Análise do Método Multi-Passos com Transformada Diferencial Generalizada na Modelagem Fracionária

ABSTRACT This paper presents a critical analysis of a numerical technique that has been used in the resolution of fractional differential equations with Caputo derivative. It is the multi-step method with generalized differential transform. It is found that the version of the method available in the literature produces erroneous solutions from the second step, this is shown in applications to the Malthus and Riccati models. The procedure error is explained in terms of the nonlocality of the Caputo derivative and the properties of the generalized differential transform.

摘要 本论文针对采用Caputo导数（Caputo derivative）的分数阶微分方程的数值求解技术展开批判性分析，所分析的方法为带广义微分变换的多步法（multi-step method with generalized differential transform）。研究发现，现有文献中该方法的版本自第二步起便会生成错误解，这一结论通过将该方法应用于马尔萨斯（Malthus）模型与里卡蒂（Riccati）模型得到验证。本文从Caputo导数的非局部性以及广义微分变换的特性两方面，对该过程中的误差进行了解释。