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Heat conduction close-to-Fourier means, that we look for a minimal extension of heat conduction theory using the usual Fourier expression of the heat flux density and modifying that of the internal energy as minimal as possible by choosing the minimal state space. Applying Liu's procedure results in the class of materials and a differential equation both belonging to the close-to-Fourier case of heat conduction. A symbolic-numerical computing method is applied to approximate the numerical solutions of 2 special heat conduction equations belonging to the close-to-Fourier class.