Solving Equations Research Articles

Combined discrete Fourier transform method for azimuthal acoustic mode analysis

Modal analysis is an effective and prevalent method for analysing and diagnosing oscillatory systems. In this study, a combined discrete Fourier transform (cDFT) method was developed in addition to the discrete Fourier transform (DFT) method and solving equation (SE) method for azimuthal acoustic mode analysis to utilize the advantages of both methods. cDFT transforms the original sampling matrix equation into a sparse matrix equation by using the aliasing property of the DFT and obtains the modal spectrum results by solving the sparse matrix equation. In this study, the relative accuracy and computational complexity of modal analysis are further demonstrated. The cDFT method improves the calculation speed by grouping the DFT and solving the sparse matrix while simultaneously influencing the relative accuracy of the analysis results by changing the condition number of the sampling matrix. Applications of the cDFT method in terms of robustness to sensor failure and constrained sampling scheme optimisation are also presented. The cDFT method broadens the sampling limitations of DFT and enhances the sampling flexibility and robustness to sensor failure. Further, it can be used to determine the optimal sampling and analysis scheme according to the actual constraints, accuracy, and speed requirements, as well as the expectations of the future state of the measurement system.

APPLICATION OF DIFFERENTIAL EQUATIONS IN SOLVING ECONOMIC PROBLEMS

APPLICATION OF DIFFERENTIAL EQUATIONS IN SOLVING ECONOMIC PROBLEMS

OBSTACLES TO GRAPHICALLY SOLVING EQUATIONS AND INEQUALITIES

ABSTRACT Graphical problem-solving techniques have recently become a more important part of undergraduate mathematics education. In particular, graphically solving equations and inequalities using technology is a growing practice today. Many instructors take it for granted that students can easily acquire these skills. But our experience suggests this may not be the case. Students face many obstacles, including a narrow understanding of functions and equations. This paper addresses difficulties students have in acquiring graphical problem-solving skills, and presents some techniques and concepts intended to help students overcome them.