebyshev wavelet function is deduced similarly.Combined with Picard iteration,the three-scale third kind of Chebyshev wavelet collocation method is used to discretize the initial value problem and boundary value problem of nonlinear fractional differential equations into algebraic equations.The numerical results show that the method is effective suitability and applicability.3.Based on the one-dimensional three-scale third kind of Chebyshev wavelet,the definition of two-dimensional three-scale third kind of Chebyshev wavelet is given,and the convergence of the function expansion of two-dimensional three-scale third kind of Chebyshev wavelet is given.Combined with Picard iteration and using the three-scale third kind of Chebyshev wavelet collocation method,the nonlinear singular differential equation is discretized into a system of linear equations,and finally solved in the form of Sylvester equation Numerical results show the high accuracy and feasibility of the algorithm.