The Green inverse problem in differential equations is a mathematical problem that involves finding the coefficients of a differential equation from the knowledge of its Green's function. This is a challenging problem as it requires finding a solution to the differential equation that satisfies certain boundary conditions.
The Green's function is a fundamental solution of a differential equation that represents the response of the system to an impulse input. In the Green inverse problem, we use the knowledge of this Green's function to determine the coefficients of the differential equation.
The Green inverse problem has applications in various fields such as physics, engineering, and finance. It is used to determine the properties of physical systems, predict the behavior of dynamic systems, and solve inverse problems in financial mathematics.
The Green inverse problem is a highly nonlinear and ill-posed problem, which means that small errors in the input can lead to large errors in the output. Additionally, the problem is computationally intensive and requires specialized numerical techniques for accurate solutions.
There are various numerical methods that can be used to solve the Green inverse problem, such as the Tikhonov regularization, the Levenberg-Marquardt method, and the Bayesian inference method. These methods involve finding the best fit solution that minimizes the error between the known Green's function and the one obtained from the differential equation.