Question

Find the one-dimensional diffusion model for a mass particulate through a given medium, governed by the partial differential equation: ∂²r(x,t) / ∂x² = 1/4 ∂f(x,t) /∂t Your task is to derive the appropriate solution for this diffusion equation.

Answer 1

Main Answer:

The one-dimensional diffusion model for a mass particulate through a given medium, governed by the partial differential equation
`∂²r(x,t) / ∂x² = 1/4 ∂f(x,t) /∂t, is given by r(x,t) = ∫[G(x - ξ, t)f(ξ, t)dξ + φ(t)], where G(x, t)` is the Green's function and φ(t) is an arbitrary function of time.

Step-by-step explanation:

The solution to the one-dimensional diffusion equation involves the use of Green's function, which acts as a fundamental solution to the homogeneous equation. In the derived solution, the integral over ξ represents the convolution of the Green's function with the source term f(x,t). This convolution accounts for the spreading of mass particulates through the medium over time.

The arbitrary function φ(t) accounts for the initial conditions of the system, allowing flexibility in describing the concentration profile at the initial moment. It encapsulates the information about the system's state at the starting point of the diffusion process.

In summary, the derived solution combines the influence of the initial conditions with the effects of the source term, providing a comprehensive representation of the mass particulate diffusion through the specified medium.