feat: add numerical laplace transform (#14602)

* feat: add numerical laplace transform

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* Update maths/laplace_transformation.py

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

* refactor: add input validation and fix doctest precision

* fix: address PR review comments

Updated module docstring, added validation for non-negative s_value, and replaced arrange with linspace for clarity.

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* Refactor docstring and remove unnecessary blank lines

Removed extra blank lines and cleaned up docstring.

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* Refactor error handling for s_value check

* Fix indentation for s_value validation

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---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>

maths/laplace_transformation.py

@@ -0,0 +1,76 @@
+"""
+Laplace Transform — Numerical Implementation.
+
+Computes the numerical Laplace Transform using the trapezoidal
+integration rule. Supports real-valued, non-negative Laplace
+parameters only.
+
+Reference: https://en.wikipedia.org/wiki/Laplace_transform
+"""
+
+import numpy as np
+
+
+def laplace_transform(
+    function_values: np.ndarray, s_value: float, delta_t: float
+) -> float:
+    """
+    Calculate the numerical Laplace Transform of a function given its values over time.
+
+    This implementation supports only real-valued, non-negative Laplace
+    parameters ``s``.
+
+    Args:
+        function_values: A numpy array of the function values f(t).
+        s_value: The real-valued Laplace parameter ``s``. Must be non-negative.
+        delta_t: The time step between samples.
+
+    Returns:
+        The approximate real-valued value of the Laplace transform at s_value.
+
+    Example: For f(t) = 1, the Laplace transform L{1} = 1/s.
+    If s = 2, L{1} should be 0.5.
+
+    >>> t = np.linspace(0, 50, 10000)
+    >>> f_t = np.ones_like(t) # f(t) = 1
+    >>> res = laplace_transform(f_t, s_value=2.0, delta_t=50/10000)
+    >>> abs(res - 0.5) < 1e-3
+    True
+
+    Example: For f(t) = e^(-t), the Laplace transform L{e^-t} = 1/(s+1).
+    If s = 1, L{e^-t} should be 0.5.
+
+    >>> t = np.linspace(0, 50, 10000)
+    >>> f_t = np.exp(-t)
+    >>> res = laplace_transform(f_t, s_value=1.0, delta_t=50/10000)
+    >>> abs(res - 0.5) < 1e-3
+    True
+    """
+    if delta_t <= 0:
+        raise ValueError("delta_t must be a positive value.")
+    if function_values.size == 0:
+        raise ValueError("function_values array cannot be empty.")
+    if s_value < 0:
+        error_msg = (
+            f"s_value must be non-negative for this implementation, got {s_value}."
+        )
+        raise ValueError(error_msg)
+
+    # Time vector corresponding to the function values
+    time_vector = np.linspace(
+        0, (len(function_values) - 1) * delta_t, len(function_values)
+    )
+
+    # The integrand: f(t) * e^(-s*t)
+    integrand = function_values * np.exp(-s_value * time_vector)
+
+    # Numerical integration using the trapezoidal rule
+    result = np.trapezoid(integrand, dx=delta_t)
+
+    return float(result)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()