A Precise Algorithm To Solve Quaternion Heat Conduction Equation

With the development of modern control theory,quaternion has becamean important tool in the research of quantum mechanics;while precisealgorithm is based on homogeneous linear autonomous system,proposed bya very famous academician named Zhong Wanxie,now, the high precisiondirect(HPD) has become the focus of research. This paper does acombination of both quaternion and precise algorithm to solve the problemof the quaternion equation of heat conduction.The article focuses on3topics:(1) This paper establishes12precise algorithm models for quaternionequation of heat conduction, including6base2-complex model and base4-real mode, there are4models for constant coefficient quaternion equationof heat conduction,4models for with incentive items quaternion heatconduction equation and other4models are for the general variablecoefficient linear quaternion heat conduction equation,the paper gives adetailed derivation of precise algorithm for base4-real model, and gives sixexamples to verify,compared with R-K method, the results from precisealgorithm are more satisfactory.while for base2-complex model,we only dosome theoretical analysis.(2) In the third chapter,this paper mainly researches the detailed erroranalysis of precise algorithms about quaternions heat conductionequation.on one hand,it is from the differential forms of the truncationerror,on the other hand, it is from the exponential matrix during theSummation Index truncation error. At the same time,the computer wordlength limit may also cause rounding errors,and we also do some analysis.(3) this paper deals with the stability analysis for each base4-realmodel, we get the following conclusions: for a constant coefficient of zero boundary conditions quaternion heat conduction equation, precise algorithmasymptotically stable unconditionally; nonzero boundary conditionsconstant coefficients quaternion equation of heat conduction and withincentive items quaternion heat conduction equation fine algorithm is alsounconditionally stable.For general variable coefficient near quaternionequation of heat conduction, whena(x)0,regardless of the quaternionheat conduction equation boundary conditions is zero,the HPD algorithm isstable. Ifa(x)0, to ensure the stability of HPD, it is desirable for the oddnumber of nodes to ensure the stability of fine algorithm.