Discovering The Laplace Transform in Undergraduate Differential Equations

This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs.

This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations.

The present book contains the 11 articles published in the Special Issue "Advances in Delay Differential Equations" of the MDPI journal Mathematics. The papers cover a wide range of topics connected to the theory of differential equations with delay. These topics include, among others, the construction of solutions, analytical and numerical methods for; dynamical properties of; and applications of DDE to the mathematical modeling of various phenomena and processes in physics, biology, ecology, and medicine.

Domain decomposition methods are divide and conquer methods for the parallel and computational solution of partial differential equations of elliptic or parabolic type. They include iterative algorithms for solving the discretized equations, techniques for non-matching grid discretizations and techniques for heterogeneous approximations.

The homotopy index theory was developed by Charles Conley for two­ sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi­ cal measure of an isolated invariant set, is defined to be the ho­ motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the "exit ramp" of N . 1 It is shown that the index is independent of the choice of the in­ dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde­ generate critical point p with respect to a gradient flow on a com­ pact manifold. In fact if the Morse index of p is k, then the homo­ topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.