一类分数阶Langevin方程block⁃by⁃block算法的数值分析

Numerical Analysis of a Class of Fractional Langevin Equations With the Block⁃by⁃Block Method

分数阶Langevin方程有重要的科学意义和工程应用价值,基于经典block⁃by⁃block算法,求解了一类含有Caputo导数的分数阶Langevin方程的数值解.Block⁃by⁃block算法通过引入二次Lagrange基函数插值,构造出逐块收敛的非线性方程组,通过在每一块耦合求得分数阶Langevin方程的数值解.在0<α<1条件下,应用随机Taylor展开证明block⁃by⁃block算法是3+α阶收敛的,数值试验表明在不同α和时间步长h取值下,block⁃by⁃block算法具有稳定性和收敛性,克服了现有方法求解分数阶Langevin方程速度慢精度低的缺点,表明block⁃by⁃block算法求解分数阶Langevin方程是高效的.

The fractional Langevin equation is of great scientific significance and engineering application value.Based on the classical block⁃by⁃block method,the numerical solution of a class of fractional Langevin equations with Caputo derivatives was obtained.Through introduction of the quadratic Lagrange basis function interpola⁃tion,the block⁃by⁃block convergent nonlinear equations were constructed,and the numerical solution of the Langevin equation was obtained by coupling in each block.Under the condition of 0<α<1,the stochastic Taylor expansion was used to prove that the block⁃by⁃block method is(3+α)⁃order convergent.Numerical ex⁃periments show that,the block⁃by⁃block method is stable and convergent under different values ofαand time step h,and overcomes the existing methods’disadvantages of slow speed and poor accuracy for solving frac⁃tional Langevin equations.