Research On The Solution Of Fractional Riccati Differential Equation Based On Cubic Spline Function

One of the most important features of fractional differential equation is its non-local nature,and it can be very effective in characterizing materials and processes that have memory and genetic properties.So it has a wide range of applications in the fields of physics,engineering,biology,finance and hydrology.The fractional Riccati differential equations has substantial importance in classical,as well as,modern science and engineering applications,like stochastic realization theory,random processes,optimal filtering,controls,robust stabilization,calculus of variations,network synthesis,diffusion problems and areas such as financial mathematics.Although great progress has been made in the study of numerical algorithms for fractional Riccati differential equations in recent years,there are still problems in terms of high computational complexity,slow convergence,and low accuracy.In recent years,foreign scholars have begun to use modern intelligent optimization algorithms to solve the fractional Riccati differential equations.Such new algorithms generally do not require the continuity and convexity of the objective function.It can overcome the limitations of traditional methods,but in China few modern intelligent optimization algorithms are used to solve fractional differential equation.The cuckoo search algorithm belongs to a swarm intelligence global optimization algorithm.It has simple operation,easy programming,fewer parameters,and strong global search ability.Therefore,this paper attempts to use the modern intelligent optimization algorithm-cuckoo search algorithm to solve the fractional Riccati differential equation.This paper mainly studies the initial value problem of fractional Riccati differential equation based on the definition of Caputo.The specific idea is: firstly,the cubic spline function is introduced to represent the solution of the equation;then,the cubic spline-based nonlinear equations model is constructed by using the known conditions and the properties of the cubic spline function;finally,the cuckoo search algorithm is used to solve the nonlinear algebraic equations.In this process,we prove the existence and uniqueness of the solution of the fractional Riccati differential equation,and we derive the convergent order of the cubic spline function approximating the analytical solution of the equation.On this basis,we use cuckoo search algorithm to solve specific numerical examples,which verifies the feasibility and effectiveness of this method in solving the initial value problem of fractional Riccati ordinary differential equations,and it also provides a new idea for future researchers to study other higher order fractional differential equations.