The classification of delay differential equations using lie symmetry analysis

Abstract

In recent years, delay differential equations (DDEs) have started to play crucial roles in natural phenomena modeling. Their solutions are essential to the determination of the behavior of such models. However, DDEs are generally difficult to be solved, especially those of higher-orders. This thesis overcomes the hurdle by the way of the method of classification, which allows us to study the solution properties of higher-order DDEs easily and accurately. Earlier researchers were unsuccessful in their attempts to classify DDEs to Lie algebra by changing the space variables. This failure was due to the absence of an equivalent transformation related to the change of variables in DDEs. Consequently, these equations were studied via Lie algebraic classification to the specific case of second-order retarded DDEs (RDDEs). The present work develops a new approach to classify the second-order RDDEs as well as neutral DDEs (NDDEs) to solvable Lie algebra without changing the space variables, and obtains one-parameter Lie groups of the corresponding DDEs to arrive at the transformation solutions. These transformation solutions then lead to solutions of the DDEs. The effectiveness of the proposed classification technique is verified by applying it on modeling the ankle joint of Human Postural Balance (HPB). The proposed model is expected to play a significant role in computational neuroscience related to accurate control of human walking. For completeness, the method is extended to classifying nth-order DDEs of retarded and neutral types. The excellent features of the results and the successful implementation of the method suggest that our new classifier may constitute a basis for classifying DDEs as solvable Lie algebras to obtain the solutions of these equations after getting the transformation solutions of DDEs.