Sensitivity Analysis of Models with Higher-Order Differential Equations

Abstract

All models considered in previous chapters are based on differential equations of first order. There exists a wide variety of models however, that are based on higher-order differential equations, such as the Poisson equation or wave equation. While application of the linearization approach to forward problems with these equations poses no substantial problems application of the adjoint approach, which needs formulation of corresponding adjoint problems, becomes more and more sophisticated with increasing the order of equations. In a nutshell, one has to apply the Lagrange identity rule as many times, as the order of the equations dictates, and this procedure becomes increasingly complicated [see, e.g., (Marchuk 1995)]. In this chapter we present an alternative approach based on the standard techniques using the reduction of the higher-order differential equation to a system of differential equations of first order. Further on, this system is represented in the form of a matrix differential equation of first order complemented by corresponding matrix initial-value conditions (IVCs) and/or boundary conditions (BCs). We present the general principles of application of this matrix approach and the results of its application to a set of problems based on selected stationary and non-stationary equations of mathematical physics.