The Wavelet Collocation Method For Differential Equations

In this paper, we study the basis of wavelet application in numerical. Wavelet method for calculating the definite integral and numerical solution of partial differential equations is very important theory significance and practical value.The reason why wavelet can be used in solving partial differential equation is that wavelet can concentrate on local characteristics of a function in both spatial and frequency domain,and it can adaptively take different sample intervals according to different frequency parts,therefore it can decompose signals into different frequency parts,and can focus any details.In this paper,we use trigonometric Hermite type interpolation wavelet,researching definite integral numerical computation of the general function and solving Heat Conduction Equation.Trigonometric Hermite type interpolation wavelet is a periodic wavelet,which has a high-level of wavelet and wavelet characteristics of the nature of localized.When calculating the integral,its original function can not be expressed by elementary function,finding the original function can not apply directly Newton-Leibniz formula evaluation;and when the original function is very complicated,it also difficults to solve by Newton-Leibniz formula evaluation.Therefore,the calculation of the definite integral of numerical method is necessary.There are many numerical methods for solving definite integral.In this paper,we use trigonometric Hermite type interpolation wavelet operator, we derive the computing formula of definite integrals for the general functions, and also establish the corresponding error estimates, then we give numerical examples and show that the numerical results computed by the formula have high degree of accuracy.Finally,we apply the trigonometric Hermite type interpolation wavelet for solving Heat Conduction Equation.When the initial of Heat Conduction Equation is a periodic function.,its solution has periodicity.So using trigonometric Hermite-type interpolation wavelets as basis function to approximate the solution,this approach is feasible.We use wavelet collocation method discreting the spatial domain of Heat Conduction Equation,establish a set of ordinary differential equations of time,then solve ordinary differential equations by fourth-order Runge-Kutta method.