In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.

中文翻译：

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使用Müntz-Legendre小波方法的时域分布阶分数阶微分方程的数值解

本文提出了一种数值方法来获取和分析时域中具有Caputo分数阶导数的一般形式的分布阶分数阶微分方程的数值解的行为。建议的方法基于Müntz-Legendre小波逼近。通过使用Laplace变换方法，我们为Müntz-Legendre小波的Riemann-Liouville分数积分导出了一个新的运算向量。在我们的方法中应用这种运算向量和并置方法，可以将问题简化为线性和非线性代数方程组。产生的系统可以通过牛顿法求解。讨论了该方法的误差界和收敛性分析。最后，考虑了七个测试问题，以将我们的结果与其他用于解决这些问题的方法进行比较。列表中的结果突出表明，所提出的方法是一种有效的数学工具，用于分析一般形式的分布式分数阶微分方程。