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If a system initially rests at its equilibrium position, from where it is acted upon by a unit-impulse at the instance t=0, i.e., p(t) in the equation above is a Dirac delta function δ(t), () = | = =, then by solving the differential equation one can get a fundamental solution (known as a unit-impulse response function)
The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution. The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function.
The early identification of self-similar solutions of the second kind can be found in problems of imploding shock waves (Guderley–Landau–Stanyukovich problem), analyzed by G. Guderley (1942) and Lev Landau and K. P. Stanyukovich (1944), [3] and propagation of shock waves by a short impulse, analysed by Carl Friedrich von Weizsäcker [4] and ...
the solution of the initial-value problem = is the convolution (). Through the superposition principle , given a linear ordinary differential equation (ODE), L y = f {\displaystyle Ly=f} , one can first solve L G = δ s {\displaystyle LG=\delta _{s}} , for each s , and realizing that, since the source is a sum of delta functions , the solution ...
Intuitively, one can think of the inhomogeneous problem as a set of homogeneous problems each starting afresh at a different time slice t = t 0. By linearity, one can add up (integrate) the resulting solutions through time t 0 and obtain the solution for the inhomogeneous problem. This is the essence of Duhamel's principle.
In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of an infinitely long plate from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problems that have an exact solution for the Navier-Stokes equations.
The impulse response of a linear transformation is the image of Dirac's delta function under the transformation, analogous to the fundamental solution of a partial differential operator. It is usually easier to analyze systems using transfer functions as opposed to impulse responses.
Infinite impulse response (IIR) is a ... Desired solutions can be transferred to the case of discrete-time filters whose ... Step invariant solves the problem of the ...