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===First-order exponential integrator method===
{{details|Exponential integrator}}
Exponential integrators describe a large class of integrators that have recently seen a lot of development.<ref name="Exponential integrators">{{harvtxt|Hochbruck|Ostermann|2010|pp=209–286}} This is a modern and extensive review paper for exponential integrators</ref> They date back to at least the 1960s.
In place of ({{EquationNote|1}}), we assume the differential equation is either of the form
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