Wavelet Numerical Solution Of Fredholm Integral Equation Of The Second Kind And Fisher Equation

With the global economy, as well as the rapid development of technology information, many physical phenomena need to be solved by the solutions of the integral equation and the differential equation in our daily life.But it’s difficult to get the solutions of those equations. So it’s necessary to obtain the numerical solution using computer technology. The most common of those method have difference transformation method, Runge-Kutta method and finite element method, etc. Those numerical methods are effective in solving the regular and smooth solutions of the equations. Those methods have their own advantages, but they also have some shorttages. For example, those numerical methods have disadvantage to solve the singularity equations. With the fast development of wavelet analysis, scholars have found that wavelet analysis has the good time-frequency localization features and some other good properties, such as orthogonality, vanishing moment andcompact suppor, etc. It is feasible and effective to solve the numerical solution of integral equations and differential equations by using the wavelet function as the basis function, which is convenient for numerical calculation. This paper mainly studies the application of wavelet in the numerical solution of integral equations and differential equations. Therefore, this paper mainly finished the following contents:(1) The development and application of wavelet analysis as well as the research progress of the wavelet numerical solution of the integral equation and differential equation are introduced.(2) The basic theory of wavelet analysis, including the definition of wavelet, continuous wavelet transform,discrete wavelet transform, reconstruction formula and wavelet multi-resolution analysis are simply described.(3)Focus on the construction method of piecewise polynomial wavelet. First, use the compression mapping,the properties of the wavelet multi-resolution analysis and the orthogonality of the wavelet to construct piecewise polynomial wavelet in bounded area. And then use the orthogonality of the wavelet and Legendre polynomial to get the explicit expression of piecewise linear polynomial wavelets and piecewise quadratic polynomial wavelets.The giving of the explicit expression brings great convenience to the numerical calculation.(4)Use the piecewise polynomial wavelets to solve the wavelet numerical solution of Fredholm integral equations of the second kind and Fisher equation. First, make the piecewise linear polynomial wavelet as basis function, and adopt the method of function approximation to the unknown function of Fredholm integral equations of the second kind for the functional approximation. We can discrete Fredholm integral equations of the second kind as piecewise linear wavelet coefficients of polynomial algebra group by the function approximation. Finally by solving discrete algebraic equations obtain the wavelet numerical solution of Fredholm integral equations of the second kind. At the same time, make the piecewise quadratic polynomial wavelet as basis function, and adopt approximation of function to Fisher equation of differential operator for functional approximation, then with the collocation method, convert Fisher equation to algebraic equations about piecewise quadratic polynomial wavelet coefficients, finally using a computer programming, wavelet numerical solution of Fisher equation can be obtained.The last numerical examples show that the wavelet method is feasible and effective.