• Numerical Solution of Ordinary Differential Equations – Initial-value problems Boundary-value problems: the shooting method, finite difference methods, Collocation method, Galerkin method.
• Numerical Solution of Partial Differential Equations – Initial/boundary value problems for parabolic and hyperbolic PDEs (one space and one time dimension) Explicit finite-difference schemes.
• Implicit finite-difference schemes. Stability Parabolic and hyperbolic PDEs in two space dimensions Boundary value problems for elliptic PDEs.

This is an advanced postgraduate-level course in numerical techniques for ordinary and partial differential equations (ODE, PDE). It is required of all graduate students in Applied Mathematics and Mathematical Physics. Using several examples, this course examines conventional numerical techniques for ODE and PDE problems, as well as the characteristics of these approaches. The course will offer applied mathematicians and applied scientists with the foundational knowledge and expertise needed to work with numerical techniques.

1. Numerical approaches for solving initial-value and boundary-value problems for ordinary differential equations,
2. Numerical solution of non-linear system of equations,
3. Method of weighted residuals and finite difference methods for ODEs,
4. Numerical solutions to initial/boundary value problems for parabolic and hyperbolic linear partial differential equations,
5. Method of weighted residuals and finite difference methods for PDEs,
6. Stability of numerical solutions, estimates of errors, and convergence.

Joe D. Hoffman, Steven FrankelMayers, Numerical Methods for Engineers and Scientists, CRC , 2001

G. D. Smith, Numerical Solutions to Partial Differential Equations, Oxford University Press , 2016

Kendall E. Atkinson, Weimin Han, David Stewart, "Numerical Solution of Ordinary Differential Equations", Wiley , 2009

S. Chakraverty, Nisha Rani Mahato, Perumandla Karunakar and Tharasi Dilleswar Rao, Advanced Numerical and Semi Analytical Methods for Differential Equations, Wiley , 2019