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Purpose: The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system. Design/methodology/approach: First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution. Findings: An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system. Originality/value: The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work. [ABSTRACT FROM AUTHOR]

•The influence of dispersion forces on the dynamic pull-in behavior of nano-beams is investigated.•The impact of amplitude of vibration on pull-in instability, pull-in voltage and fundamental frequency of nano-actuators is studied.•The second-order frequency–amplitude relationship is introduced by employing PEM.•The phase portraits of the nano-beam actuator exhibit periodic and homoclinic orbits.