Theoretical Principles & Computational Physics Guide
Hermite Polynomial: Computational Physics Modeling & Analysis
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Core Concept: Computational formulation and simulation of Hermite Polynomial within the study of Special Functions.
Algorithmic Translation: Translates theoretical governing equations into clean, discrete Python procedures and verified numerical solvers.
Numerical Verification: Evaluates numerical accuracy, parameter sensitivity, and error bounds for reliable computational results.
2. Mathematical Formulation & Algorithmic Pipeline
Governing computational formulation for Hermite Polynomial:
$$ \mathcal{F}_{\text{model}}(\mathbf{x}, t; \mathbf{p}) = 0 \implies \mathbf{y}_{n+1} = \mathcal{S}(\mathbf{y}_n, \Delta t, \mathbf{p}) $$
1. Define Physical Parameters & Boundaries →
2. Execute Numerical Simulation Steps →
3. Evaluate Observables & Plot Results
3. Numerical Mechanics & Implementation Logic
- Parameter Initialization: Set physical constants, initial states, and coordinate boundary bounds for Hermite Polynomial.
- Step Execution: Advance the computational state using optimized vector operations and scientific libraries.
- Convergence Control: Monitor stability and ensure simulation metrics remain physically meaningful.
4. Relevant Computational Practice Exercises
Exercise 1
Implement Hermite Polynomial and observe how the output responds to variations in primary physical parameters.
Exercise 2
Refine the calculation grid or step size to observe convergence and reduction in numerical approximation error.
Exercise 3
Compare the simulated computational results against an analytical benchmark or limiting physical case.
- Mark Newman, "Computational Physics", University of Michigan (2013).
- Hans Petter Langtangen, "A Primer on Scientific Programming with Python", Springer (2016).