In this study, we present the new generalized derivative and integral operators which are based on the newly constructed new generalized Caputo fractal–fractional derivatives (NGCFFDs). Based on these operators, a numerical method is developed to solve the fractal–fractional differential equations (FFDEs). We approximate the solution of the FFDEs as basis vectors of shifted Legendre polynomials (SLPs). We also extend the derivative operational matrix of SLPs to the generalized derivative operational matrix in the sense of NGCFFDs. The efficiency of the developed numerical method is tested by taking various test examples. We also compare the results of our proposed method with the methods existed in the literature In this paper, we specified the fractal–fractional differential operator of new generalized Caputo in three categories: (i) different values in $\rho$ and fractal parameters, (ii) different values in fractional parameter while fractal and $\rho$ parameters are fixed, and (iii) different values in fractal parameter controlling fractional and $\rho$ parameters.

在这项研究中，我们介绍了新的广义导数和积分算子，这些算子基于新构建的新的广义Caputo分形-分形导数（NGCFFDs）。基于这些算子，开发了一种数值方法来求解分形-分数阶微分方程（FFDE）。我们将FFDE的解近似为移位的Legendre多项式（SLP）的基向量。在NGCFFD的意义上，我们还将SLP的导数运算矩阵扩展为广义导数运算矩阵。通过采用各种测试示例来测试开发的数值方法的效率。我们还将我们提出的方法的结果与文献中已有的方法进行比较。在本文中，我们在三类中指定了新的广义Caputo的分形-分形微分算子：$\rho$ 和分形参数；（ii）分形参数的不同值，而分形和 $\rho$ 参数是固定的；以及（iii）分形参数中控制分数值和分数值的不同值 $\rho$ 参数。