Euler Wavelet Numerical Solutions For Three Kinds Of Differential Equations

In many fields such as signal processing,biochemistry,economics and control theory,differential equation plays an important role as a mathematical model of complex system.For most differential equations,it is very difficult to solve their analytical solutions,so it is particularly important to solve the differential equations numerically.Nowadays,there are many methods to solve the numerical solution of differential equations,such as collocation method,finite difference method,B-spline,Adomian decomposition method,variational iteration method,homotopy analysis method and so on.In recent years,the study of numerical solution of differential equations based on a wavelet collocation method in spectral method is very popular,such as Haar wavelet,Legendre wavelet,Chebyshev wavelet,Bernoulli wavelet,etc.In this paper,Euler wavelet method is used to solve differential equations.In the study of Euler wavelet,it is found that it is not orthogonal through verification.In order to make it orthogonalized,the orthogonal Euler wavelet with orthogonality is derived and applied to the numerical solution of several kinds of differential equations.This is also the innovation point of this paper.Through the function family of orthogonal Euler wavelet constructed,the fractional order integral formula is derived by using Laplace transform,and several kinds of differential equations are solved by using the integral formula,and the high precision numerical solutions are obtained.In this paper,the application of orthogonal Euler wavelet with orthogonal Euler wavelet in differential equations is mainly expounded.The main content is as follows:Firstly,the Euler wavelet function arbitrary fractional order integral formula is derived,and the numerical solution of Lane-Emden equation with three boundary constraints is carried out by Euler wavelet function.The corresponding error chart is given through numerical experiments,and the comparison with Haar wavelet method shows the high precision of the proposed method.Secondly,the orthogonal Euler polynomials are derived by the Gram-Schmidt orthogonalization method,and then the orthogonal Euler wavelet with orthogonality is constructed by stretching and translating the orthogonal Euler polynomials.Then,the fractional order integral formula is derived by using the definition and properties of Laplace transform,and the fractional order Riccati differential equation is solved numerically by using the deduced integral formula,and the convergence analysis and error estimation of this method are given.Compared with other wavelet methods,the numerical results show the feasibility and high precision of the proposed method.Finally,on the basis of one dimensional Euler orthogonal wavelet was deduced two-dimensional Euler orthogonal wavelet,shows its convergence analysis and error estimation,and using two dimensional Euler orthogonal wavelet for wave equation with nonlocal conservation conditions for numerical solution,and related numerical example is given,and the numerical results show that this method is of high precision,verify the effectiveness of the proposed method.