Inverse Coefficient Problems for Dynamic Equations on Time Scales

In this article, we have worked toward developing the theory for the inverse coefficient problems on an arbitrary time scale. We considered a general $n$th order linear dynamic equation $y^{{\Delta }^n}\left(t\right)+{\sum }_{i=1}^n{p}_i\left(t\right)y^{{\Delta }^{n-i}}\left(t\right)=f\left(t\right)$ on arbitrary time scale $𝕋$. We investigated inverse problems of determining the regressive coefficients $p_i$ and $f$ when one or more solutions are known or some additional information regarding the solution is provided. We proved the existence and uniqueness of results for well-posed problems. We have solved some problems explicitly on isolated time scale domains. We provided several examples to validate our results and some interesting applications in finance, electro-prospecting, and epidemiology.