Contractivity And Dissipativity Of Several Numerical Schemes For Fractional ODEs

The dissipativity and contractivity theory of Caputo fractional nonlinear systems are recently established by scholars.They've studied the dissipativity and contractivity of two popular numerical methods,Grünwald-Letnikov formula and L1 method,and have given strict proof of the long time algebraic contractivity and dissipativity decay rates of numerical solutions.In this paper,the author first improves the above numerical results so that to study the contractivity and dissipativity by initial value correction technique of the time-fractional numerical method.Thus,the descending order of numerical methods can be avoided.The numerical examples have confirmed our theoretical results.For the four numerical methods of fractional ordinary differential equations with second order accuracy proposed in the literature: p-fractional linear multistep methods,fractional backward differential formula,fractional trapezoid formula,and fractional Newton-Gregory formula,by numerical simulation of fractional Lorenz model,fractional Fitz-Hugh-Nagumo equation and subdiffusion equation,it is proved that the four methods all are dissipative and contractive.And the preliminary discussion of the theoretical analysis is given,which provide good numerical support for our next strict theoretical proof.