In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

中文翻译：

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一类分数阶非线性偏微分方程的最小二乘微分正交方法

本文介绍了一种称为最小二乘微分正交方法（LSDQM）的新方法，它是一种直接有效的方法，用于计算带有分数时间导数的非线性偏微分方程的解析近似多项式解。LSDQM是微分求积法和最小二乘方法的组合，在本文中，它被用于找到非常通用的一类非线性偏微分方程的近似解，其中分数导数在Caputo意义上进行了描述。本文包含该方法的清晰分步介绍和收敛定理。为了强调LSDQM的准确性，我们引入了两个以前通过其他已知方法解决的测试问题，并观察到我们的解决方案不仅呈现出较小的误差，而且呈现出更为简单的表达方式。我们还包括了一个尚无确切解决方案的问题，并且LSDQM计算出的解决方案与以前的解决方案非常一致。