变阶分数阶微分方程的伯努利多项式数值解
Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials
摘要：
介绍一种新的数值方法，基于伯努利多项式，求解多项变阶分数阶微分方程。在Caputo意义下考虑了变阶分数阶导数，而Riemann-Liouville积分算子用于给出未知函数及其变阶导数的近似.对伯努利函数引入了变阶分数次积分的运算矩阵。通过假设问题的解是足够光滑的，我们用伯努利多项式近似了它的导数的给定阶。然后，我们使用引入的操作矩阵，以找到未知函数及其导数的一些近似。利用这些近似和一些配置点，该问题被简化为一个非线性代数方程的解决方案。给出了用该方法得到的近似解的误差估计。最后，五个说明性的例子被认为是证明所提出的技术的适用性和高精度，比较我们的结果与现有的方法在文献中得到的，明确的新颖性的工作。数值结果表明，新方法是有效的，即使在基函数数目较少的情况下，以及当问题的解不是无穷可微时，也能给出高精度的近似解，与现有方法相比，所用基函数数目较少，结果更好.
英文摘要：
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann--Liouville integral operator was used to give approximations for the unknown function and its variable-order derivatives. An operational matrix of variable-order fractional integration was introduced for the Bernoulli functions. By assuming that the solution of the problem is sufficiently smooth, we approximated a given order of its derivative using Bernoulli polynomials. Then, we used the introduced operational matrix to find some approximations for the unknown function and its derivatives. Using these approximations and some collocation points, the problem was reduced to the solution of a system of nonlinear algebraic equations. An error estimate is given for the approximate solution obtained by the proposed method. Finally, five illustrative examples were considered to demonstrate the applicability and high accuracy of the proposed technique, comparing our results with the ones obtained by existing methods in the literature and making clear the novelty of the work. The numerical results showed that the new method is efficient, giving high-accuracy approximate solutions even with a small number of basis functions and when the solution to the problem is not infinitely differentiable, providing better results and a smaller number of basis functions when compared to state-of-the-art methods.
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