**ABSTRACT**

In this work, the differential transformation approach with after-treatment technique is used to evaluate the nonlinear vibration of a single-walled nanotube operating in a multi-layer elastic media in a thermal-magnetic environment. Hamilton’s principle is used to calculate the equation of motion for the nanotube based on nonlocal elasticity theory and the Euler-Bernoulli beam model. The nonlinear vibration model is then broken down into its spatial and temporal components using the Galerkin decomposition technique. Using the differential transformation approach in conjunction with the cosine after-treatment technique, the resulting temporal ordinary differential equation is solved. The effect of nonlocal, elastic media and thermal-magnetic factors on the dynamic behavior of the nanotube is further examined using the established analytical solution. The analytical analysis demonstrates that the frequency ratio rises as the dimensionless amplitude grows as the frequency ratio increases, as do the values of the dimensionless nonlocal, quadratic, and cubic elastic medium stiffness parameters. However, as the temperature change, magnetic force, Winkler, and Pasternak layer stiffness factors increase, the frequency ratio falls. The study, design, and uses of nanotubes in thermal and magnetic settings are greatly aided by the current work.

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