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Ian Turner
Ian Turner

Handbook Of Differential Equations: Ordinary Di...


Content: How do you reconstruct a curve given its slope at every point? Can you predict the trajectory of a tennis ball? The basic theory of ordinary differential equations (ODEs) as covered in this module is the cornerstone of all applied mathematics. Indeed, modern applied mathematics essentially began when Newton developed the calculus in order to solve (and to state precisely) the differential equations that followed from his laws of motion.




Handbook of Differential Equations: Ordinary Di...



However, this theory is not only of interest to the applied mathematician: ideas from the theory of ODEs prove invaluable in various branches of pure mathematics, such as geometry and topology. The first half of this module will focus on ordinary differential equations - how to understand them and how to solve them. The second half of the module covers topics from multivariable calculus - partial derivatives, div, grad, curl, and some differential geometry and integration needed for subsequent modules on differential equations.


Aims: To introduce simple differential equations and methods for their solution and to provide a solid foundation in the calculus needed to study future modules involving ordinary and partial differential equations.


The optimization of systems which are described by ordinary differential equations (ODEs) is often complicated by the presence of nonconvexities. A deterministic spatial branch and bound global optimization algorithm is presented in this paper for systems with ODEs in the constraints. Upper bounds for the global optimum are produced using the sequential approach for the solution of the dynamic optimization problem. The required convex relaxation of the algebraic functions is carried out using well-known global optimization techniques. A convex relaxation of the time dependent information is obtained using the concept of differential inequalities in order to construct bounds on the space of solutions of parameter dependent ODEs as well as on their second-order sensitivities. This information is then incorporated in the convex lower bounding NLP problem. The global optimization algorithm is illustrated by applying it to four case studies. These include parameter estimation problems and simple optimal control problems. The application of different underestimation schemes and branching strategies is discussed.


In this paper we give the general solutions of a class of first order nonlinear Fuchs ordinary differential equations. This leads us to show by an example that the necessary conditions of Fuchs' theorem are not sufficient. 041b061a72


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