Numerical Algorithm For Fractional Differential Algebraic Equations

Differential-algebraic equations are mathematical model for describing thecomplex engineering system, which are composed of differential and algebraicequations, these equations are widely existed in electric power system, chemicalprocess and soft multi-body dynamics system. At present many numerical algorithmsfor solving this kind of equations, such as Augmented method, Null space methodand Condensed method, etc. The common feature of these numerical algorithms isthat the system is converted into ordinary differential equation by dealing with thealgebraic equations. Fractional calculus is that means non-integer order derivation orintegral of the variable function. It is the generalization of the classical calculustheory. In engineering, in order to better describe some physical characteristics ofthe system, the fractional constitutive relationships are used to instead of thetraditional constitutive relationships, thus the fractional differential-algebraicequations (FDAEs) are obtained. Up to now, the numerical algorithms for fractionaldifferential-algebraic equations are still very little over the world, so the study onthe subject is an interesting work.The main contents of this thesis are organized as follows:Firstly, the concept of differential-algebraic equations, development and itsnumerical algorithms are introduced; Definition of fractional calculus theory and itsdevelopment history, related properties and its numerical methods are introduced too;Some theoretical knowledge about the sliding mode control, some reaching laws andtheir advantages and disadvantages are discussed detailly in chapters1-3.Secondly, a numerical algorithm for a class of FDAEs, namely fractional term ison the left in the fractional differential equations is presented in chapter4. The ideaof sliding mode control is introduced here, the algebraic equations are regarded asthe sliding surface, while the control item is added to the fractional differentialequations to control the errors of integral. So the FDEAs have been converted intofractional differential equations. The jitter phenomenon of numerical solution maybe caused from the switching function in the equivalent transformation. In order toeliminate the jitter, the parameter of the exponential reaching law is adjustable; Forthe equivalent fractional differential equations, the predictor-corrector method isused for computation and violation values are corrected. A numerical example isused to verify the validity of the algorithm; Meanwhile the numerical iterativeconvergence and the stability of predictor-corrector method are studied.Thirdly, another class of FDAEs, namely fractional term is on the right in thefractional differential equations are studied in chapter5. Sliding mode control method is introduced for the equivalent transformation of FDAEs. Numericalalgorithm is derived according to the principle of precise integration, while theviolation correction method of constraint equations is given. Then, the numericalresults are obtained by different methods to verify the accuracy of the proposedalgorithm.Finally, a rear wheel compliance steering car as an engineering example isapplied. Fractional constitutive relationship is used to describe the characteristics ofrubber component used in compliance steering system. According to the principle ofmutil-rigid-body dynamics, the dynamics equations of rear-wheel compliancesteering car are established and the equations are converted into the standard form.The validity of the proposed numerical algorithm is verified by numerical results.