analytical.hpp

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#pragma once
 
#include "types.hpp"
#include <vector>
 
/**
 * @brief Analytical solution for the 1D heat equation.
 *
 * Problem:
 *   dT/dt = D * d^2T/dx^2
 *   IC: T(x, 0) = Tin
 *   BC: T(0, t) = T(L, t) = Tsur
 *
 * Solution (Series expansion):
 *   T(x,t) = Tsur + (Tin - Tsur) * 4/pi * Sum_{k=0,1,...} [ 1/(2k+1) * sin((2k+1)*pi*x/L) *
 * exp(-D*((2k+1)*pi/L)^2 * t) ]
 */
class Analytical
{
  public:
    /**
     * @brief Compute the exact temperature at a given time and position.
     * @param t Time [h]
     * @param x Position [cm]
     * @param phys Physical parameters (L, D, Tin, Tsur)
     * @return Temperature [°C]
     */
    static double solve(double t, double x, const PhysParams& phys);
 
    /**
     * @brief Generate a full temperature profile for a set of grid points.
     * @param t Time [h]
     * @param x_grid Vector of positions
     * @param phys Physical parameters
     * @return Vector of temperatures corresponding to x_grid
     */
    static std::vector<double>
    get_profile(double t, const std::vector<double>& x_grid, const PhysParams& phys);
};