In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given.

Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial

本文利用切比雪夫多项式的性质和高斯-勒让德求积法则构造了一个新的分布阶分数阶导数运算矩阵。该运算矩阵适用于求解分布阶次分数阶微分方程、分布阶次时间分数阶扩散方程和分布阶次时间分数阶波动方程等问题。我们的方法很容易将所有这些问题的解简化为一组代数方程的解。我们还讨论了使用该运算矩阵进行近似分布阶分数阶导数的误差分析。最后，为了说明所提出技术的效率和有效性，给出了五个例子。

缩写： DFDEs：分布阶次分数阶微分方程；DTFDEs：分布阶次时间分数扩散方程；DTFWEs：分布阶次时间分数波动方程；OMFD：分数阶导数运算矩阵；SCP：移位切比雪夫多项式