In this paper, a novel iterative algorithm is developed to solve second-order nonlinear singular boundary value problem, whose solution exactly satisfies the Robin boundary conditions specified on the boundaries of a unit interval. The boundary shape function is designed such that the boundary conditions can be fulfilled automatically, which renders a new algorithm with the solution playing the role of a boundary shape function. When the free function is viewed as a new variable, the original singular boundary value problem can be properly transformed to an initial value problem. For the new variable the initial values are given, whereas two unknown terminal values are determined iteratively by integrating the transformed ordinary differential equation to obtain the new terminal values until they are convergent. As a consequence, very accurate solutions for the nonlinear singular boundary value problems can be obtained through a few iterations. The present method is different from the traditional shooting method, which needs to guess initial values and solve nonlinear algebraic equations to approximate the missing initial values. As practical applications of the present method, we solve the Blasius equation for describing the boundary layer behavior of fluid flow over a flat plate, where the Crocco transformation is employed to transform the third-order differential equation to a second-order singular differential equation. We also solve a nonlinear singular differential equation of a pressurized spherical membrane with a strong singularity.

本文提出了一种新颖的迭代算法来求解二阶非线性奇异边值问题，其求解恰好满足了单位区间边界上的Robin边界条件。边界形状函数的设计使得可以自动满足边界条件，这提出了一种新的算法，其解决方案起着边界形状函数的作用。当自由函数被视为新变量时，原始奇异边值问题可以适当地转换为初始值问题。对于新变量，将给出初始值，而两个未知的终极值将通过积分变换后的常微分方程以获得新的终极值进行迭代确定，直到它们收敛为止。作为结果，非线性奇异边值问题的非常精确的解决方案可以通过几次迭代来获得。本方法不同于传统的射击方法，传统的射击方法需要猜测初始值并求解非线性代数方程以近似缺少的初始值。作为本方法的实际应用，我们求解了描述流体在平板上流动的边界层行为的Blasius方程，其中使用Crocco变换将三阶微分方程转换为二阶奇异微分方程。我们还求解了具有奇异性的加压球面膜的非线性奇异微分方程。本方法不同于传统的射击方法，传统的射击方法需要猜测初始值并求解非线性代数方程以近似缺少的初始值。作为本方法的实际应用，我们求解了描述流体在平板上流动的边界层行为的Blasius方程，其中使用Crocco变换将三阶微分方程转换为二阶奇异微分方程。我们还求解了具有奇异性的加压球面膜的非线性奇异微分方程。本方法不同于传统的射击方法，传统的射击方法需要猜测初始值并求解非线性代数方程以近似缺少的初始值。作为本方法的实际应用，我们求解了描述流体在平板上流动的边界层行为的Blasius方程，其中使用Crocco变换将三阶微分方程转换为二阶奇异微分方程。我们还求解了具有奇异性的加压球面膜的非线性奇异微分方程。我们求解Blasius方程，以描述平板上流体流动的边界层行为，其中使用Crocco变换将三阶微分方程转换为二阶奇异微分方程。我们还求解了具有奇异性的加压球面膜的非线性奇异微分方程。我们求解Blasius方程，以描述平板上流体流动的边界层行为，其中使用Crocco变换将三阶微分方程转换为二阶奇异微分方程。我们还求解了具有奇异性的加压球面膜的非线性奇异微分方程。