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This paper considered the Schrödinger equation with different types of nonlocal integral term coefficients. By using the algebraic analysis method and the spectrum theory of differential equations, the existence of solutions was transformed into the existence of roots for some algebraic equations, the shape of the corresponding curve for the solution of the one-dimensional case. Compared with the results on the existence of solutions in the references, improved results were derived with the help of simpler methods. In order to promote the combination of theory and practice, as the auxiliaries, some curves of solutions were set in the progress of proving main results. [ABSTRACT FROM AUTHOR]

This paperanalyzed the relationship between the frequency phase difference of the ultrasonic transducer and the offset value.Considering the fact thatthe classic controller constructed bythe time domain model has not high applicability, this research proposed a frequency tracking system of ultrasonic power supply based on the fuzzy PI control.Also, a time-domain model for molding the current and voltage at both ends of the transducer was presented.The phase difference between the two can be obtained by spectrum analysis methodandthe simulation analysis of ultrasonic power frequency trackingcan bemade.The ultrasonic power frequency tracking system was constructed and analyzed.The initial resonance frequency wasmeasured under normal working voltage.Simulation analysis and experimental data showed the effectiveness of the method.This study found that compared with the traditional control method, the fuzzy PI method can improve the control accuracy of the system, and consequently it can improve the service life and operating efficiency of the ultrasonic transducer. [ABSTRACT FROM AUTHOR]