A fast numerical solution method for two dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation

作者：Liang Fen; Lin Fu Rong^*

来源：Applied Mathematics and Computation, 2010, 216(10): 3073-3088.

DOI：10.1016/j.amc.2010.04.027

• Integral equation
• Polynomial interpolation
• Approximate matrix
• Residual correction scheme
• Kronecker tensor product

In this paper we consider fast numerical solution methods for two dimensional Fredholm integral equation of the second kind
f(x,y) - integral(beta)(alpha) integral(beta)(alpha) a(x, y, u, v)f(u, v)dudv =g(x, y), (x, y) is an element of [alpha, beta] x [alpha, beta],
where a(x, y, u, v) is smooth and g(x, y) is in L(2)[alpha, beta](2). Discretizing the integral equation by certain quadrature rule, we get a linear system. To deduce fast approximate solution methods for the resulted linear system, we study the approximation of the four-variable kernel function a(x, y, u, v) by piecewise polynomial: partition the domain [alpha, beta](4) into subdomains of the same size and interpolate the kernel function a(x, y, u, v) in each subdomain. Fast matrix-vector multiplication algorithms and efficient iterative methods are derived. Numerical results are given to illustrate the efficiency of our methods.

• 出版日期2010-7-15
• 单位汕头大学