2-D Spectral Collocation Method Of The Volterra-Fredholm Equation And Error Analysis

With the development of science and technology, the mathematical model of many engineering problems can be written into integral equation forms, such as fluid mechanics, mechanics of elasticity, biology and population problem, and so on. So it is meaningful to study the integral equation. Integral equation of the study provides great help to solve the pratical problems, at the same time, the integral equation of different types of various models corresponding to the real problems. Fredholm and Volterra types are common forms of integral equations. However the collocation method is used when we come across the one dimensional or two dimensional equations, while two dim-ensional Fredholm- Volterra integral equation’s study is not complete.The solution of integral equation has quadrature method, degradation of nuclear method, collocation method and so on. In this thesis, we study the method of the two-di-mensional Fredholm- Volterra integral equations. This thesis introduce four collocation methods to us, such as Taylor collocation method, Fibonacci collocation method, Lagra-nge collocation method and Chebyshev collocation method. A kind of numerical metho-d in chapter four introduced enrich the two-dimensional hybrid integral equation numer-ically. Based on different basis functions and different collocation points, we can take the collocation of the solution to the original equation. Finally when solving matrix equation,we can get the solution of the coefficients. Take the coefficients into the collocation solution to get the final result. Behind each section will give a method of error analysis, comparison between various methods of benefits. Collocation method of the coefficient matrix of the condition number for the collocation for short quantity of law. The size of the condition number of affecting the computation of discrete equations,is an important index of evaluating algorithm, we also study collocation method of the foundation.In addition to the four collocation methods given in chapter four, thesis also gives a Chebyshev- Legendre numerical solving method. The third chapter mainly gives the Chebyshev-Legendre collocation method on the integral equation,which is extended to two dimensional on Fredholm-Volterra integral equation. Here, the thesis use the Gauss points solution on the integral parts. In the practical solution, we choose method accordi-ng to the equation of concrete forms. For the Chebyshev-Legendre spectral collocation method. This thesis gives a kind of the most common form, because there are otherforms of Chebyshev polynomials. If the weight function is different, the whole matrix changes, so this method also has its limitation.