Numerical ODEs: Euler, RK4, Multistep & Applications

TAKEAWAYS:

• ODEs model real-world rates of change (e.g., population growth, heat transfer, bank interest).
• Euler’s method approximates solutions iteratively but is less accurate; RK4 improves precision by averaging slopes.
• Multistep methods (e.g., Adams-Bashforth) use past points for better accuracy but require initial conditions.
• Stability and error depend on step size $h$: smaller $h$ reduces truncation error but increases round-off error.
• Applications in Nepal: Ncell’s network traffic modeling, Daraz’s inventory forecasting, and NTC’s power demand prediction.
• Exam focus: Derive formulas, compute $y(x)$ for given $h$, and compare methods’ accuracy/errors.

1. Ordinary Differential Equations (ODEs): Definition and Types

ODEs describe how a quantity changes over time or space. They are fundamental in physics, engineering, and economics.

Types of ODEs

classDiagram
    class ODE {
        <<abstract>>
        +dy/dx = f(x, y)
    }
    class FirstOrder {
        +dy/dx = f(x, y)
        +Example: Population growth
    }
    class SecondOrder {
        +d²y/dx² = f(x, y, dy/dx)
        +Example: Spring-mass systems
    }
    ODE <|-- FirstOrder
    ODE <|-- SecondOrder

Example: Newton’s Law of Cooling (first-order ODE): $$\frac{dT}{dt}=-k(T-T_{\text{env}})$$ where $T$ = temperature, $t$ = time, $k$ = cooling constant.

2. Euler’s Method: The Simplest Numerical Approach

Euler’s method approximates solutions by stepping forward using tangent lines.

Formula

Given $\frac{dy}{dx}=f(x,y)$ with initial condition $y(x_0)=y_0$, the update rule is: $$y_{n+1}=y_n+h\cdot f(x_n,y_n)$$ where $h$ = step size.

Worked Example: Population Growth

Problem: Solve $\frac{dy}{dx}=0.1y$, $y(0)=100$ for $x=0.5$ using $h=0.1$.

Solution:

1. Start at $(x_0,y_0)=(0,100)$.
2. Compute $f(x_n,y_n)=0.1\cdot y_n$.
3. Iterate:
   • $y_1=100+0.1\cdot (0.1\cdot 100)=110$
   • $y_2=110+0.1\cdot (0.1\cdot 110)=121$
   • Continue until $x=0.5$.

Graph of Euler’s Approximation vs. Exact Solution:

Error Analysis:

• Truncation Error: $O(h)$. Smaller $h$ → better accuracy but more computations.
• Round-off Error: Accumulates with many steps.

3. Runge-Kutta 4th Order (RK4): Higher Accuracy

RK4 improves accuracy by averaging slopes at intermediate points.

Formula

$$\begin{array}{rl}{k}_1& =h\cdot f(x_n,y_n)\\ k_2& =h\cdot f\left(x_n+\frac{h}{2},y_n+\frac{k_1}{2}\right)\\ k_3& =h\cdot f\left(x_n+\frac{h}{2},y_n+\frac{k_2}{2}\right)\\ k_4& =h\cdot f(x_n+h,y_n+k_3)\\ y_{n+1}& =y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4)\end{array}$$

Worked Example: Loan Interest Calculation (Nepal’s Banks)

Problem: Solve $\frac{dy}{dx}=0.05y$, $y(0)=10000$ for $x=1$ (year) using $h=0.25$.

Solution:

1. Compute $k_1,k_2,k_3,k_4$ for each step.
2. After 4 steps ($x=1$):
   • $y_4\approx 10511.4$ (vs. exact $y=10000e^{0.05}\approx 10512.7$).

Comparison Table:

4. Multistep Methods: Using Past Points

Methods like Adams-Bashforth use previous $y$-values for better accuracy.

Adams-Bashforth 2nd Order

$$y_{n+1}=y_n+\frac{h}{2}\left(3f(x_n,y_n)-f(x_{n-1},y_{n-1})\right)$$

Worked Example: Traffic Flow (Kathmandu Roads)

Problem: Model car density $\frac{dy}{dx}=-0.1y+5$, $y(0)=100$, $h=0.1$.

Solution:

1. Use Euler for first step: $y_1=100+0.1(-0.1\cdot 100+5)=94.5$.
2. Apply Adams-Bashforth: $$y_2=94.5+\frac{0.1}{2}\left(3(-0.1\cdot 94.5+5)-(-0.1\cdot 100+5)\right)=90.025$$

Graph of Multistep vs. Euler:

5. Stability and Error Control

• Stability: Small errors should not grow (e.g., Euler is unstable for stiff ODEs).
• Error Types:
  • Truncation Error: Due to method approximation.
  • Round-off Error: Due to finite precision (e.g., floating-point arithmetic).

Rule of Thumb:

• For non-stiff ODEs: RK4 is preferred.
• For stiff ODEs: Use implicit methods (e.g., backward Euler).

6. Real-World Applications in Nepal

Example 1: Ncell’s Network Traffic

• ODE Used: $\frac{dN}{dt}=\lambda -\mu N$ (M/M/1 queue model).
• Purpose: Predict call drops during peak hours.
• Method: RK4 for dynamic traffic simulation.

Example 2: Daraz’s Inventory Forecasting

• ODE Used: $\frac{dI}{dt}=-rI+s$ (inventory decay + supply).
• Purpose: Optimize stock levels to avoid shortages.
• Method: Adams-Bashforth for long-term trends.

Example 3: NTC’s Power Demand

• ODE Used: $\frac{dP}{dt}=kP(1-P/P_{\text{max}})$ (logistic growth).
• Purpose: Plan grid capacity for monsoon seasons.
• Method: Euler for real-time monitoring.

7. Exam Tip: How to Score Full Marks

1. Derive Formulas: Show all steps for Euler, RK4, or multistep methods.
2. Compute Iteratively: For $y(x)$, list all intermediate $y_n$ values.
3. Compare Methods: Discuss accuracy vs. computational cost (e.g., "RK4 is more accurate but slower than Euler").
4. Real-World Link: Relate ODEs to Nepalese examples (e.g., "This ODE models Ncell’s call traffic").
5. Error Analysis: Always mention truncation/round-off error for numerical methods.

Common Pitfalls:

• Forgetting to update $x_n$ in iterations.
• Misapplying $h$ in RK4’s intermediate steps.
• Ignoring initial conditions.

Visual Summary:

flowchart TD
    A["ODE Problem"] --> B["Choose Method"]
    B --> C["Euler: Simple but Low Accuracy"]
    B --> D["RK4: High Accuracy, More Work"]
    B --> E["Multistep: Uses Past Points"]
    C --> F["Truncation Error: O(h)"]
    D --> G["Truncation Error: O(h⁴)"]
    E --> H["Stability Depends on h"]
    F --> I["Small h → Better but Slower"]
    G --> J["Best for Smooth ODEs"]