The proposed project aims to investigate and compare numerical methods for solving differential equations and their applications in engineering and scientific fields. The results of this project are expected to contribute to the development of improved numerical methods for solving differential equations, which are essential for modeling and predicting physical phenomena in various fields.

1. Title: Numerical Analysis of Differential Equations
Background: Differential equations are fundamental mathematical models used to describe various
physical phenomena, such as heat transfer, fluid mechanics, and electromagnetic fields. Analytical
solutions of differential equations are often difficult or impossible to obtain, making numerical
methods essential for their solution. Numerical methods allow for the approximation of the solution of
differential equations through iterative computations, making them a powerful tool in engineering and
scientific applications.

2. Objective: The main objective of this project is to investigate and compare numerical methods for solving
differential equations and their applications in different engineering and scientific fields.
Methodology: The proposed methodology includes the following steps:
Literature review: Conduct a comprehensive literature review to identify the current state of the art of
numerical methods for solving differential equations, their accuracy, convergence, and stability properties.
Implementation of numerical methods: Implement numerical methods for solving ordinary and partial
differential equations, including finite difference, finite element, and spectral methods.
Verification and validation: Verify the accuracy of the implemented numerical methods using benchmark
problems with known solutions and validate their performance using real-world engineering and scientific
applications.
Sensitivity analysis: Conduct sensitivity analysis of the numerical methods with respect to parameters such as
time step size, spatial discretization, and boundary conditions.
Optimization: Optimize the numerical methods for improved accuracy, convergence, and computational
efficiency.

3. Expected outcomes: The project is expected to provide the following outcomes:
Comparison of numerical methods for solving differential equations based on their accuracy, convergence,
stability, and computational efficiency.
Implementation and optimization of numerical methods for solving ordinary and partial differential equations.
Verification and validation of the implemented numerical methods using benchmark problems and real-world
engineering and scientific applications.
Sensitivity analysis of the numerical methods with respect to parameters such as time step size, spatial
discretization, and boundary conditions.
Recommendations for the most suitable numerical methods for different types of differential equations and
their applications.
Conclusion: The proposed project aims to investigate and compare numerical methods for solving differential
equations and their applications in engineering and scientific fields. The results of this project are expected to
contribute to the development of improved numerical methods for solving differential equations, which are
essential for modeling and predicting physical phenomena in various fields.