In this thesis, we demonstrate a method outlined by Dr. Mikhail Klibanov for solving a 1-D coefficient inverse problem by the convexification method. Our inverse problem in question concerns finding buried bombs, where the dielectric constants of the bomb and the sand in which it is buried are represented by the coefficient function $ c(x) $. The goal of our method is to approximate $ c(x) $. In the method demonstrated in this thesis, we compute an orthonormal basis from the set $ \{k^{n}e^{k}\}_{n=0}^{\infty} $ consisting of $ N $ vectors. Then we derive a series of boundary value problems from our coeficient inverse problem. Then we get a functional $ J_{\lambda,\gamma}(V) $ that we wish to minimize. Then we find the unique minimizer $ V_\text{min} $. And finally, having our $ V_\text{min} $, we use it to compute an approximate solution $ c_\text{approx}(x) $ for our coefficient inverse problem.