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Simultaneous Linear Equations
Published in Bilal M. Ayyub, Richard H. Mccuen, Numerical Analysis for Engineers, 2015
Bilal M. Ayyub, Richard H. Mccuen
Elimination methods like the Gaussian elimination procedure are often called direct equation-solving methods because the solution is found after a fixed, predictable number of operations. As an alternative, however, we can solve sets of simultaneous equations by using a trial-and-error procedure. In this procedure, we assume a solution—that is, a set of estimates for the unknowns—and successively refine our estimate of the solution through some set of rules. This approach is the basis for iterative methods for solving simultaneous equations. Iterative methods, in general, produce an exact solution for the problem; the precision of the solution depends on our patience in performing the iterations—that is, on the number of trial-and-error or iteration cycles that we perform. Obviously, the number of operations required to obtain a given solution using iterative methods is not fixed; this, too, is a function of the number of iteration cycles that we choose to perform. However, a major advantage of iterative methods is that they can be used to solve nonlinear simultaneous equations, a task that is not possible using direct elimination methods. The use of these methods for solving nonlinear simultaneous equations is discussed in Section 4.10 in Chapter 4.
Soil-Structure-Interaction using Cone Model in Time Domain for Horizontal and Vertical Motions in Layered Half Space
Published in Journal of Earthquake Engineering, 2020
Mehran Khakpour, Masoud Hajialilue Bonab
The right side of Equations (40) and (42) are known in the time interval t for a specified seismic excitation. Numerical integration methods can be employed for solving the motion equations of block and one degree-of-freedom superstructure. Considering the quantity of the system’s available degrees of freedom, various numerical methods including central difference, Runge Kutta, Howbelt, Wilson-theta, and Newmark-theta can be applied. In these procedures, equation solving is performed in time intervals and equation response in a specific time interval is dependent on the response of the previous time intervals. In this study, Newmark-beta method is applied for solving Equations (40) and (42). In this method it is assumed that acceleration changes linearly between two time intervals. Displacement and velocity values are formulated as follows:
How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge
Published in International Journal of Mathematical Education in Science and Technology, 2019
The results of international comparative studies concerning student achievement on these exams have led to disappointment for certain countries. In recent years, studies that have sought the reasons for ongoing failure have altered their focus from students to teachers, or more specifically, to teacher candidates. Some of the studies have focused on specific mathematical concepts in terms of content knowledge [4], pedagogical content knowledge (PCK) [5] or both knowledge bases [6]; while another tendency in recent years is to address a specific content domain [7,8]. For instance, Li [8] investigated secondary-school mathematics teachers’ algebra teaching knowledge in equation solving and identified certain challenges in algebra teaching. The current study aligns with this tendency through an examination of prospective mathematics teachers’ PCK in the numbers content domain.
Traveling waves in an SEIR epidemic model with a general nonlinear incidence rate
Published in Applicable Analysis, 2020
Xin Wu, Baochuan Tian, Rong Yuan
Let and . Obviously, hold. Given , , , and , then . Obviously, Π is a quartic equation of . We use the formula of roots of quartic equation solving and get the roots: , and . (We write the roots in decimals for simplicity). The unique positive real root of is . The minimal wave speed is the asymptotic propagation speed of infectious diseases and the explicit expression of is very complicated. We plot the curves when c takes different values in Figure 1 and draw the surface of about and in Figure 2.