Finding the Integral-Equation-Based Linear Renewal Density Equation and Analytical Solutions

In this study, the linear renewal equation is obtained by using the integral equation, the renewal function and the Fourier–Stieltjes transform. It is proven that the linear renewal equation can be obtained by taking the derivative of the integral equation. Analytical methods for the solution of the obtained linear renewal equation are discussed. It is shown that the linear renewal equation is a powerful tool that can model the direct relationship between stochastic processes and density functions. It is shown that the Fourier–Stieltjes transform allows the equation to be simplified in the frequency domain and analytical solutions to be obtained, and the Laplace transform provides an effective analytical solution method, especially for uniform distribution and exponential density functions. The integral equation-based linear renewal density equation derived in this study preserves the temporal and structural symmetries of the system, allowing for the analytical derivation of symmetric forms in the solution space. In the light of the findings, predictions were made about what kind of studies would be done in the future.