SOLVE_IVP

Overview

The SOLVE_IVP function numerically integrates a system of ordinary differential equations (ODEs) given an initial value. It is a wrapper around the scipy.integrate.solve_ivp method, simplified for use in Excel.

The general form of an initial value problem for a system of ODEs is:

Here, $t$ is a 1-D independent variable (time), $y(t)$ is an N-D vector-valued function (state), and $f(t, y)$ is an N-D vector-valued function that determines the differential equations. The goal is to find $y(t)$ approximately satisfying the differential equations, given an initial value $y(t_0)=y_0$.

This function simplifies the f(t, y) callable by assuming a linear system of the form $dy/dt = A \cdot y$, where $A$ is a matrix provided as fun_coeffs. This means the right-hand side function f does not depend on $t$ explicitly and is linear in $y`. Only the most commonly used parameters are exposed.

For more details on the underlying Scipy function, refer to the official SciPy documentation .

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:

=SOLVE_IVP(fun_coeffs, t_start, t_end, y_zero, t_eval, [method])

• fun_coeffs (2D list, required): A 2D list representing the matrix A in the ODE system dy/dt = A @ y.
• t_start (float, required): The initial time t0 for the integration interval.
• t_end (float, required): The final time tf for the integration interval.
• y_zero (2D list, required): A 2D list (column vector) representing the initial state y(t0).
• t_eval (2D list, required): A 2D list (column vector) of times at which to store the computed solution. Must be sorted and lie within t_start and t_end.
• method (string, optional, default=‘RK45’): The integration method to use. Supported methods are ‘RK45’, ‘RK23’, ‘DOP853’, ‘Radau’, ‘BDF’, ‘LSODA’.

The function returns a 2D list (matrix) where the first column is the time points from t_eval and subsequent columns are the corresponding values of the solution y at those time points. Returns an error message (string) if inputs are invalid.

Examples

Example 1: Simple Exponential Decay

This example solves the simple exponential decay ODE $dy/dt = -0.5y$ with an initial condition $y(0)=2$. The solution is evaluated at specific time points.

Inputs:

Live Notebook

Edit this function in a live notebook .

Live Demo