高阶Haar小波配置法求解五阶微分方程

The Solution of Fifth-Order Differential Equations by the High-Order Haar Wavelet Collocation Method

摘要: 本文利用高阶Haar小波方法求解具有不同边界条件的五阶微分方程。对线性微分方程，使用高阶Haar小波配置法，将微分方程转化为线性代数方程组求解；对于非线性微分方程，则使用拟线性化方法将其转化为线性微分方程后求解。通过计算方程组系数矩阵的条件数，判断出方法的稳定性。数值实验表明，高阶Haar小波方法比经典的Haar小波方法有着更高的数值精度，可以用更少的配置点获得更小的误差，并且增加尺度误差下降得更快，通过求解最大绝对误差和均方根误差，得到了高阶Haar小波方法具有四阶精度的结论，数值计算结果与其他方法进行了比较。

Abstract: This paper utilizes the high-order Haar wavelet method to solve fifth-order differential equations with different boundary conditions. For linear differential equations, the high-order Haar wavelet collocation method is employed to transform the differential equation into a system of linear algebraic equations for solution. For nonlinear differential equations, the quasi-linearization method is used to convert them into linear differential equations before solving. The stability of the method is determined by calculating the condition number of the coefficient matrix of the equation system. Numerical experiments show that the high-order Haar wavelet method has higher numerical accuracy than the classical Haar wavelet method, achieving smaller errors with fewer collocation points. Moreover, as the scale increases, the error decreases more rapidly. By solving the maximum absolute error and the root mean square error, it is concluded that the high-order Haar wavelet method has a fourth-order accuracy. The numerical results are compared with those of other methods.