Numerical Analysis Questions Medium
The shooting method is a numerical technique used to solve boundary value problems (BVPs). BVPs involve finding a solution to a differential equation subject to specified boundary conditions. The shooting method is particularly useful when the BVP cannot be solved analytically or when other numerical methods, such as finite difference or finite element methods, are not applicable.
The shooting method involves transforming the BVP into an initial value problem (IVP) by introducing an additional parameter, often called the shooting parameter. This parameter is used to adjust the initial conditions of the IVP until the desired boundary conditions of the BVP are satisfied.
To apply the shooting method, the BVP is first converted into a system of first-order ordinary differential equations (ODEs) by introducing new variables. The initial conditions for the IVP are then set based on a guessed value for the shooting parameter. The resulting ODE system is then solved numerically using a suitable ODE solver.
After solving the IVP, the obtained solution is evaluated at the boundary points. If the boundary conditions are not satisfied, the shooting parameter is adjusted and the process is repeated until the desired accuracy is achieved. This adjustment of the shooting parameter is typically done using root-finding algorithms, such as the bisection method or Newton's method.
The shooting method is an iterative process that converges to the solution of the BVP by refining the guessed value of the shooting parameter. It is a versatile technique that can be applied to a wide range of BVPs, including linear and nonlinear problems. However, it may require some trial and error to find an appropriate initial guess for the shooting parameter and can be computationally expensive for complex problems.