How to Numerically Solve Differential Equations: Methods & Examples

1 Numerical Solution of Differential Equations

In many practical problems we need to determine how a variable changes with another variable t . The exact functional relationship is often unavailable, but we can often formulate a differential equation that the variables satisfy, allowing us to explain the problem by solving the equation.

While some special differential equations have analytical solutions, most are too complex for closed‑form answers. Numerical methods provide approximate solutions in such cases.

2 Initial Value Problem for First‑Order ODEs

Consider the initial value problem for a first‑order ordinary differential equation

y' = f(t, y), y(t_0) = y_0

where f is a given continuous function on an interval and y_0 is the prescribed initial value. The exact solution is denoted by y(t) . A numerical solution consists of approximations y_n ≈ y(t_n) at a set of points t_n = t_0 + n h with step size h (usually uniform).

2.1 Approximate Derivative with Difference Quotient

Using the forward difference (y_{n+1} - y_n)/h to replace y' in the differential equation yields a recurrence relation that can be solved step by step, producing a discrete analogue known as a difference equation.

2.2 Numerical Integration Methods

Express the solution in integral form and apply a numerical integration rule, such as the rectangle or trapezoidal rule, to approximate the integral. This leads to a formula for advancing the solution from one step to the next.

2.3 Taylor Polynomial Approximation

Expand the unknown function in a Taylor series about a point and retain only the first‑order term. Substituting this approximation into the differential equation gives another discrete update formula, and the truncation error can be estimated from higher‑order terms.

All three approaches discretize the differential equation, each yielding a specific computational formula. The Taylor expansion method not only provides a numerical scheme but also facilitates error analysis.

Reference

Python数学实验与建模 / 司守奎, 孙玺菁, 科学出版社