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In this study, a stochastic differential equation capable of describing (using the motion function) the automatic manufacturing process of a lithium battery with a sleeve shell is introduced. The boundary-condition modeling method for this type of motion is an ordinary differential equation. The nonlinear equation is found using a dynamic method. The equations of the motions for the assembly process are derived by reducing the order of terms and separating the variables. Both the derivatives and integrals of the motion functions are derived and applied to describe the assembly process for a lithium battery. By using a characteristic constitutive assembly model, a new method to calculate the nonlinear terms is proposed. The main advantage of this method is that it can simplify the problem, which is done by finding the algebraic error in the region space of the motion function. The reliability and the applicability of the method were confirmed using a real-life example. Finally, by solving and improving the Euler model, the fourth-order Kutta method and Taylor series are applied to verify the correctness of the algorithm, and the convergence order of the function and the optimization of the assembly model are discussed. Through a one-dimensional search of the convergence domain, a method is found to make the analytical solution approximate the exact solution and reduce the calculation cost. This method in dynamics and mathematical engineering.

• We derived the differential equation for the assembly process of a lithium battery. • A new mathematical method to solve assembly dynamics is proposed. • The new method performs generally better than other algorithms. In this study, a stochastic differential equation capable of describing (using the motion function) the automatic manufacturing process of a lithium battery with a sleeve shell is introduced. The boundary-condition modeling method for this type of motion is an ordinary differential equation. The nonlinear equation is found using a dynamic method. The equations of the motions for the assembly process are derived by reducing the order of terms and separating the variables. Both the derivatives and integrals of the motion functions are derived and applied to describe the assembly process for a lithium battery. By using a characteristic constitutive assembly model, a new method to calculate the nonlinear terms is proposed. The main advantage of this method is that it can simplify the problem, which is done by finding the algebraic error in the region space of the motion function. The reliability and the applicability of the method were confirmed using a real-life example. Finally, by solving and improving the Euler model, the fourth-order Kutta method and Taylor series are applied to verify the correctness of the algorithm, and the convergence order of the function and the optimization of the assembly model are discussed. Through a one-dimensional search of the convergence domain, a method is found to make the analytical solution approximate the exact solution and reduce the calculation cost. This method in dynamics and mathematical engineering. [ABSTRACT FROM AUTHOR]