Shooting Method for Boundary Value Problem: Example-1 - Computational Physics Formulation & Simulation

Program 1

Example-1

Python 3.11 Ready

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Given the second-order differential equation:

\[ y'' = -y + 2 \frac{\left(y'\right)^2}{y} \]

Boundary Conditions

The equation has the following boundary conditions:

\[ y(-1) = \frac{1}{e} + \frac{1}{e}, \quad y(1) = \frac{1}{e} + \frac{1}{e} \]

Exact Solution

We want to find an approximate numerical solution for $ y(x) $ using the shooting method and compare it with the exact solution:

\[ y(x) = \frac {1}{e^x + e^{-x}} \]

The solution will be evaluated over the interval:

\[ x \in [-1, 1] \]

Theoretical Background & Computational Mechanics

Convert to a System of First-Order Differential Equations

Let $ y = y(x) $ and $ z = y'(x) $. Then the second-order equation:

\[ y'' = -y + \frac{2(y')^2}{y} \]

can be expressed as two first-order equations:

\[ y' = z \]

\[ z' = -y + \frac{2z^2}{y} \]

This system can be represented as a function $ f(u, x) $, where $ u = [y, z] $.

Initial Guess for the Derivative at the Start Point

Start with two initial guesses for $ z(-1) $, denoted $ z_1 $ and $ z_2 $. For example:

\[ z_1 = 0.1, \quad z_2 = 0.25 \]

Iterative Shooting Method

For each guess $ z_i $, solve the system of equations from $ x = -1 $ to $ x = 1 $.

Let $ w_1 $ and $ w_2 $ represent the values of $ y(1) $ obtained with initial values $ z_1 $ and $ z_2 $, respectively.

Convergence Check and Update of $ z $ Values

Check if the error $ |y_1 - w_2| $ is within a tolerance, e.g., 0.001. If so, stop.

Otherwise, update the guess for $ z $ as follows:

\[ z_{\text{new}} = z_2 + (z_2 - z_1) \cdot \frac{y_1 - w_2}{w_2 - w_1} \]

Set $ z_1 = z_2 $ and $ z_2 = z_{\text{new}} $ and repeat the process.

Theoretical Principles & Computational Physics Guide

Example-1: 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 Example-1 within the study of Shooting Method for Boundary Value Problem.

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 Example-1:

$$ \mathcal{F}_{\text{model}}(\mathbf{x}, t; \mathbf{p}) = 0 \implies \mathbf{y}_{n+1} = \mathcal{S}(\mathbf{y}_n, \Delta t, \mathbf{p}) $$

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 Example-1.
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 Example-1 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.

Standard Literature & References:

Mark Newman, "Computational Physics", University of Michigan (2013).
Hans Petter Langtangen, "A Primer on Scientific Programming with Python", Springer (2016).

Ch. 18.1 Example-1