Research On Solutions Of Fractional Order Differential Equations

In recent years,fractional calculus theory has been widely used in physics,chemistry,biology,finance and other fields,especially in signal and image processing,robot control,polymer material chain breaking and other aspects.Fractional differential equations describe phenomena and describe dynamic processes more accurately and carefully than integer differential equations.In addition,because of the characteristics of fractional order calculus operators,such as singularity,nonlocality,etc.,fractional order differential equations are more suitable than integer order differential equations to describe the dynamic process of materials with memory and heredity,and can more accurately describe materials,medical and engineering phenomena.Because the fractional order model has a wide range of applications,this paper will study Caputo type fractional order stochastic neural network model.Random neural networks are widely used in fields such as physics,chemistry,and biology,so studying random neural networks has high practical application value.In practical applications,due to factors such as machinery and equipment,time delays often occur,leading to instability in the entire system.Therefore,adding delays to the research system can better simulate such phenomena.In addition,due to factors such as the conduction speed and delay of signals in neural networks,so discrete delay is not suitable for modeling.Therefore,this paper will study Caputo type fractional order stochastic network models with discrete and distributed delays.In this paper,the deterministic fractional differential equation is reasonably extended,and the Caputo type fractional order stochastic neural network model with discrete and distributed delays is established.Using Laplace transform,the equivalent form of the solution of Caputo fractional order stochastic differential equation is obtained.By using the Mittag-Leffler function to describe the solution of the equation and estimating the Mittag-Leffler function,the existence,uniqueness and asymptotic stability of solutions of Caputo fractional stochastic differential equation with discrete and distributed delays in infinite intervals are obtained.This article uses computers to simulate the solutions of special fractional order differential equations and verify the stability of the solutions.In this paper,we prove the polynomial stability of solutions of Caputo fractional stochastic differential equation with discrete and distributed delays in infinite intervals.In a stronger norm sense,this paper proves the existence and uniqueness of solutions of Caputo type fractional stochastic differential equation with discrete and distributed delays in a finite time interval.