Finding the unknowns in directly inaccessible areas is one of the challenging problems in engineering applications. The fluid flow or thermal conditions of the flow inside the pipe are examples of such unknowns. Developing analytical models that provide explicit mathematical formulas is a mathematically efficient and desirable way of dealing with these challenges. This dissertation focuses on developing an analytical solution that can find the temperature distribution in radial and axial directions in the pipe wall. Partial heating along with Green’s functions is used to develop this solution. Working with Greens functions sparked the idea of using them as building blocks for a novel numerical method for solving the heat conduction equation. In the second chapter, the transient temperature response inside the pipe to the partial heating on it is found using GFs and the solution is verified using a few intrinsic verification principles. This analytical solution, then, is used in developing a non-invasive method for measuring the flow rate in pipes by measuring the temperature at a single point. Optimal experiment design methods are used to find the optimal location and time duration for performing the measurement. Chapter 3 is about developing a novel numerical solution to the linear heat conduction equation. This method uses superposition of exact solutions (SES), obtained using GFs, to evaluate the temperature and heat flux at any point of the 1-D domain. The SES method is not sensitive to the size of the grids and is much more accurate than the conventional Crank Nicolson (CN) method. This method is extended to the cases in which the thermal properties vary with temperature, later, in Chapter 4. The results confirm the grid independence of the SES and show high accuracy in its prediction when the time step is not large. One of the applications of the analytical solution obtained herein is using them in solving inverse problems. Inverse heat conduction problems (IHCPs) are used to estimate unknown heat flux functions through measuring temperatures far from active surfaces. The second part of this dissertation (Chapters 5 and 6) focuses on generalizing and optimizing two IHCP solution methods.