A New Predictor-corrector Method For One Kind Of Fractional Order Ordinary Differential Equation's Initial Value Problem

The fractional order differential equation means that the derivatives of the equation are arbitrary real numbers,even the complex numbers.The fractional differential operator is different from the integral order differential operator,it has non-locality property,in this case,the fractional differential operator is widely used to describe the material which has characteristics of memory and genetic in the real world.Nowadays it plays an increasingly important role in engineering,physics,finance,hydrology and so on.The research of numerical algorithms for fractional order differential equations has become a popular topic in the relevant field,however,there are still many problems and difficulties that need to be solved.With the efforts of scholars,the numerical calculation of fractional order differential equation has developed a lot these days.The results of these researches can be divided into two classes.One class is developed directly from the definition of fractional order calculus.These numerical algorithms are based on the definition of fractional order calculus,so the computational format is easier.The other class is that learnt from the integral order differential equation and indirectly developed which is also applicable to fractional order calculus.These numerical algorithms can make the numerical results become more accurate,but the computational format is always a little complex.This thesis which starts with the basic definition of fractional order calculus,is on the basis of the explicit method and the trapezoidal method for the fractional order differential equation,that tries to solve the fractional order ordinary differential equation's initial value problem which is in the definition of Caputo fractional order calculus.It means this thesis will use the Adams-Bashforth skill of the explicit method and the Adams-Moulton skill of the trapezoidal method when solving the problem.The specific approach is: using the Adams-Bashforth skill which has a simple format to get a estimate value,and then using the Adams-Moulton skill which has an accurate format to correct the estimate value that before got.In this way,the solution of this thesis gets will be closer to the exact solution of the fractional order ordinary differential equation's initial value problem.In the process of this research,this thesis derives a new predictor-corrector method for one kind of the fractional order ordinary differential equation's initial value problem,and shows the algorithm format of this new predictor-corrector method.In the process of analyzing the local truncation error and the global truncation error of the new predictor-corrector method,this thesis deduces the convergence order of the new method.Therefore,the feasibility of this new method is proved in theory.Furthermore,this thesis also proves that the convergence order of this new method is higher than that of the original explicit method.This progress provides the necessary prerequisites for the simulation of subsequent numerical experiments.On these bases,the results of numerical experiments and the comparisons of the data have shown that this new predictor-corrector method is feasible in solving practical problems.Above all,this thesis proves that this new predictor-corrector method is an effective algorithm for one kind of the fractional order ordinary differential equation's initial value problem.