Spectral Method Analysis and Condition Number Assessment in Solving Water-Rich Allocation Problems Using Partial Differential Equations in Ethiopia

Water-rich allocation problems in Ethiopia are critical for sustainable development. Partial differential equations (PDEs) are used to model these systems, but their solutions can be challenging due to complex geometries and varying boundary conditions. A spectral method based on Chebyshev polynomials was employed to solve a specific set of PDEs representing water distribution systems in Ethiopia. The condition number of the resulting linear system was analysed, ensuring robustness and reliability of the solution process. The analysis revealed that the spectral method is effective in solving complex PDEs for water-rich allocation problems in Ethiopia. A key finding is that the condition number of the linear systems derived from the PDE solutions remained within acceptable bounds, confirming the numerical stability and accuracy of the approach. This study demonstrates the applicability of a spectral method to solve PDEs related to water-resource allocation in Ethiopia with a focus on ensuring robustness through condition-number analysis. The findings suggest that further research should explore the use of adaptive spectral methods for more intricate and variable conditions, potentially leading to even more accurate solutions. water-rich allocation, partial differential equations, spectral method, condition number, stability, Ethiopia Under standard regularity and boundary assumptions, the forecast state is modelled by ∂ₜ u(t,x)=κ\,∂ₓₓu(t,x)+f(t,x), and stability follows from bounded perturbations.