In [[applied mathematics]], '''explicitExplicit and implicit methods''' are approaches used in [[computernumerical simulationanalysis]]s for obtaining numerical solutions of time-dependent [[physicsordinary differential equation|physicalordinary]] and [[Processpartial (science)differential equation|processespartial differential equations]], oras inis otherrequired words, they arein [[numericalcomputer analysissimulation|numericalcomputer methodssimulations]] for solving time-variableof [[ordinaryProcess differential equation(science)|ordinary]]physical and [[partial differential equationprocesses]]s.
<strong>Explicit methods</strong> calculate the state of a system at a later time from the state of the system at the current time, while an <strong>implicit methodmethods</strong> findsfind a itsolution by solving an equation involving both the current state of the system and the later one. Mathematically, if <math>Y(t)</math> is the current system state and <math>Y(t+\Delta t)</math> is the state at the later time (<math>\Delta t</math> is a small time step), then, for an explicit method
: <math>Y(t+\Delta t) = F(Y(t))\,</math>
while for an implicit method one solves an equation
Line 33:
: <math>y_{k+1}=\frac{-1+\sqrt{1+4\Delta t y_k}}{2 \Delta t}. \quad \quad (4)</math>
In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no exact solution exists. Then one uses [[root-finding algorithm|root-finding algorithms]]s, such as [[Newton's method]].
