Dissertation (MSc Mathematics)In this dissertation, it is considered, the investigation of the Fourier Transform and the fundamental solutions of mechanical models with internal body forces. Although, there are many models, only two models are considered on the review, namely heat conduction and linear isotropic elasticity. For the investigation, the statement of the problem bases only on the linear isotropic elasticity problem. In working with calculations, the Fourier Transform, Inverse Fourier Transform, Residue Theorem, Jordan‟s Lemma, are applied in particular as well as the complex analysis and integration are applied in general. The obtained results of the statement of the problems having two cases, namely first and second cases, satisfy both their relevant equations and corresponding conditions accordingly. Therefore, they are identified as the fundamental solutions of their corresponding cases according to the problem. In the conclusion, the reviews also show that the Fourier Transform is widely applicable in variety of fields such as electromagnetic fields, magnetic resonance imaging (MRI) and quantum physics. Generally, it is also important in mathematics, engineering, and physical sciences. Finally, the investigator of this writing advises more application of the Fourier Transform since there is a lot of problems which are difficult/nearly impossible to solve directly, become easy after using a Fourier Transform. Also, it is reported that the mathematical operations on functions, like derivatives or convolutions, become much more manageable on the far side of a Fourier Transform.