Sensitivity Analysis of Models with Higher-Order Differential Equations

Part of the SpringerBriefs in Earth Sciences book series (BRIEFSEARTH)


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.


Higher-order differential equations Boundary conditions Initial-value conditions Final-value conditions Indefinite conditions 


  1. The material presented in this chapter was partly developed by the author during preparation of the course on sensitivity analysis presented at the Jet Propulsion Laboratory some time ago. The rest of the material is new and is published in this book for the first time. The technique of reducing the higher-order differential equations to systems of first-order differential equations used in this chapter is well-known in the literature and can be found in any suitable book on differential equations. The conventional approach to formulation of adjoint problems with higher-order differential equations is presented in detail in the monograph written by Gury Marchuk (1995) Google Scholar
  2. Marchuk GI (1995) Adjoint equations and analysis of complex systems. Kluver Academic Publishers, Dordrecht, Boston, LondonGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaUSA

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