Two reliable wavelet-based algorithms for solving nonlinear differential equations in water quality assessment models

In this research work, two reliable methods based on wavelets to elucidate water quality valuation models which follow nonlinear differential equations as a basis are presented. The characterizing equation of the steady reservoir in the form of an Advection-Diffusion equation (ADE) or Drift Diffusion equation for variable coefficients is focussed. ADE elucidates a physical phenomenon inside a physical system due to diffusion and convection which leads to the transport of mass, energy or any other physical attributes. To assess a normal channel model's water quality, compound dispersion computation in the channel is required, provided the velocity of the water is known. The main objective of the proposed research work is that the nonlinear differential equations can be converted into a system of algebraic equations using an operational matrix of derivatives convert these complex equations into a set of simple algebraic equations of unknowns equal to the order of the matrices which is then solved at the collocation point to arrive at the approximate analytical solution. The Lucas wavelet and Hermite wavelet methods or transforms (LWM, HWM) have been employed to assess the Chemical Oxy-gen Demand (COD) in a given river.A few numerical experiments are provided illustrating the reliability and applicability of the proposed methods.