MQ函数在偏微分方程中的应用

Application of the Multiquadric Method for Numerical Solution of Partial Differential Equation

讨论了用径向基multiquadric(MQ)函数φ(r)=(r2+c2)^(1/2)作为基函数解一类偏微分方程,给出方法步骤,并通过一个数值算例,说明这个方法是可行的.针对数值算例,比较了在相同步长时,用径向基函数在不同的形状参数时绝对误差的差异,说明微分方程数值解的精确程度与径向基函数形状参数的取值密切相关,得出节点越密时,数值解的精度不一定越高.同时也论证了在插值过程中所得到的矩阵方程解的存在唯一性.

An algorithm for partial differential equation based onthe multiquadric （MQ） function φ（r）=√r2＋c2 as basis function approximation scheme is presented. A fairly exphcit scheme is used to approximate the solution. One model problem of the algorithm is given. The comparison is made with the exact solutions of the problem by different shape parameter and different nodal distance. Numerical results show that the method offers a very high accuracy in computation of the partial differential equation. It is debated that numerical results may not be better when nodal distance is smaller. It is proved that the matrix equation we obtain has a solution.