Canonical Euler Splitting Methods For Two Classes Of Nonlinear Composite Stiff Differential Algebraic Equations

Functional differential-algebraic equations are mainly used in scientific problems in the fields of mechanics,control science,and biology.Differential algebraic equations are different from differential equations and algebraic equations.The behavior of the equation solution has changed because they contain differential processes and algebraic constraints.So the numerical methods of differential equations cannot be directly applied to differential algebraic equations.However,functional differential-algebraic equations also contain delayed effects,which brings difficulties of numerically solving this kind of equations.Therefore,numerically solving of functional differential algebraic equations have important theoretical significance and application value.In this paper,we mainly use the canonical Euler splitting method to solve1-index nonlinear composite stiff functional differential algebraic equations and2-index variable delay differential algebraic equations: for the 1-index nonlinear composite stiff functional differential-algebraic equation,the right-hand side function of the equation is firstly decomposed into two sub-problems with weak and strong stiffness,which satisfy the classical and one-sided Lipschitz conditions respectively,and then use the piecewise constant or piecewise linear interpolation operator proves the stability of the method;secondly,the linear interpolation operator is used to prove the consistent and convergence of the method;finally,the validity of the method is further verified by the constructed numerical examples.In particular,the nonlinear composite stiff functional differential-algebraic equations mentioned here can be more special delay differential-algebraic equations and integro-differential-algebraic equations,etc..It can be seen that the proposed canonical Euler splitting method has a very broad application prospect.For the 2-index variable-delay differential-algebraic equation,similarly,we firstly decompose the right-side function of the equation into two subproblems with weak and strong stiffness;then,the consistent and convergence of the canonical Euler splitting method are proved by the index reduction technique,Lipschitz condition and the linear interpolation operator;finally,the theoretical results of the method are confirmed by numerical examples.