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Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
[1] [2] The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. [2] The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general ...
The absorption coefficient is fundamentally the product of a quantity of absorbers per unit volume, [cm −3], times an efficiency of absorption (area/absorber, [cm 2]). Several sources [2] [12] [3] replace nσ λ with k λ r, where k λ is the absorption coefficient per unit density and r is the density of the gas.
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.
D is the diffusion constant of the solute unit m 2 ⋅s −1, t is time unit s, c 2, c 1 concentration should use unit mol m −3, so flux unit becomes mol s −1. The flux is decay over the square root of time because a concentration gradient builds up near the membrane over time under ideal conditions.
In the anisotropic case where the coefficient matrix A is not scalar and/or if it depends on x, then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative ...
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation [1] occurring in various areas of applied mathematics, such as fluid mechanics, [2] nonlinear acoustics, [3] gas dynamics, and traffic flow. [4]
These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical properties are the same at every point in space. [5] [6] In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this ...