Numerical analysis second-order differential equations

In [211] the authors obtained a new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nystrom (RKN) method to integrate second-order differential equations with oscillating solutions. The proposed method has high phase-lag order with small principal local truncation error coefficient. The authors given the stability analysis of the proposed method. Numerical comparisons of this new obtained method to problems with oscillating and/or periodical behavior of the solution show the efficiency of the method. 

Combining Eq. (24) with Eqs. (12) (14), Chen obtained f(a) for given N and c by the following numerical analysis. Firstly, a zero-th approximation to U(a) was calculated by numerical integration of Eq. (24) using a properly chosen zero-th approximation to f(a), secondly the calculated U(a) was used to obtain a first-order approximation to f(a) by solving the differential equation, Eq. (12), with Eqs. (13) and (14), and finally the process was iterated until the mean-square relative difference between the two successive approximations toT(a) became less than a prescribed small value. 

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

Mathematically, Eq. 4 represents a system of linear differential equations of second order and the solution of this system can be obtained by standard procedures for the solution of differential equations. In practical finite element analysis, we are mainly interested in a few effective methods and we will concentrate in the next sections on the presentation of those techniques and in particular on the direct integration ones. In direct integration the system of linear differential equations in Eq. 4 is integrated using a numerical step-by-step procedure the term direct means that no transformation of the equations is carried out prior to the numerical integration. 

The most commonly encountered partial differential equations in chemical engineering are of first and second order. Our discussion in this chapter focuses on these two categories. In the next two sections, we attempt to classify these equations and their boundary eonditions, and in the remainder of the chapter we develop the numerical methods, using finite difference and finite element analysis, for the numerical solution of first- and second-order partial differential equations. 

Among all the known methods of numerical simulation, we use the most versatile method of spectral collocation to analyze classical breathers in our system of ferroelectrics. This method is not only the latest numerical technique with ease of implementation, but also gives rise to a minimum of errors in the analysis. Spectral methods are a class of spatial discretizations for differential equations. In order to prepare the equation for numerical solution we introduce the auxiliary variable Q. = p, = — -. This reduces the second order Eq. (2) to the first order system ... 

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