Boundary value problems (BVPs) for ordinary differential equations arise naturally in many areas of physics, engineering, and applied mathematics. Classical examples include the vibration of strings, heat conduction in solids, and quantum mechanical bound states. Unlike initial value problems (IVPs), where all conditions are specified at a single point, BVPs impose constraints at different points of the domain, making them significantly more challenging to solve both analytically and numerically.

The shooting method is one of the most intuitive and historically rooted techniques for tackling such problems. Its central idea is simple: transform a boundary value problem into an initial value problem by guessing the missing initial conditions, then iteratively refine this guess until the solution satisfies the boundary conditions at the other end. The method is often illustrated through a ballistic analogy—one “shoots” from the initial point and adjusts the trajectory until the target is hit. Although the shooting method was formalized only in the 20th century, its conceptual foundations can be traced back much earlier. The study of differential equations in the 18th and 19th centuries by mathematicians such as Leonhard Eulerand Joseph-Louis Lagrange already revealed the difficulty of boundary value problems in mechanics and astronomy. At that time, analytical solutions were often unavailable, and practitioners relied on approximation strategies that implicitly resembled “trial-and-error” approaches. A decisive step toward the modern shooting method came with the development of reliable numerical solvers for initial value problems around 1900, notably through the work of Carl Runge and Martin Kutta. Their methods provided the computational backbone needed to integrate differential equations accurately from a given starting point. This made it feasible to implement the idea of repeatedly “shooting” with different initial guesses. The method gained wider recognition and systematic treatment in the mid-20th century, alongside the emergence of numerical analysis as a distinct discipline. Influential mathematicians such as Richard Courant contributed to the theoretical understanding of boundary value problems, while the increasing availability of digital computers transformed the shooting method into a practical and widely used computational tool.

Today, the shooting method remains a cornerstone in the teaching of numerical methods due to its conceptual clarity and direct connection to physical intuition. While more robust techniques—such as finite difference and finite element methods—are often preferred for complex or stiff problems, the shooting method continues to play an important role in applications ranging from classical mechanics to quantum physics, where it is frequently used to determine eigenvalues and admissible solutions.

In this blog, I will give an example of the application of the method to the solution of the Thomas-Fermi and Thomas-Fermi-Dirac equations.