Euler’s differential equation

Hence, one arrives at Euler s differential equation for the radial part of the stream function ... 

Upon substitution into Euler s differential equation,... 

Euler s differential equations in conjunction with the introduction of La-grangian multipliers constitute the necessary conditions for a minimum, see Courant and Hilbert [56] or Denn [62]. Thereby the integrand Uq of Eq. (6.29) is extended by the product of appropriate parameters known as Lagrangian multipliers and integrands of the side conditions. In the vectorial representation to be given here, this results in Uq + with the vector of Lagrangian multipliers A and respective vector of integrands < from Eqs. (6.30). To obtain Euler s differential equations, the variation of this expression is equated to zero ... 

The minimization with respect to a subset of functions is permissible insofar as the examination of a subset of Euler s differential equations has no limiting implications on an eventually succeeding complete solution of the problem. The imposed side conditions will be given in the form of the right-hand side of Eqs. (6.30b) and are supposed to be linear with respect to the vector x involving a proportionality matrix P ... 

Thus, for the problem at hand, Euler s differential equations, given in general form by Eq. (6.31), reduce to... 

It is a property of this family of differential equations that the sum or difference of two solutions is a solution and that a constant (including the constant i = / ) times a solution is also a solution. This accounts for the acceptability of forms like A (t) = Acoscot, where the constant A is an amplitude factor governing the maximum excursion of the mass away from its equilibrium position. The exponential form comes from Euler s equation... 

This set of ordinaiy 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. If Euler s method is used for integration, the time step is hmited by... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

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... 

The Cauchy problem for a system of differential equations of first order. Stability condition for Euler s scheme. We illustrate those ideas with concern of the Cauchy problem for the system of differential equations of first order... 

Most branches of theoretical science can be expounded at various levels of abstraction. The most elegant and formal approach to thermodynamics, that of Caratheodory [1], depends on a familiarity with a special type of differential equation (Pfaff equation) with which the usual student of chemistry is unacquainted. However, an introductory presentation of thermodynamics follows best along historical lines of development, for which only the elementary principles of calculus are necessary. We follow this approach here. Nevertheless, we also discuss exact differentials and Euler s theorem, because many concepts and derivations can be presented in a more satisfying and precise manner with their use. 

Nonlinear Dynamic Simulation The nonlinear ordinary differential equations are numerically integrated in the Matlab program given in Figure 4.2. A simple Euler integration algorithm is used with a step size of 2 s. The effects of several equipment and operating parameters are explored below. 

Depending on the numerical techniques available for solving optimal control or optimisation problems the model reformulation or development of simplified version of the original model was always the first step. In the Sixties and Seventies simplified models represented by a set of Ordinary Differential Equations (ODEs) were developed. The explicit Euler or Runge-Kutta methods (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981) were used to integrate the model equations and the Pontryagin s Maximum Principle was used to obtain optimal operation policies (Coward, 1967 Robinson, 1969, 1970 etc.). 

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. 

ORDINARY DIFFERENTIAL EQUATIONS, I.G. Petrovski. Covers basic concepts, some differential equations and such aspects of the general theory as Euler lines, Ariel s theorem, Beano s existence theorem, Osgood s uniqueness theorem, more. 45 figures. Problems. Bibliography. Index, xi + 232pp. 5X 8H. 

Numerical solutions have also been obtained by Brooks (1980) using his three-state model (see above). The relevant simultaneous differential equations were solved Euler s method. Brooks has included in his examples the case where o decays exponentially with time. In all the cases investigated. he finds that allowance must be made for bimolecular mutnal termination of radicals and for the re-entry of desorbed radicals into reaction loci he concludes that failure to take account of these possibilities can lead to serious errors. 

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. 

From the set of differential equations, enter formulas in spreadsheet cells using Euler s method, Fill Down and make sure that the results make sense. 

Point-Slope Methods. Euler s method follows directly from the initial condition as a starting point and the differential equation as the slope (Fig. 3). Consider the simple model of a single differential Eq. (13) with one first-order rate process ... 

Euler s methods can be derived from a more general Taylor s algorithm approach to numerical integration. Assuming a first-order differential equation with an initial value such as [dy/dx] = / = function of x, and y = f(x,y) with y(xo) = yo. if the f(x,y) can be differentiated with respect to x and y, then the value of y at X = (xo + h) can be found from the Taylor series expansion about the point x = xq with the help ofEq. (16) ... 

This is Euler s theorem. Furthermore it follows from the theory of partial differential equations that conversely any function/(a , y, z. ..) which satisfies (1.12) is homogeneous of the mth degree in x, y, z. ... ... 

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... 

While the examples given here have dealt only with chemical reaction kinetics, the method illustrates how one can, in general, solve single as well as coupled differential equations. Euler s explicit method is useful as a qualitative tool it is easily implemented, and can provide a reasonably close result when Af is sufficiently small. The latter requirement, however, may make the explicit method impractical on a spreadsheet. For quantitative work, an implicit method is usually required, as it provides a better approximation given the limited number of iterations practical on a spreadsheet. 

When we apply Euler s reciprocity relation to obtain the total differential of entropy by using Equation (176) for a constant interfacial flat boundary area, A, between them, we have... 

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