The parameter estimation problem or the inverse problem of ordinary differential equations is prevalent in many process models in chemistry, molecular biology, control system design and many other engineering applications. It concerns the re-construction of auxillary parameters by fitting the solution from the system of ordinary differential equations( from a known mathematical model) to that of measured data obtained from observing the solution trajectory. Some of the traditional techniques (for example, initial value technques, multiple shooting, etc.) used to solve this class of problem have been discussed. A new algorithm, motivated by algorithms proposed by Childs and Osborne(1996) and Z.F.Li's dissertation(2000), has been proposed. The new algorithm inherited the advantages exhibited in the above-mentioned algorithms and, most importantly, the parameters can be transformed to a form that are convenient and suitable for computation. A statistical analysis has also been developed and applied to examples. The statistical analysis yields indications of the tolerance of the estimates and consistency of the observations used.

In this work, using one parameter Lie groups, we analyze differential equations with partial fluxions, of second and third rank. Find equation symmetry groups of heat conductivity, undulatority, gas flow, waves spreading in shallow water and deducting appropriate formulas.

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