Chapter 16: Differential Equations

Jets and Differential Invariants

Transformation groups figure prominently in Lie’s theory of symmetry groups of differential equations, which we discuss in Chapter 4. They will act on the basic space coordinatized by the independent and dependent variables relevant to the system of differential equations under consideration. Since we are dealing with differential equations we must be able to handle the derivatives of the depen...

Chapter 17 Dynamics of PlanarMedia

In this chapter, we continue our ascent of the dimensional ladder for linear systems. In Chapter 6, we embarked on our journey with equilibrium configurations of discrete systems — mass–spring chains, circuits, and structures — which are governed by certain linear algebraic systems. In Chapter 9, the dynamical behavior of such discrete systems was modeled by systems of linear ordinary different...

Numerical Solution of Stochastic Differential Equations in Finance

This chapter is an introduction and survey of numerical solution methods for stochastic differential equations. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial systems. We include a review of fundamental concepts, a description of elementary numerical methods and the concepts of convergence and order for stochas...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

A Survey of Time-Differencing Schemes for the Oscillation and Decay Equations

In atmospheric dynamics, the governing equations are usually non-linear partial differential equations. Some knowledge of finite-difference approximations to ordinary differential equations (especially first order) is needed, however. In fact, if we linearize a governing partial differential equation and assume a wave form for the solution, the equation simply reduces to an ordinary differentia...