Numerical Solution of Nonlinear Multi Proportional Delay Differential Equations

In this paper, we propose a new method to solve multi proportional delay differential equations. The Haar wavelet collocation method (HWCM) is introduced and applied to proportional delay differential equations involving two or more delay terms. The proportional delay differential equations find applications in population dynamic model where the present system may depend on the gestation period and the maturation period simultaneously. In epidemiological model, the incubation and the immunity period coexist. Individually, the effect of each delay occurs at different pace. However, their combined effect may lead to instability which may be difficult to capture with a single delay. Thus, it has become important to study DDEs with multiple delays.

To show reliability and robustness of the method few numerical examples are presented. Further, numerical solutions obtained by HWCM are compared with the existing exact solutions. The comparison is done by calculating the maximum absolute error and the relative error. Further, to verify the accuracy of the results obtained, rate of convergence is calculated to be approximately 2. It is observed that with the increase in the resolution level, the error decreases and hence the accuracy increases. The findings suggest that HWCM can be extended to neutral time dependent DDE with multiple delays.