## Explicit and implicit methods, stiff equations, and matlab solvers

Explicit and implicit method in integrating differential equations.

Explicit method calculates yn+1 when we already know yn. In other words, yn+1 is explicitly shown as a function of yn: yn+1 = f(yn).

Explicit Euler method:

Implicit method calculates yn+1 from an equation involving both yn+1 and yn. It considers yn+1 as an unknown variable.

Implicit method requires extra computation and harder. It is used to solve stiff equations.

Implicit Euler method: . It’s called implicit because it requires information at xn+1.

Stiff equations

Stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable (for a change of independent variable x will produce a very high value of dependent variable y) unless the step size is taken to be extremely small (very small change of x).

When an explicit method is used to solve stiff equations, it requires impractically small time steps (∆t). Hence, it needs longer computation time.

Matlab solvers

 Solver Implicit/explicit Used for Stiff/Non-Stiff Equations Ode45 Explicit Non-stiff Ode23 Explicit Non-stiff Ode113 Explicit Non-stiff Ode15s Implicit Stiff Ode23s Implicit Stiff Oe23tb implicit Stiff