Study On The Legendre Wavelet Method Of Differential And Integral Equations

Wavelet analysis has an important role in scientific and engineering calculation, so the methods of solving differential and integral equations by wavelets have also been a wide development and application. In most practical problems, the solutions to the problems are defined on a finite interval, therefore, interval wavelets attract people's attention. In this paper, the interval wavelets methods of differential and integral equations are studied and a series of results are obtained. The main contents of the paper are as follows:(1) The Legendre multi-wavelets basis functions are constructed based on the multi-resolution analysis theory on L2 ([0,1]). The wavelets have the orthogonality, symmetry, small support set and the computability, so the calculation is more simple and accurate than other wavelets, it is more suitable for scientific computing applications.(2) The Legendre wavelets method of the parabolic equations is studied. Firstly, the time is discreted and ordinary differential equations with space variable are obtained. Then the ordinary differential equations are discreted, the unknown functions and their derivatives are approached by Legendre wavelets and integral operator matrix. Finally, the equations are solved by collocation method. The algorithm is simple, at the same time, a large quantity of calculation and slow convergence with other classical methods are overcomed.(3) The Legendre multi-wavelets method of parabolic partial differential equations is studied. In this chapter, one-dimensional nonlinear Burgers equation is solved. Firstly, the equation is discreted in the scale space and wavelet space, it will be converted to ordinary differential equation, then we use the direct integration method, the unknown functions and their derivatives are expressed with Legendre multi-wavelets, and finally the equations are solved by collocation method. In regard to nonlinear term of the burgers equation, we use the Taylor series to approched and the calculation is more convenient. Experimental results show that the accuracy by wavelets approximation is higher than the scale functions approximation, at the same time, multi-wavelets can perfectly combine the orthogonality and symmetry, so the calculation is more convenient.(4) The Fredholm integral equations of the first kind are studied by the Legendre muli-wavelets. In the process of solving, the wavelets coefficients are selected adaptively by threshold and the "best" wavelets basis functions are selected to solve the equation. Finally, the Fredholm integral equations of the first kind are solved, the results show that the calculation is simple and effective.