In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of...In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.展开更多
Soft matters are observed anomalous viscosity behaviors often characterized by a power law frequency dependent attenuation in acoustic wave propagation. Recent decades have witnessed a fast growing research on develop...Soft matters are observed anomalous viscosity behaviors often characterized by a power law frequency dependent attenuation in acoustic wave propagation. Recent decades have witnessed a fast growing research on developing various models for such anomalous viscosity behaviors among which one of the present authors proposed the modified Szabo＇s wave equation via the positive fractional derivative. The purpose of this study is to apply the modified Szabo＇s wave equation to simulate a recent ultrasonic imaging technique called the clinical amplitude- velocity reconstruction imaging （CARI） of breast tumors which are of typical soft tissue matters. Investigations have been made on the effects of the size and position of tumors on the quality of ultrasonic medical imaging. It is observed from numerical results that the sound pressure along the reflecting line, which indicates the detection results, varies obviously with sizes and lateral positions of tumors, but remains almost the same for different axial positions.展开更多
基金Foundation item:The NSF(11071097,11101217)of Chinathe NSF(BK20141476)of Jiangsu Province of China
文摘In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.
基金supported by National Basic Research Program of China (973 Project No. 2010CB832702)the National Science Funds for Distinguished Young Scholars (11125208)+2 种基金the R&D Special Fund for Public Welfare Industry (Hydrodynamics, Project No. 201101014)Wen Chen is grateful of the Alexander von Humboldt Foundation, Germany, for an Experienced Researcher fellowshipXiaodi Zhang would like to thank China Scholarship Council (CSC) for the financial support
文摘Soft matters are observed anomalous viscosity behaviors often characterized by a power law frequency dependent attenuation in acoustic wave propagation. Recent decades have witnessed a fast growing research on developing various models for such anomalous viscosity behaviors among which one of the present authors proposed the modified Szabo＇s wave equation via the positive fractional derivative. The purpose of this study is to apply the modified Szabo＇s wave equation to simulate a recent ultrasonic imaging technique called the clinical amplitude- velocity reconstruction imaging （CARI） of breast tumors which are of typical soft tissue matters. Investigations have been made on the effects of the size and position of tumors on the quality of ultrasonic medical imaging. It is observed from numerical results that the sound pressure along the reflecting line, which indicates the detection results, varies obviously with sizes and lateral positions of tumors, but remains almost the same for different axial positions.
