**4. Conclusions**

We studied the heat conduction equations with the Caputo fractional derivatives in two joint half-planes under conditions of perfect thermal contact (the equality of temperatures and heat fluxes at the contact surface). It should be emphasized that due to the constitutive Equation (8), the proper boundary conditions should be stated in terms of the heat fluxes, not in terms of the normal derivatives of temperature alone. Introducing the auxiliary function *φ*(*<sup>x</sup>*, *t*) allows us to use the cos-Fourier transforms in two contact regions. The fundamental solutions permit obtaining various solutions to the Cauchy and source problems in the convolution form.

**Author Contributions:** Both authors have equally contributed to this work. Both authors read and approved the final manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank the reviewers for their helpful comments to improve the quality of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
