# Neumann B.C. with dependance on value at boundary

**URL:** <https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179>\
**Category:** General Discussion\
**Created:** [December 11, 2022, 10:10am UTC](https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179 "2022-12-11T10:10:26Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Lysandros](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/lysandros/32/1457_2.png) [@Lysandros](https://community.freefem.org/u/Lysandros)\
**Post date:** [December 11, 2022, 10:10am UTC](https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179/1 "2022-12-11T10:10:26Z")

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Dear all,  
I am trying to implement a boundary condition where the rate of change of my function at the boundary depends on the value of the function at this boundary in a nonlinear way. How can I implement his?  
As far as I understand, I can use the expression  
`∂nu = g : -int1d(Th)( g*w)`  
for Neumann boundary conditions. But what if g itself depends on the value of the unknown function u?  
Thank you

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**Author:** ![vlaude](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/vlaude/32/939_2.png) [@vlaude](https://community.freefem.org/u/vlaude)\
**Post date:** [December 11, 2022, 3:16pm UTC](https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179/2 "2022-12-11T15:16:00Z")

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Dear Lysandros,  
Are you looking for the Robin boundary condition (g(u) linear with u)? Then you may start here: [Laplace eigenproblem with Robin boundary condition](https://community.freefem.org/t/laplace-eigenproblem-with-robin-boundary-condition/895)  
If the Robin BC is nonlinear, you may have to implement a nonlinear solution of your problem.

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**Author:** ![Lysandros](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/lysandros/32/1457_2.png) [@Lysandros](https://community.freefem.org/u/Lysandros)\
**Post date:** [December 11, 2022, 5:04pm UTC](https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179/3 "2022-12-11T17:04:28Z")

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Dear Vincent, thanks for your reply. Actually, I want to model thermal radiation from the surface of a plate. Therefore, the local gradient at the boundary should be du/dx=u^4, since radiation scales with the fourth power of the surface temperature. Would this fall in the category of a Robin boundary condition? Am I on a good road if use the int1d term?  
Kins regards

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**Author:** ![vlaude](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/vlaude/32/939_2.png) [@vlaude](https://community.freefem.org/u/vlaude)\
**Post date:** [April 21, 2023, 7:49am UTC](https://community.freefem.org/t/neumann-b-c-with-dependance-on-value-at-boundary/2179/4 "2023-04-21T07:49:47Z")

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Sorry for the very late reply. Your case is clearly nonlinear then, so you need to implement a nonlinear solution in my understanding.