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Numerical Methods Worked Sums

This post walks through key Numerical Methods with fully worked solutions, including:

• Newton–Raphson root finding
• Euler’s method for ODEs
• Trapezoidal and Simpson’s integration
• Finite difference derivatives
• Heat equation discretization

Each method is explained step-by-step with clear mathematical reasoning and computed results.

Below are standard worked examples covering root finding, integration, differential equations, and numerical approximation techniques.

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1. Newton–Raphson Method (Root Finding)

Solve:

\[ x^3 - x - 2 = 0 \]

Iteration formula:

\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Result:

\[ x \approx 1.521 \]

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2. Euler’s Method (ODE Solver)

\[ \frac{dy}{dx} = x + y,\quad y(0)=1 \]

Approximation:

\[ y(0.2) \approx 1.22 \]

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3. Trapezoidal Rule

\[ \int_0^1 x^2 dx \approx 0.34375 \]

Close to exact value \( \frac{1}{3} = 0.3333 \)

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4. Simpson’s Rule

\[ \int_0^1 x^2 dx = 0.3333 \]

Matches exact solution exactly.

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5. Finite Difference Approximation

\[ f'(1) \approx 2.1 \quad (\text{Exact } = 2) \]

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6. Heat Equation (FDM)

\[ u_j^{n+1} = u_j^n + \lambda (u_{j+1}^n - 2u_j^n + u_{j-1}^n) \]

Models diffusion processes such as heat transfer and financial volatility.