Initial value problems Euler method

In this chapter we will consider only ordinary differential equations, that is, equations involving only derivatives of a single independent variable. As well, we will discuss only initial-value problems — differential equations in which information about the system is known at f = 0. Two approaches are common Euler s method and the Runge-Kutta (RK) methods. 

Lin et al. outline the procedure, which is first to determine x and t as functions of the arc length of the homotopy trajectory. Then Eq. (L.19) is differentiated with respect to the arc length to yield an initial value problem in ordinary differential equations. Starting at Xo and o, the initial value problem is transformed by using Euler s method to a set of linear algebraic equations that yield the next step in the trajectory. The trajectory may reach some or all of the solutions of F(x) = 0 hence several starting points may have to be selected to create paths to all the solutions, and many undesired solutions (from a physical viewpoint) will be obtained. A number of practical matters to make the technique work can be found in the review by Seydel and Hlavacek.j... 

The set of ODEs represented by Equations 8.30 can be solved by various means. They are first-order, initial-value problems of the type introduced in Chapter 2 for multiple reactions. We use Euler s method. Appling it to Equations 8.31 and 8.33 gives... 

Equations that arise in modeling the dynamics of homogeneous systems are initial value problems, generally approached with techniques of the Euler type. Initial value problems involve derivatives with respect to time these must be discretized, which can be done using the forward Euler method... 

In order to correctly model the different possible states of the system, it will be necessary to cover a large part of the accessible phase space, so either trajectories must be very long or we must use many initial conditions. There are many ways to solve initial value problems such as (2.1) combined with an initial condition z(0) = 5. The methods introduced here all rely on the idea of a discretization with a finite stepsize h, and an iterative procedure that computes, starting from zo =, a sequence zi,Z2,..., where z z(nh). The simplest scheme is certainly Euler s method which advances the solution from timestep to timestep by the formula ... 

This brief discussion of Euler s method raises two important issues in all methods for solving the initial value problem in order to improve the approximate solution to the initial value problem, we need to pay close attention to how the solution curve is modeled (straight line vs. polynomials, etc.) and the size of the step to use h). How well these issues are resolved by a particular method is measured by a comparison with the Taylor series for the expected solution F t). 

The solution of a set of coupled differential equations (DEs) describing a reaction is an initial value problem, which means that all c,s are known at some starting value Cj t=0). The basic principle of numerical solutions of DEs can be illustrated using Euler s stepping method on the three state... 

To be clear, let us first define the correct solution to the initial-value problem asj (x). The approximative solution using a single-step numerical method, e.g. the Euler method, with an increment function /(x , y ) and step size h is hf Xn,yn)r. and it... 

We now return to the consideration of the truncation and roundoff errors of the Euler method and develop a difference equation, which describes the propagation of the error in the numerical solution. We work with the nonlinear form of the initial-value problem... 

The Runge-Kutta method is widely used as a numerical method to solve differential equations. This method is more accurate than the improved Euler s method. This method computes the solution of the initial value problem. 

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