Systems of Ordinary Differential Equations

In Section 9.4.1, selected numerical methods are examined for solving the initial value problems associated with first-order differential equations. Those methods are also applicable to higher-order differential equations following the reduction to a system of first-order equations. For example, the second-order differential equation 

In the particular case of the second-order equation, a procedure to approximate the solution to a system of two first-order equations 

Following Boyce and DiPrima, Euler method would be extended to 

Similarly the fourth-order Runge-Kutta method applied to Equation 9.83 and Equation 9.84 would become 

The predictor-corrector method described in Section 9.4.1, which involves the use of Equation 9.61 and Equation 9.62 would become 

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Smith, I. M., J. L. Siemienivich, and I. Gladweh. A Comparison of Old and New Methods for Large Systems of Ordinary Differential Equations Arising from Parabolic Partial Differential Equations, Num. Anal. Rep. Department of Engineering, no. 13, University of Manchester, England (1975). 

Thus for Hamiltonians of finite dimension the effective action functional can be found by immediately integrating a system of ordinary differential equations. The simplest yet very important case is a bath of two-level systems. 

Hindmarsh A. C. (1976) Preliminary Documentation of GEARIB. Solution of Implicit Systems of Ordinary Differential Equations with Banded Jacobians, Rep. UCID - 30130, Lawrence Livermore Laboratory, Livermore. 

Hie quasi steady state approximation can be conveniently applied to equations 19 to 21, without any significant loss of accuracy, due to tlie high reactivity of tlie reacting species in aqueous solution. Hms, the system of ordinary differential equations is readily reduced to a system of algebraic non linear equations. 

Thus, in sumnary, the solution of the PEE requires the eval iatlon of the first M moments of the PSD. This can be done by integrating the original PEE to give the following system of ordinary differential equations ... 

The accuracy of this method increases nhen increasing M in equation 33, i. e. the dimension of the system of ordinary differential equations 36. Usually, due to the monomiodal sh ie of the PSD considered in this work, M = 3 provides a satisfactory approxinatlon of the solution for the same reason, a low mmher of quadrature points (<5) is required in the evaluation of the integral terms in equations 20, 21 and 36. 

In what follows one possible example demonstrates for a system of ordinary differential equations that there is an implicit scheme which is rather economical than the explicit ones requiring the additional operations. 

Donnely, J.K. and D. Quon, "Identification of Parameters in Systems of Ordinary Differential Equations Using Quasilinearization and Data Pertrubation", Can. J. Chem. Eng, 48, 114 (1970). 

Briefly the idea behind this method is to delineate families of curves in the x-t plane, called characteristic curves, along which the partial differential equations [(123) and (128)] become a system of ordinary differential equations which could then be integrated with greater ease. However, only hyperbolic partial differential equations possess two families of characteristics curves required by the method. 

The book explains how to solve coupled systems of ordinary differential equations of the kind that commonly arise in the quantitative description of the evolution of environmental properties. All of the computations that I shall describe can be performed on a personal computer, and all of the programs can be written in such familiar languages as BASIC, PASCAL, or FORTRAN. My goal is to teach the methods of computational simulation of environmental change, and so I do not favor the use of professionally developed black-box programs. 

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. 

Numerical integration of systems of ordinary differential equations, including... 

The method of lines reduces a partial differential equation to a system of ordinary differential equations which can be solved by readily available software. It is applicable to PDEs that have only the first derivative of one of the variables, for example,... 

DNS of homogeneous turbulence thus involves the solution of a large system of ordinary differential equations (ODEs see (4.3)) that are coupled through the convective and pressure terms (i.e., the terms involving T). 

For this reaction, we can write the following system of ordinary differential equations (ODEs) ... 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

Although in this chapter we have chosen to linearize the mathematical system after reduction to a system of ordinary differential equations, the linearization can be performed prior to or after the reduction of the partial differential equations to ordinary differential equations. The numerical problem is identical in either case. For example, linearization of the nonlinear partial differential equations to linear partial differential equations followed by application of orthogonal collocation results in the same linear ordinary differential equation system as application of orthogonal collocation to the nonlinear partial differential equations followed by linearization of the resulting nonlinear ordinary differential equations. The two processes are shown ... 

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

Now we turn to the problem of constructing conformally invariant ansatzes that reduce systems of partial differential equations invariant under the group C(1,3) to systems of ordinary differential equations. 

Inserting (52) into (46) yields a system of ordinary differential equations for the functions ( ). If we succeed in constructing its general or particular solution, then substituting it into (52) gives an exact solution of the Yang-Mills equations (46). However, the so-constructed solution will have an unpleasant feature of being asymmetric in the variables , while Eqs. (46) are symmetric in these. 

Thanks to Assertion 9, the problem of symmetry reduction of Yang-Mills equations by subalgebras of the algebrap(l, 3) reduces to routine substitution of the corresponding expressions for , > into (61). We give below the final forms of the coefficients (60) of the reduced system of ordinary differential equations (59) for each subalgebras of the algebra p(l, 3) ... 

Clearly, efficiency of the symmetry reduction procedure is subject to our ability to integrate the reduced systems of ordinary differential equations. Since the reduced equations are nonlinear, it is not at all clear that it will be possible to construct their particular or general solutions. That it why we devote the first part of this subsection to describing our technique for integrating the reduced systems of nonlinear ordinary differential equations (further details can be found in Ref. 33). 

In a similar way we have reduced some other systems of ordinary differential equations (59) to systems of two or three equations. Below we list the substitutions for ( ) and corresponding systems of ordinary differential equations. Numbering of the systems below reflects numbering of the corresponding subalgebras Lj of the algebra p 1,3) ... 

So, combining symmetry reduction by the number of independent variables and direct reduction by the number of the components of the function to be found, we have reduced the SU(2) Yang-Mills equations (46) to comparatively simple systems of ordinary differential equations (80). 

In a previous work [33] we suggest an effective approach to study of conditional symmetry of the nonlinear Dirac equation based on its Lie symmetry. We have observed that all the Poincare-invariant ansatzes for the Dirac field i(x) can be represented in the unified form by introducing several arbitrary elements (functions) ( ), ( ),..., ( ). As a result, we get an ansatz for the field /(x) that reduces the nonlinear Dirac equation to system of ordinary differential equations, provided functions ,( ) satisfy some compatible over-determined system of nonlinear partial differential equations. After integrating it, we have obtained a number of new ansatzes that cannot in principle be obtained within the framework of the classical Lie approach. 

The choice of the functions co(x), 0 ( ) is determined by the requirement that substitution of ansatz (53) into the Yang-Mills equations yield a system of ordinary differential equations for the vector function ( ). By direct check, one can prove the validity of the following statement [33,49]. 

Assertion 10. Ansatz (53),(54) reduces the Yang-Mills equations (46) to a system of ordinary differential equations, if and only if the functions m(x), 0 ( ) satisfy the following system of partial differential equations ... 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

Consider, as an example, system of ordinary differential equations (87) with coefficients given by the formula 2 from (93). We look for its solutions of the form... 

Hence, it follows that C( 1, 3) x -invariant ansatzes for the Maxwell fields, which reduce (99) to systems of ordinary differential equations, can be represented in the form (22), namely,... 

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