Mass Diffusion Equation (Fick's Second Law)
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Solving the Mass Diffusion Equation $$fracpartial rhopartial t = D_mfracpartial ^2 rhopartial x^2$$ for $$rho (x,t) = frac1sqrtD_mtfleft(fracxsqrtD_mtright)$$ and $$int_-infty^infty rho(x,t) dx = M$$ where M is the total mass of the diffusing particles, I obtained a differential equation of the form $$2f''(u) + uf'(u) + f(u) = 0$$ Above, $u = fracxsqrtD_mt$. How do I analytically solve this differential equation to find the functional form of $f(u)$?
fluid-dynamics differential-equations
migrated from physics.stackexchange.com Aug 26 at 22:04
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Solving the Mass Diffusion Equation $$fracpartial rhopartial t = D_mfracpartial ^2 rhopartial x^2$$ for $$rho (x,t) = frac1sqrtD_mtfleft(fracxsqrtD_mtright)$$ and $$int_-infty^infty rho(x,t) dx = M$$ where M is the total mass of the diffusing particles, I obtained a differential equation of the form $$2f''(u) + uf'(u) + f(u) = 0$$ Above, $u = fracxsqrtD_mt$. How do I analytically solve this differential equation to find the functional form of $f(u)$?
fluid-dynamics differential-equations
migrated from physics.stackexchange.com Aug 26 at 22:04
This question came from our site for active researchers, academics and students of physics.
Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
1
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Solving the Mass Diffusion Equation $$fracpartial rhopartial t = D_mfracpartial ^2 rhopartial x^2$$ for $$rho (x,t) = frac1sqrtD_mtfleft(fracxsqrtD_mtright)$$ and $$int_-infty^infty rho(x,t) dx = M$$ where M is the total mass of the diffusing particles, I obtained a differential equation of the form $$2f''(u) + uf'(u) + f(u) = 0$$ Above, $u = fracxsqrtD_mt$. How do I analytically solve this differential equation to find the functional form of $f(u)$?
fluid-dynamics differential-equations
Solving the Mass Diffusion Equation $$fracpartial rhopartial t = D_mfracpartial ^2 rhopartial x^2$$ for $$rho (x,t) = frac1sqrtD_mtfleft(fracxsqrtD_mtright)$$ and $$int_-infty^infty rho(x,t) dx = M$$ where M is the total mass of the diffusing particles, I obtained a differential equation of the form $$2f''(u) + uf'(u) + f(u) = 0$$ Above, $u = fracxsqrtD_mt$. How do I analytically solve this differential equation to find the functional form of $f(u)$?
fluid-dynamics differential-equations
asked Aug 18 at 5:02
Divyansh Khurana
migrated from physics.stackexchange.com Aug 26 at 22:04
This question came from our site for active researchers, academics and students of physics.
migrated from physics.stackexchange.com Aug 26 at 22:04
This question came from our site for active researchers, academics and students of physics.
Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
1
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25
 |Â
show 1 more comment
Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
1
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25
Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
1
1
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25
 |Â
show 1 more comment
1 Answer
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$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so
$$2f'+uf=C$$From the boundary conditions, C = 0. So, $$fracf'f=-fracu2$$The solution to this differential equation is:
$$f=Ae^-fracu^24$$where A is a constant of integration. So,
$$rho=fracAsqrtD_mte^-fracu^24$$To determine A, all you need to do is satisfy the condition $$int_-infty^+inftyrho du=fracMsqrtD_mt$$or$$Aint_-infty^+inftye^-fracu^24du=M$$
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so
$$2f'+uf=C$$From the boundary conditions, C = 0. So, $$fracf'f=-fracu2$$The solution to this differential equation is:
$$f=Ae^-fracu^24$$where A is a constant of integration. So,
$$rho=fracAsqrtD_mte^-fracu^24$$To determine A, all you need to do is satisfy the condition $$int_-infty^+inftyrho du=fracMsqrtD_mt$$or$$Aint_-infty^+inftye^-fracu^24du=M$$
add a comment |Â
up vote
1
down vote
$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so
$$2f'+uf=C$$From the boundary conditions, C = 0. So, $$fracf'f=-fracu2$$The solution to this differential equation is:
$$f=Ae^-fracu^24$$where A is a constant of integration. So,
$$rho=fracAsqrtD_mte^-fracu^24$$To determine A, all you need to do is satisfy the condition $$int_-infty^+inftyrho du=fracMsqrtD_mt$$or$$Aint_-infty^+inftye^-fracu^24du=M$$
add a comment |Â
up vote
1
down vote
up vote
1
down vote
$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so
$$2f'+uf=C$$From the boundary conditions, C = 0. So, $$fracf'f=-fracu2$$The solution to this differential equation is:
$$f=Ae^-fracu^24$$where A is a constant of integration. So,
$$rho=fracAsqrtD_mte^-fracu^24$$To determine A, all you need to do is satisfy the condition $$int_-infty^+inftyrho du=fracMsqrtD_mt$$or$$Aint_-infty^+inftye^-fracu^24du=M$$
$$2f''(u) + uf'(u) + f(u) = 2f''+(uf)'=0$$so
$$2f'+uf=C$$From the boundary conditions, C = 0. So, $$fracf'f=-fracu2$$The solution to this differential equation is:
$$f=Ae^-fracu^24$$where A is a constant of integration. So,
$$rho=fracAsqrtD_mte^-fracu^24$$To determine A, all you need to do is satisfy the condition $$int_-infty^+inftyrho du=fracMsqrtD_mt$$or$$Aint_-infty^+inftye^-fracu^24du=M$$
answered Aug 19 at 12:32
Chester Miller
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Use Mathematica.
– Samuel Weir
Aug 18 at 5:07
i don’t see how you get this differential equation?. To solve this differential equation i will try this ansatz $rho=w(t) v(x)$
– Eli
Aug 18 at 8:17
Mathematica gives me the solution, alright, but I was curious if we have a way to analytically arrive at that solution.
– Divyansh Khurana
Aug 18 at 9:37
Secondly, ÃÂ=w(t)v(x) is basically the "Separation of Variables" method and is perfectly fine. But the above question specifically mentions the form of ÃÂ that is to be assumed.
– Divyansh Khurana
Aug 18 at 9:40
1
Might Mathematics be better suited for this math question?
– Kyle Kanos
Aug 18 at 10:25