What is the difference between single and multistep methods

What is the difference between single and multi-step methods for solving first order ordinary differential equations (ODE). What is the difference between implicit and explicit method for solving first order ordinary differential equations. We discussed two methods for solving, Boundary value problem (BVP), namely the \"shooting\" method and the finite difference method. Briefly describe each method.

Solution

Ans 3:

One-step algorithms are characterised by no ``memory\'\'. To compute the solution point Y_j+1, a one-step algorithm uses the solution point Y_j as initial value, and does not use any previously computed solution points. In other words, it treats each new time step computation as an initial value problem.

Ex: Range-Kutta Method

Multistep algorithms, in contrast to one-step algorithms, have ``memory\'\', that is they use k+1 previously computed solution values Y_j, Y_j-1,..Y_j-kto compute the next solution value Y_j+1

Ex: Adams Method

Ans 4:

Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.

Ans 5:

In shooting method the solution starts from one of the boundary value problem and shoot to the other end with an initial value solver until the boundary condition at the other end converges to its correct value using iterative methods.

In Finite difference method the variables, differential equation and the boundary conditions are all discretised and then the boundary condition are embedded into diffrential equation to form the set of linear systems and the solution is obtained from system of linear equations.