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In 2D Poisson Equation I have example in electrostatics, $${\Delta ^2}\phi = - \frac{{{\rho _{el}}}}{\varepsilon }.$$ And I need an example of 1D Poisson Equation in daily life. In Engineering field, Physics field, etc. I will try to use a numerical method in that example. Can you help me please?

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It would perhaps not formally be called Poisson's equation in 1D but the question is still valid.


There are many examples within electrostatics and heat transfer. For electrostatics let's say an infinite dielectric plate with one side grounded (Dirichlet condition) and a charge density at the other end (Neumann condition). To get a source term you can imagine having bound charges throughout the volume- this is a bit artificial but still physically valid. To get a modern touch in this you can pretend that this is some part of a touchscreen device.


In heat transfer the corresponding problem would be an infinite plate with a fixed temperature at one end, such as room temperature, and then a heat flux at the other end, for example Newton cooling. Then you can have a volumetric source term such as from some kind of electric or chemical heating. No matter the source term, in the simplest case the unit of this source term would be the same: power per unit volume; so you wouldn't need to worry about its physical reason.

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