Ordinary differential equations Euler method

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. When Euler s method is used to integrate in time, the equations become... 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

Ordinary differential equations the Euler method Quite commonly, differential equations appear in the form... 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

Euler s method [15, 28] represented in Figure 7.12 is the simplest way to perform this task. Because of its simplicity it is ideally suited to demonstrate the general principles of the numerical integration of ordinary differential equations. 

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. 

The method of zonation was applied to the energy and material conservation equations. Based on centered finite difTerence approximations, this method can transform three partial differential equations in radial distance and time to ordinary differential equations in time only. Following this, the ordinary differential equations were solved by using Crank-Nicholson algorithm. On the basis of this, the volumetric fluxes of those tar-phase and total volatile phase components were integrated with time by using in roved Euler method to evaluate overall pyrolysis product yields, and afterwards the gas yield can be deduced. 

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... 

In the previous section we solved linear ordinary differential equations analytically, obtaining general solutions in terms of the parameters in the equations. Numerical methods can also be used to obtain solutions, using a computer. In Chapter 1 we looked at the dynamic responses of several processes by using numerical integration methods (Euler integration-see Table 1.2). 

And finally, the new values of the concentrations at the time r + Ar can be calculated using a numerical solution of this latter ordinary differential equation. In this case, Euler s method (Perry, Green, and Malone, 1984) is used due to its simplicity, although errors are proportional to AL Other method of high order, as Runge-Kutta (Perry, Green, and Malone, 1984), can be used if needed ... 

To integrate the ordinary differential equations resulting fi om space discretization we tried the modified Euler method (which is equivalent to a second-order Runge-Kutta scheme), the third and fourth order Runge-Kutta as well as the Adams-Moulton and Milne predictor-corrector schemes [7, 8]. The Milne method was eliminated from the start, since it was impossible to obtain stability (i.e., convergence to the desired solution) for the step values that were tried. 

The mathematical equations used for each numerical int ration method [7, 8] are summarized in this section, where y stands for a nonlinear ordinary differential equation and h for the step size. Modified Euler integration formula ... 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

Euler and Runge-Kutta methods with constant interval of solving an ordinary differential equation - http //twt.mpei.ac.ru/mas/worksheets/Euler.mcd Runge-Kutta method with variable interval of solving an ordinary differential equation - http //twt.mpei.ac.rn/mas/worksheets/rkadapt.mcd... 

CLQA = corrected local quadratic approximation DDRP = dynamically defined reaction path DRP = dynamic reaction path ES = Euler stabilization method GS = Gonzalez and Schlegel method IMK = Ishida-Morokuma-Kormomicld method LQA = local quadratic approximation MB = Miillar-Brown method MEP = minimum energy path ODE = ordinary differential equations SDRP = steepest descent reaction path VRl = valley-ridge inflection. 

One of the earliest techniques developed for the solution of ordinary differential equations is the Euler method. This is simply obtained by recognizing that the left side of Eq. (5.55) is the first forward finite difference of y at position v. 

Program Description Five general MATLAB functions are written for the solution of a set of simultaneous nonlinear ordinary differential equations. They are Euler.m, MEuler.m, RK.m, Adarrts.m, and AdamsMoulton.m. All these functions consist of two main sections. Tiie first part is initialization, in which specific input arguments are checked, and some vectors to be used in the second part are initiated. The next section of the function is solution of the set of nonlinear ordinary differential equations according to the specified method, which is done simultaneously in vector form, Brief descriptions of the method of. solution of these five functions are given below ... 

EULER Solves a set of ordinary differential equations by % the Euler method. 

X,Y]=MEULER( F, XI,XP,H,YI) solves a Set of ordinary differential equations by the modified Euler (the Euler predictor-corrector) method, from XI to XF. 

Solution of a set of nonlinear ordinary differential equations by the explicit Euler method. 

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