In this manuscript, we investigate the approximate solutions to the tangent nonlinear packaging equation in the context of fractional calculus. It is an important equation because shock and vibrations are unavoidable circumstances for the packaged goods during transport from production plants to the consumer. We consider the fractal fractional Caputo operator and Atangana–Baleanu fractal fractional operator with nonsingular kernel to obtain the numerical consequences. Both fractal fractional techniques are equally good, but the Atangana–Baleanu Caputo method has an edge over Caputo method. For illustrations and clarity of our main results, we provided the numerical simulations of the approximate solutions and their physical interpretations. This paper contributes to the new applications of fractional calculus in packaging systems.

在这篇手稿中，我们研究了分数阶微积分背景下正切非线性包装方程的近似解。这是一个重要的方程式，因为冲击和振动是包装货物在从生产工厂运输到消费者的过程中不可避免的情况。我们考虑分形分数 Caputo 算子和具有非奇异核的 Atangana-Baleanu 分形分数算子来获得数值结果。两种分形分数技术同样出色，但 Atangana-Baleanu Caputo 方法优于 Caputo 方法。为了说明和清楚我们的主要结果，我们提供了近似解的数值模拟及其物理解释。本文有助于分数微积分在包装系统中的新应用。