| Integral-differential equation is an useful modeling tools.The study of the prop-erties of Integral-differential models can improve the understanding of the sys-tem.By analyzing the stability of Integral-differential equations,the prediction and control of the system can be guided.But there are many kinds of Integral-differential equations,The form of difference is large,and relatively lack of ready-made analysis methods or implementation techniques.To a certain extent,it limits its application in the field of system science.Therefore,the stability of integral differential model is analyzed as well as digital and Analog Realization of the model.It will provide reliable guarantee for modeling,control,performance opti-mization and fault diagnosis of complex systems.It has important practical sig-nificance.In this paper,several typical Integro differential equations are described in detail,and some typical solutions of Integro differential equations are giv-en.Among them,the type of weakly singular kernel functions belongs to fractional(non integer order)Integro differential equations.Compared with the integer or-der model,The fractional order model has more abundant adjustable parameters,can describe the model more accurately,and is very important for revealing the general phenomenon of fractional order in nature.Therefore,the study of integral differential equations with weakly singular kernel functions has certain frontier and practical value.Stability is one of the basic requirements of the control system,and it is also the basic condition to ensure the normal operation of the system.However,stability research is a difficult problem in integral diffe.rential equations.Even for linear time invariant systems,there is still some difficulty in the stability analysis,In particular,Lyapunov method is an important method in the study of integral differential equation stability.For fract,ional Integro differential equations,it is difficult to find its Lyapunov function,Especially nonlinear equations.In this paper,Lyapunov analysis method of integer order is given,and two different conditions,one dimension and multi dimension,are analyzed respectively,Then the Lyapunov analysis method in the form of fractional order Caputo is given,which provides a theoretical support for the stability analysis of fractional Integro differential systems.In the fractional order circuit,the integral differential equation is used to model it.Not only the time domain solution of the variable in the circuit can be obtained,It is also important to analyze the stability of the circuit and ana-lyze the time frequency domain by Lyapunov method,which is very important to the study of the fractional order equivalent circuit model.However,since the exploration of Lyapunov functions is a difficult task,even in linear time invariant fractional differential systems,It is very difficult to study the stability of fractional order equivalent circuit models by Lyapunov method.Therefore,it is very impor-tant to find a simple and efficient method for the stability analysis of fractional order circuits.In this paper,the fractional order stability analysis method is intro-duced and applied to the fractional order equivalent circuit model in biomedicine,and a path integration method is proposed to analyze the stability of fractional order circuits.If the inverse Laplace transformation method is used to solve the system solution,the stability problem of the system can not be determined.If Lyapunov method is used to analyze the stability of the system,then the exact solution can not be obtained,And the exploration of Lyapunov functions is also a difficult process.The complex path integral method proposed in this paper can not only obtain the system solution,but also determine the stability of the solu-tion.Therefore,this method can be used in circuit analysis and circuit modeling and identification.A large number of numerical schemes and their analyses verify the accuracy and practicability of the above results. |
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