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The governing equations of the body which undergoes the heat and/or force shocks are the Navier's equation and energy one, which in coupled form, they are named coupled thermoelasticity equations. In this paper these equations are solved for a long axisymmetric cylinder. The boundary conditions are heat flow and/or surface stress on the inner boundary and that the heat isolated outer surface is free stress. Using the eigen values and eigen vectors of the matrix differential operator, the coupled
. thermoelasticity equations transform to uncoupled one. Using laplace transform the uncoupled equations are solved and the analytical solutions are obtained in laplace space. The inverse lap lace transform will be obtained in the companion paper using the numerical methods.