The solution of linear ODEs with measured right-hand side by the means of the Laplace transform and the inverse problems theory

The initial value problem for arbitrary order linear ordinary differential equations with constant coefficients is considered. The right-hand side of the equation and the initial conditions are affected by noise. Moreover, the values of the right-hand side are given only on a sparse set of discrete points. This type of problem arises in applications when the right-hand side or the initial conditions are results of some measurement. Although there are many papers devoted to linear ordinary differential equations, the described problem was not addressed in previous research. We suggested a solution method based on the Laplace transform and the inverse problems theory. The influence of noise level on the performance of the proposed method was studied and numerical experiments were realized to show the computational efficiency of the method. The obtained results were compared with the performance of a commonly used numerical method for ordinary differential equations.