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In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order.
First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.
Exponential growth. Malthusian catastrophe; Exponential response formula; Simple harmonic motion. Phasor (physics) RLC circuit; Resonance. Impedance; Reactance
If f is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used. If, more generally, f is a linear combination of functions of the form x n e ax , x n cos( ax ) , and x n sin( ax ) , where n is a nonnegative integer, and a a constant (which need not be the same in each term), then the ...
Exponential response formula; Finite difference ... of two similar systems. The green curve is the response of the system with impulse response () = ...
Exponential response formula; Finite difference ... The function u-v then satisfies homogeneous boundary condition, and can be solved with the above method.
The time constant of an exponential moving average is the amount of time for the smoothed response of a unit step function to reach / % of the original signal. The relationship between this time constant, τ {\displaystyle \tau } , and the smoothing factor, α {\displaystyle \alpha } , is given by the following formula:
Feynman–Kac formula. Black–Scholes equation; Affine term structure modeling [9] Fokker–Planck equation. Dupire equation (local volatility) Hamilton–Jacobi–Bellman equation. Merton's portfolio problem; Optimal stopping; Malthusian growth model; Mean field game theory [10] Optimal rotation age; Sovereign debt accumulation