Mathematics- Numerical Methods-
Root Finding- Bracketing Methods, Open Methods, Root Bracketing and Polishing Methods, Global Optimization Methods, Homotopy Methods, Multidimensional Root Finding.
Interpolation and Approximation- Linear Interpolation, Polynomial Interpolation, Spline Interpolation, Piecewise Interpolation, Hermite Interpolation, Barycentric Interpolation, Inverse Distance Weighting (IDW), Least Squares Approximation, Polynomial Approximation, Chebyshev Approximation, Trigonometric Approximation, Rational Approximation, Piecewise Approximation, Splines Approximation, Wavelet Approximation.
Numerical Integration- Rectangular Rule, Trapezoidal Rule, Simpson’s Rule, Gaussian Quadrature, Composite Methods, Monte Carlo Integration, Adaptive Quadrature.
Numerical Solution of Ordinary Differential Equations (ODEs)- Euler’s Method, Improved Euler Method (Heun’s Method), Runge-Kutta Methods, Multistep Methods (e.g., Adams-Bashforth and Adams-Moulton), Finite Difference Methods, Finite Element Methods, Boundary Value Methods, Spectral Methods, Shooting Methods, Adaptive Step-size Methods.
Numerical Linear Algebra- Matrix Factorization, Linear System Solvers, Eigenvalue and Eigenvector Computations, Least Squares Problems, Sparse Matrix Computations, Iterative Methods, Numerical Stability and Conditioning, Matrix Norms and Condition Numbers.
Numerical Solution of Partial Differential Equations (PDEs)- Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), Boundary Element Method (BEM), Spectral Methods, Meshless Methods, Lattice Boltzmann Method (LBM).
Error Analysis and Stability- Numerical Error Analysis, Statistical Error Analysis, Algorithmic Error Analysis, Modeling Error Analysis, Numerical Stability, Stability in Control Systems, Stability in Differential Equations, Stability in Optimization, Stability in Machine Learning.
Optimization- Mathematical Optimization, Metaheuristic Optimization, Gradient-Based Optimization, Multi-objective Optimization, Combinatorial Optimization, Black-Box Optimization.
Special Topics- Emerging Technologies, Environmental Sustainability, Social Justice and Equity, Health and Wellness, Cultural Studies, Global Affairs and Politics, Ethics and Philosophy, Education and Pedagogy, Business and Economics, Media and Communication, History and Heritage, Science and Innovation, Literature and Writing, Technology and Society, Psychology and Behavior.