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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]
Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low ...
The coefficient α in the equation takes into account the thermal conductivity, ... Depicted is a numerical solution of the non-homogeneous heat equation. The ...
The undetermined coefficients method is a method of appropriately selecting a solution type according to the form of the non-homogeneous term and determining the undetermined constant, so that it satisfies the non-homogeneous equation. [4] On the other hand, the ERF method obtains a special solution based on differential operator. [2]
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas.The reactant entities are given on the left-hand side and the product entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow that points towards the products to show the direction of the reaction. [1]
where ϕ(r, t) is the density of the diffusing material at location r and time t and D(ϕ, r) is the collective diffusion coefficient for density ϕ at location r; and ∇ represents the vector differential operator del. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.
The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other. Typically, a compound's diffusion coefficient is ~10,000× as great in air as in water. Carbon dioxide in air has a diffusion coefficient of 16 mm 2 /s, and in water its diffusion coefficient is 0.0016 mm 2 /s. [1] [2]
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 ...