What is Diffusion equation: Definition and 120 Discussions

The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's laws of diffusion). In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of convection–diffusion equation, when bulk velocity is zero.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Modified diffusion equation PDE

    Hi I'd appreciate any help on identifying the type of PDE the following equation is... *This is NOT homework, it is part of research and thus the lack my explanation of what this represents and boundary conditions. I have a numerical simulation of the solution but I'm looking to have a math...
  2. T

    Diffusion equation with an external force

    I thought I would do a little physics for fun, since it has been over 20 years since I last did any. I picked up Sethna's Stat Mech book. He gives a derivation of the diffusion equation that goes as follows: A particle makes random steps, ie x(t+Δt)=x(t) + l(t). The steps l(t) are given...
  3. K

    Scaling Problem in Diffusion Equation

    Hello everyone, I have a question that is bothering me a bit. I would be happy if you could give an idea or tell me a specific point to look at. Lets say that we have an arbitrary function that obeys diffusion equation: f = f(η), here η is the scaling parameter for pde which equals to...
  4. C

    What is the Geometric and Material Buckling in Diffusion Equation?

    Hi, I have read a lot about Diffusion Equation and solving neutron flux problems in different mediums, planes and groups, but I can't grasp this topic. In other words, I don't know why they mention: 1. Infinite/finite medium 2. Homogeneous/non-Homogenous medium 3. One/two or multi-group...
  5. P

    Convection diffusion equation 1D exact solution

    I'm looking for the analytical solution for the 1D convection diffusion equation with a constant heat flux. Boundary conditions: The domain I'm looking at is x from 0 meters to 1 meter. The temperature at x=0 is T=0 degrees Celsius. At x=1, T=100 C. I'm given the equation...
  6. B

    Boundary condition problem for diffusion equation

    BOUNDARY CONDITION PROBLEM I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation: δc/δt=D*((δ^2c)/(δx^2))-kc assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is channel's length...
  7. B

    Boundary condition problem for diffusion equation

    Homework Statement BOUNDARY CONDITION PROBLEM I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation: δc/δt=D*((δ^2c)/(δx^2))-kc assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is...
  8. M

    Biosensor diffusion equation solution

    Hi In this online lecture (click "View Presentation"), Prof. Allam discusses the solution of a diffusion-capture problem. On slide 8, he formulates two equations I = C_0 (\rho_0 - \rho_S)\\ I = A \frac{dN}{dt} = A k_F N_0 \rho_S from which he arrives at an expression for N(t), namely...
  9. O

    Diffusion equation with Dirichlet BC

    Homework Statement Solve ut=kuxx u(x,0)=0 (ψ(x)) u(0,t)=1 on the half line 0<x<infinity (exercise 2, 3.1, Strauss) Homework Equations The Attempt at a Solution There are two bits I don't get: First, I know I have to make an odd or even extension to the whole line. But for...
  10. D

    MHB Diffusion Equation Invariant to Linear Temp. Transform

    Show the diffusion equation is invariant to a linear transformation in the temperature field $$ \overline{T} = \alpha T + \beta $$ Since $\overline{T} = \alpha T + \beta$, the partial derivatives are \begin{alignat*}{3} \overline{T}_t & = & \alpha T_t\\ \overline{T}_{xx} & = & \alpha T_{xx}...
  11. S

    Diffusion Equation on a plate - 2 dimensions

    Homework Statement The edges of a thin plate are held at the temperature described below. Determine the steady-state temperature distribution in the plate. Assume the large flat surfaces to be insulated. If the plate is lying along the x-y plane, then one corner would be at the origin...
  12. T

    Black-Scholes equation (a type of diffusion equation)

    Homework Statement The equation for the probability distribution of the price of a call option is \frac{\partial P}{\partial t} = \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + rS\frac{\partial P}{\partial S} - rP with the conditions P(0,t) = 0, P(S,0) = \max(S-K,0), and...
  13. Z

    Solving Diffusion Equation with Separation of Variables Method

    Homework Statement Solve the diffusion equation with the boundary conditions v(0,t)=0 for t > 0 and v(x,0) = c for t=0. The method should be separation of variables. Homework Equations The separation of variables method. The Attempt at a Solution Attempting a solution of the form...
  14. Z

    PDE for Heat Diffusion Equation

    Homework Statement The one-dimensional heat diffusion equation is given by : ∂t(x,t)/∂t = α[∂^2T(x,t) / ∂x^2] where α is positive. Is the following a possible solution? Assume that the constants a and b can take any positive value. T(x,t) = exp(at)cos(bx) Homework Equations...
  15. fluidistic

    PDE, inhomogeneous diffusion equation

    Homework Statement Mathews and Walker problem 8-2 (page 253): Assume that the neutron density n inside U_{235} obeys the differential equation \nabla ^2 n+\lambda n =\frac{1}{\kappa } \frac{\partial n }{\partial t} (n=0 on surface). a)Find the critical radius R_0 such that the neutron density...
  16. K

    Analytical solution to the diffusion equation with variable diffusivity

    Hi, I'm trying to find an analytical solution (if one exists) to the 1d diffusion equation with variable diffusivity κ(x); \partial_t u(x,t) = \partial_x[\kappa(x) \partial_x u(x,t)] Could someone point me in the right direction to solve this if its possible to do so analytically. I've...
  17. X

    Problem solving Heat Diffusion Equation

    Homework Statement One face of a thick uniform layer is subject to a sinusoidal temperature variation of angular frequency ω. SHow that the damped sinusoidal temperature oscillation propagate into eh layer and give an expression for the decay length of the oscillation amplitude. A cellar...
  18. P

    Perona-Malik Diffusion Equation

    Homework Statement Implement the non-linear de-noising algorithm of Perona-Malik. Consider a noisy image, u, with pixel values referenced by u(i,j). Non-linear de-noising can be achieved by solving the following non-linear diffusion equation: ∇ · (g(∇u)∇u) = 0 with g(s) = ((K^2)v) /...
  19. X

    Diffusion coefficient in diffusion equation and random walk ?

    Hi all: Now I have a question about the concept of diffusion coefficient in two cases: the diffusion equation (J=DdT/dx) and the random walk (tao^2=6Dt). My quesion is the two D in two equations are the same or different. If they are different, is there any relationship between them? Best Xu
  20. T

    Diffusion equation, boundary conditions

    EDIT: The subscripts in this question should all be superscripts! Homework Statement I'm trying to solve a temperature problem involving the diffusion equation, which has led me to the expression: X(x) = Cekx+De-kx Where U(x,y) = X(x)Y(y) and I am ignoring any expressions where...
  21. A

    Time dependent PDEs - mathematical modelling - diffusion equation

    Homework Statement Porous membranes are used to separate mixtures in industry, because smaller compounds permeate through them more easily than larger ones. KnoGas Pty Ltd are trialling an experimental separation process, using a membrane to sep- arate compounds A and B: compound A...
  22. J

    Can You Help Me Solve a 1D Diffusion Equation with a FTCS Scheme?

    I have a 1_D diffusion equation dc/dt = D*d^2c/dx^2-Lc where L,D = constants I am trying to solve the equation above by following b.c. by FTCS scheme -D*dc/dx = J0*delta(t); where delta(t)= dirac delta function ----(upper boundary) I have written the code for it but i just...
  23. J

    How to Solve a 1D Diffusion Equation with Initial and Boundary Conditions?

    Homework Statement I have a 1D diffusion equation as du/dt = D*d^2u/dx^2-K*u; where D and k = constants the initial condition is u(t=0)=0 B.C. is u(x=0,t=0)= u0*delta(t); (a pulse like input at x=0 and delta(t)= dirac delta function) where u = contaminant in a semi infinite slab...
  24. M

    Diffusion Equation and variable of Similitude

    I am studying fluid dynamics by using adimensional analysis to obtain an expression for a dynamic (thermal) boundary layer in a poiseuille flow. The fluid nearest the boundary (y = 0) is said to have adimenional temperature T = 1 and as y tends to infinity, the temperature is T = 0. The final...
  25. N

    Nondimentional Diffusion Equation

    Hello, Here is my problem: I need to turn my time dependent neutron diffusion equation into dimensionless one.. So, I have a couple of questions: 1. As neutron flux is a : "A measure of the intensity of neutron radiation in neutrons/cm2-sec" per definition, is "neutron" a unit? If yes, then...
  26. J

    1 D steady state diffusion equation in the atmosphere

    Hello, I have a 1-D steady state (dc/dt=0) differential equation in the atmosphere. It looks like follows, K*C'' + (K'+K/H)*C' + (1/H*K'- (K/H^2)*H'- (L+Si))C + S = 0 where, C = concentration of the contaminant in the atmosphere at different heights z K = vertical diffusion...
  27. D

    Diffusion equation without initial conditions

    Hi, How can I solve the diffusion equation in one dimension: u_t=ku_{xx} ; -\infty < t < \infty , 0<x<\infty With the boundary conditions: u(0,t)= T_0 +Acos(wt) u(x\rightarrow \infty,t) \rightarrow T_0 Thanks!
  28. T

    Adjoint flux for multi-group diffusion equation for criticality problem?

    Hi, I'm new to the entire neutronics field. I've learned about adjoints as a physics student in undergrad and I'm doing nuclear engineering for my graduate studies. I understand how to derive the adjoint operator for the diffusion equation, but I'm a bit confused as to how to calculate the...
  29. A

    Understanding the Diffusion Equation for Water Pressure in Filtration Beds

    Water pressure in a filtration bed is given by the following diffusion equation: \frac{\partial p}{\partial t} = -k
  30. P

    Solving Diffusion equation with Convection

    The problem is as follows: \frac{\partial u}{\partial t}=k\frac{\partial^{2}u}{\partial x^{2}}+c\frac{\partial u}{\partial x}, -\infty<x<\infty u(x,0)=f(x) Fourier Transform is defined as: F(\omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(x)e^{i\omega x}dx So, I took the Fourier...
  31. J

    Steady state Vertical diffusion equation

    Hallo everyone, I am trying to find the way to solve the vertical diffusion equation for a spicies X in the atmosphere for steady state conditions (dc/dt=0). The equation has the form, dJ/dz -P+S=0 where J =(vs*C(z)) + rho*kz*d/dz(C(z)/rho(z)) is the flux of spicies X at height z...
  32. K

    Solution to the Diffusion equation using a two dimensional fourier transform

    Homework Statement at t=0, x=0, a schoolboy sets off a stink bomb halfway down a corridor that is long enough to be considered infinite. The dispersion of the particles obey the modified diffusion formula: \frac{\partial \rho (x,t)}{\partial t} - D\frac{\partial^2...
  33. C

    Solving Diffusion Equation By Finite difference Method in fortran

    Hey, I want to solve a parabolic PDE with boundry conditions by using FINITE DIFFERENCE METHOD in fortran. (diffusion) See the attachment for the problem The problem is that there is a droplet on a leaf and it is diffusing in the leaf the boundry conditions are dc/dn= 0 at the upper...
  34. J

    Time evolution of a diffusion equation

    Hallo everyone, I have a 1-D diffusion equation with decay as dA/dt = d2A/dx2-L*A with initial condition C(x,0)=C0=exp(-ax) and boundary condition= -Ddc/dx = I0 where L= decay constant A = certain concentration the concentration A is not in equilibrium. We can solve the above...
  35. S

    Diffusion equation from random motion

    hi all, i am a new member here. nice to meet you all. i remember seeing a derivation of diffusion equation from the equation of motion \dot{x}=\eta(t) where \eta(t) is white noise. i can't remember where i saw this... could anyone please help me on that? it should be something like...
  36. S

    Schrodinger's equation and the Diffusion equation

    I've been combing over websites and papers on this, but I can't get a handle on trying to explain or visualize the similarity between them. The wave equation "smears out" over time as does the diffusion equation?
  37. B

    How Is the Steady State Assumed in Solving the Thermal Diffusion Equation?

    thermal diffusion equation - URGENT Homework Statement Please see attached q Homework Equations The Attempt at a Solution Ok so thermal diffusion equation is DT/Dt = D del squared T apparently this is a steady state problem, so DT/dt = 0, how am i meant to know? any hints on...
  38. T

    Can S2 Solve the Diffusion Equation in 2D?

    Show that S2 = S(x,t)S(y,t) solves St = k (Sxx + Syy) well St = St(x,t)S(y,t) + S(x,t)St(y,t) Sxx = Sxx(x,t)S(y,t) Syy=Syy(y,t)S(x,t) but what do i do from there?
  39. G

    Flicks diffusion equation - metabolism rate

    So i have an organic chemical in a bio-film reactor being diffused into a bio-film and also being metabolized at constant rate R by bacteria. the concentration into the reactor Cin = 2mg/L with f-in at 20 m^3/hr the concentration in the reactor is C1 C3 is the concentration in the...
  40. S

    Diffusion Equation PDE: Solving for u(x, t) with Initial Condition e^(-x^2)

    Homework Statement Solve u_{tt} - 4u_{xx} = 0, x \in \mathbb{R}, t > 0 u(x, 0) = e^{-x^2} , x \in \mathbb{R} Homework Equations General solution to the diffusion equation: u(x, t) = \frac{1}{\sqrt{4\pi kt}} \int\limits_{-\infty}^{\infty} e^\frac{{-(x - y)^2}}{4kt} \varphi(y) \, dyThe...
  41. R

    Solving the diffusion equation finite difference technique

    Hi everyone. I developed a Matlab program to solve the diffusion equation, a partial differential equation, using the finite difference method. I solved it first with boundary conditions of C = 0, at x=0 and C = 0, at x = 1. I now want to change the boundary condition C = Co (some constant)...
  42. L

    Diffusion equation in sperical coordinates

    I have the following diffusion equation \frac{\partial^{2}c}{\partial r^{2}} + \frac{2}{r}\frac{\partial c}{\partial r} = \frac{1}{\alpha}\frac{\partial c}{\partial t} where \alpha is the diffusivity. The solution progresses in a finite domain where 0 < r < b, with initial...
  43. C

    Diffusion Equation: Flux Must Be Finite Explained

    Hi Everyone, on page 238 of lamarsh, section 5.5, first paragraph, it says "flux must also be finite". What does it mean?
  44. M

    Solving the 'modified diffusion equation' using fourier transform

    Homework Statement A the density of a gas \rho obeys the modified diffusion equation \frac{\partial \rho(x,t)}{\partial t}-D\frac{\partial^2 \rho(x,t)}{\partial x^2}=K\delta(x)\delta(t) A) Express \rho in terms of its 2D Fourier transform \widetilde{\rho}(p,\omega) and express the right...
  45. T

    Solving Diffusion Equation with Boundary Conditions

    Homework Statement Obtain the solution of the diffusion equation, u(t,x) a) satisfying the boundary conditions: u(t,0)=u1, u(t,l)=u2, u(0,x)=u1+(u2-u1)x/l+a.sin(n\pix/l); b) in the semi-plane x > 0, with u(t,0)=u0+a.sin(\omegat). Homework Equations Wish I knew... The...
  46. H

    Heat diffusion equation solutions for semi-infinite slab

    Homework Statement http://img42.imageshack.us/img42/1082/clipboard01lx.jpg Homework Equations (see solution) The Attempt at a Solution I literary just spent 5 hours trying to apply those boundary conditions, trying exponentials, sines, cosines, hyperbolic function etc... I...
  47. O

    Diffusion equation with variable dissipation

    Homework Statement Solve the equation U{t}=kU{xx}+cU Hint:use an integrating factor Homework Equations The Attempt at a Solution
  48. O

    Solving the 1D Diffusion Equation: Boundary Conditions & Eigenvalues

    Technically speaking, this is not a homework problem; it is something extra that I want to explore using data from my biomedical engineering class. We recently finished a lab involving testing the design of a micromixer; now comes the analysis of the data. No where in the lab are we instructed...
  49. C

    Introductory PDE (diffusion equation)

    Homework Statement u_t = -{{u_{x}}_{x}} u(x,0) = e^{-x^2} Homework Equations The Attempt at a Solution The initial state is a bell curve centred at x=0. The second partial derivative of u at t=0 is {4x^2}{e^{-x^2}}, which is a Gaussian function, which means nothing to me other than its...
  50. I

    Diffusion Equation with source term

    I have been asked to solve a diffusion equation with a source term using finite differences method. I need to numerically integrate the following equation either in MATLAB or C++. The equation is dT/dt = d2T/dx2 + S(x) The form of S(x) is some function given by a Gaussian profile...
Back
Top