Numerical Solutions Of Higher-order Ordinary Differential Equations And Poisson Equations And Biharmonic Equations By Using Trigonometric Wavelets

The trigonometric series plays an important role in the theoretical study of harmonic analysis.In the meantime,trigonometric wavelets also occupy a pivotal position in the history of wavelet analysis.Trigonometric wavelets,which is the perfect combination of trigonometric series and wavelets,not only have the advantage of good approximation characteristics of trigonometric series but also have local characteristics of wavelet.It has been widely used in various fields.The aim of this paper is detailed integral collocation approach on trigonometric wavelets for numerically solving high-order ODEs,Poisson equations and biharmonic equations.This approach not merely broadens the fields of numerical solution of differential equation but also provides a new method for numerical solution of differential equation.In this paper,firstly,the integral formulas of trigonometric wavelets are studied by using trigonometric wavelets definition,the partial integration and mathematical induction,which has made first-phase preparations for the numerical solution of partial differential equations.Secondly,the integral collocation approach on trigonometric wavelets is introduced for numerically solving four ordinary differential equations and six 2D Poisson equations and 2D biharmonic equations which satisfy different boundary conditions.The basic idea is that the highest order derivative function appearing in differential equation is approximated by truncated trigonometric wavelet series.And the other low-order derivative equations and unknown function are obtained by multiple integrals and using boundary conditions.All these integral formulas are substituted into the differential equations and convert differential equations to the system of linear equations by choosing the collocation points.The trigonometric wavelets coefficients are obtained by solving the system of linear equations,then the numerical solution of trigonometric wavelets are obtained.Finally,the feasibility and effectiveness of trigonometric wavelets for solving the differential equations in this paper are fully verified by the error between the numerical solution and the exact solution,which obtains through Matlab programming.