Question

Which of the following numerical method is a multistep method to solve IVP or BVP's (IVP: Initial Value Problem, BVP: Boundary Value Problem. Multistep methods are typically for IVPs of ODEs.)

• A. Successive approximation method
• B. Adams - Bashforth method
• C. Runge - Kutta method
• D. Euler's method

Hint:

• Single-step methods (for \(y'=f(t,y)\)): \(y_{n+1}\) depends only on \(y_n\) and values from the interval \([t_n, t_{n+1}]\). Examples: Euler, Modified Euler, Runge-Kutta.
• Multistep methods: \(y_{n+1}\) depends on \(y_n, y_{n-1}, \dots, y_{n-k+1}\) and possibly derivatives at these points. Examples: Adams-Bashforth (explicit), Adams-Moulton (implicit).

Answer 1

The Correct Option is B

Numerical methods for solving ordinary differential equations (ODEs) can be classified as single-step or multistep methods.

• Single-step methods: Use information only from the previous point (\(y_n\), \(t_n\)) to compute the next point (\(y_{n+1}\), \(t_{n+1}\)). Examples include Euler's method and Runge-Kutta methods.
• Multistep methods: Use information from several previous points (e.g., \(y_n, y_{n-1}, y_{n-2}, \dots\)) to compute the next point \(y_{n+1}\). Examples include Adams-Bashforth methods (explicit) and Adams-Moulton methods (implicit).

Let's analyze the options:
(a) Successive approximation method (e.g., Picard's iteration): This is an iterative method for solving ODEs or integral equations, generally considered single-step in its basic form of progression. 
(b) Adams-Bashforth method: This is a family of explicit multistep methods for solving ODEs. For example, the two-step Adams-Bashforth method uses \(y_n\) and \(y_{n-1}\) (and their derivatives) to find \(y_{n+1}\). 
(c) Runge-Kutta method: This is a family of single-step methods. They achieve higher accuracy by evaluating the derivative function \(f(t,y)\) at multiple intermediate points within a single step. 
(d) Euler's method: This is the simplest single-step method: \(y_{n+1} = y_n + h f(t_n, y_n)\). Therefore, the Adams-Bashforth method is a multistep method. (Note: While some methods listed can be adapted for BVPs, multistep methods are primarily developed for IVPs of ODEs. The question wording is a bit broad.) \[ \boxed{\text{Adams - Bashforth method}} \]