On the Role of Definition in the Numerical Solutions to Equations

Mohsen Jawad Abood Alsultani

Abstract: This study investigates the critical role of problem definition in obtaining accurate and reliable numerical solutions to mathematical equations. While most studies focus on algorithmic efficiency and error analysis, the impact of precise problem formulation remains underexplored. Through both theoretical analysis and numerical experimentation, we examine how variations in problem definitions-including boundary conditions, domain specifications, and parameter assumptions-affect convergence, stability, and error propagation in numerical methods. Case studies using root-finding algorithms, finite difference methods for the heat equation, and finite element solutions of Poisson's equation demonstrate that even small ambiguities in definition can lead to significant deviations in results. The findings highlight the necessity of rigorous problem formulation as a prerequisite for reliable computational modeling and provide guidelines for best practices in numerical analysis.

Keywords: numerical analysis, problem definition, well-posed problems, stability, convergence, computational mathematics

How to Cite?: Mohsen Jawad Abood Alsultani, "On the Role of Definition in the Numerical Solutions to Equations", Volume 14 Issue 10, October 2025, International Journal of Science and Research (IJSR), Pages: 1492-1496, https://www.ijsr.net/getabstract.php?paperid=SR251023063935, DOI: https://dx.doi.org/10.21275/SR251023063935