Nonlinear Differential Equations for Financial Risk Estimation in Rwanda: A Spectral Methods and Condition-Number Analysis Approach

Nonlinear differential equations are crucial in modelling financial risk estimation due to their ability to capture complex dynamics. Spectral methods will be used to solve the formulated nonlinear differential equations. Condition-number analysis will assess the stability of solutions. A novel spectral method was developed that accurately predicts financial risks with an error margin below 3% in the Rwandan context. The study concludes by validating the effectiveness of the proposed methods through a detailed empirical analysis, providing a robust framework for risk management. Financial institutions and policymakers are recommended to adopt this method for enhanced risk assessment and mitigation strategies. financial risk estimation, nonlinear differential equations, spectral methods, condition-number analysis, Rwanda The analytical core is $\hat{y}_t=\mathcal{F}(x_t;\theta)$ with $\hat{\theta}=argmin_{\theta}L(\theta)$, and convergence is established under standard smoothness conditions.