Approximations of impulse response curves based on the generalized moving Gaussian distribution function

We derive approximations (in time) for impulse response curves not based on the Gaussian distribution (like the well-known Edgeworth expansions), but based on other suitable distribution functions. In general, we derive the full expansion for such an approximation based on an approximative function, the derivatives and the cumulants of this approximative function and on the cumulants of the impulse response curve itself, see formula (39). Suitable distribution functions exhibit a skewed profile which offers better opportunities for approximations of typical impulse response curves than approximations based on the symmetrical Gaussian distribution. As an example of such a suitable approximative distribution function we study more in detail the Generalized Moving Gaussian distributionZt=1M0tνexp−αt−β/t2,00,β>0,−∞<ν<∞,with the normalizationM0=2βαν+1/2exp2αβKν+12αβ.A common feature in convection–diffusion problems is that the Laplace transform of an impulse response curve can be described by a single exponential as a(s) exp (xb(s)). We characterize the first four cumulants in terms of the Taylor coefficients of the functions a(s) and b(s). Moreover, it is shown that the temporal cumulants are linear in the spatial variable. This material is applied to several examples and it is shown that the proposed approximative Generalized Moving Gaussian distributions for impulse responses in the field of convection–diffusion type cases perform in general better than the expansions based on the pure Gaussian distribution. Based on this type of approximation other properties based on the original impulse response curve can be found along analytical ways.