基于径向基函数插值的积分方程求解

Solving integral equations based on radial basis function interpolation

将径向基函数(radial basis function,RBF)插值引入积分方程的求解中,具体将待求函数表示为RBF的线性组合,再通过配点法将积分方程离散为线性或非线性方程组,求得权系数后给出待求函数的近似表示.论文选用的RBF是插值性能优异的多重二次曲面(multiquadric,MQ)函数,能在较少节点下取得较高的近似精度;而且RBF定义为距离的函数,在三维或高维插值时仅需改变距离公式,因而便于推广到高维积分方程求解中.在RBF插值矩阵的构造中,元素的积分计算分别通过高斯积分或基于区域剖分的数值求积完成,实现了一维、二维下Fredholm和Volterra方程的求解.算例结果表明:论文方法具有实施方便和精度较高的优点,是一种适合积分方程求解的新方法.

In order to acquire a numerical method with high precision character- istic and multi-dimensional adaptability for solving integral equations, the radial basis function （RBF） is introduced. Specifically, the unknown function is firstly expressed as a linear combination of RBF, then the integral equation is transformed to discrete linear or nonlinear equations nally, an approximate representation for through the collocation method, and ti- the unknown function is obtained after determining the weights of RBF. The multi-quadric （MQ） function is adopted due to its superior interpolation performance. In addition, solving three or higher di- mensional integral equations are easily implemented because the RBF is a function of distance and the only changing is the distance formula. The Gauss quadra- ture formula, the geometric splitting and numerical integral methods are employed in calculating the integral in RBF interpolation matrix. Numerical examples as Fredholm or Volterra equations in one or two-dimension have been carried out. The results show that the proposed method is easily to implement, and it has the advantage of high precision and convenience. Therefore, it is a new and suitable method for solving integral equations.