【摘要】 为解决在物理学中有着广泛应用的一种非线性双曲Sine-Gordon(SG)方程的数值解问题,提出了移位的Chebyshev多项式与分数阶微分性质相结合的高效数值算法.首先,我们推导出移位的Chebyshev多项式一阶微分算子矩阵和分数阶微分算子矩阵,然后将Sine-Gordon(SG)方程转化为线性代数方程组的形式,进而得到分数阶非线性SG方程的数值解.根据所提出的误差校正相关理论,对数值解进行校正以达到更高的精确度.最后用数值算例及收敛阶数对算法进行验证,表明了本文所提方法的有效性和实用性.更多还原

【Abstract】 In order to solve the numerical solution problem for a class of nonlinear hyperbolic Sine-Gordon(SG) equation which has been widely used in physics. An efficient numerical algorithm combining the shifted Chebyshev polynomial with the fractional differential property is proposed. First, we derive the first-order differential operator matrix and fractional differential operator of the shifted Chebyshev polynomials. Then, we transform the Sine-Gordon(SG) equation into a form of linear algebraic equations. And the numerical solution of fractional nonlinear SG equation is obtained. Moreover, correct the numerical solution to achieve higher accuracy according to the proposed error correction related theory. Finally, numerical examples and convergence order are given to show the effectiveness and practicability of the proposed method.更多还原