Impara Nonlinear Equation Solving | Mathematical Modeling and Simulation

Nonlinear equations frequently appear in engineering scenarios where relationships between variables are not simply proportional. For instance, calculating the flow rate of a fluid through a pipe often involves nonlinear relationships due to friction, pipe geometry, or fluid properties. Similarly, chemical reaction rates or heat transfer calculations may require solving equations where the unknown variable appears with exponents, roots, or inside transcendental functions. Unlike linear equations, these cannot be solved with basic algebra and usually require numerical methods.


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import numpy as np
from scipy.optimize import root

# Define the nonlinear equation for flow rate Q in a pipe (example: Darcy-Weisbach equation form)
def flow_equation(Q):
    # Given constants for illustration
    L = 10      # pipe length (m)
    D = 0.1     # pipe diameter (m)
    f = 0.02    # friction factor (dimensionless)
    g = 9.81    # gravity (m/s^2)
    h = 2       # head loss (m)
    # Nonlinear equation: h = (f*L/(2*g*D)) * (Q/(A))^2, where A = area
    A = np.pi * (D/2)**2
    return h - (f*L/(2*g*D)) * (Q/A)**2

# Initial guess for Q (m^3/s)
Q_guess = 0.1

# Solve for Q
sol = root(flow_equation, Q_guess)

print("Estimated flow rate (Q):", sol.x[0], "m^3/s")

The root-finding process involves finding a value for the unknown variable that makes the function output zero. In the flow rate example, you defined a function representing the difference between the actual head loss and the calculated head loss for a given flow rate. The scipy.optimize.root function numerically searches for the flow rate that makes this difference zero, indicating a solution to the engineering problem. This is especially useful when the equation cannot be rearranged to solve for the unknown directly.


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# Check the solution by substituting back into the original equation
Q_solution = sol.x[0]
residual = flow_equation(Q_solution)
print("Residual (should be close to zero):", residual)

1. What is a nonlinear equation in the context of engineering?

2. Which scipy function is used for root finding?

3. Why might a nonlinear equation have multiple solutions?

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Sezione 2. Capitolo 4

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Suggested prompts:

Can you explain how the Darcy-Weisbach equation is used in this example?

What other engineering problems can be solved using similar numerical methods?

Can you walk me through how the root-finding algorithm works in this context?

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