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I have tried to solve the heat equation with a Fourier-Bessel approach but I fail to implement the boundary condition, which is a Neumann condition. Every text book that I have available treats the corresponding Dirichlet problem but not the Neumann one. Below I have tried to summarize the problem and my findings so far in the hope that someone could show me how to solve the problem with Neumann boundary conditions. Please be aware that my skills in solving PDEs with analytical methods are limited.

The problem is to calculate the temperature in a unit 2D disc as a function of time, starting with a known temperature distribution at time 0 (t=0) and with heat exiting through the boundary of the circle. Thus,

[tex]

\frac{\partial u}{\partial t} = a \left[ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \varphi^2} \right]

[/tex]

with boundary condition (BC)

[tex]

\frac{\partial u}{\partial n} +h(u-u_c)=0

[/tex]

where [tex] u_c [/tex] is the temperature of the surrounding material, and initial condition (IC)

[tex]

u(r,\varphi,t=0)=T_0

[/tex]

where we can assume that [tex] T_0 [/tex] is constant.

Separation of variables leads to 3 differential equations. We assume now also that the problem is axisymmetric leaving only 2 diff. eqs. according to

[tex] T'(t)=-a \lambda^2 T(t) [/tex]

[tex] r^2 R''(r) + r R'(r) + \lambda^2 r^2 R(r) - \mu R(r) =0 [/tex]

Solution to the first equation will be on the form

[tex]

T(t)=A e^{-a \lambda^2 t}

[/tex]

and by making the variable substitution [tex] x=\lambda r [/tex] the second equation turns in the well known Bessel equation with solutions according to

[tex]

X=c_1 J_0(x)

[/tex]

where [tex] J_0(x) [/tex] is a Bessel function of first kind and zero order. Thus, the solution is on the form

[tex]

u(r,t)=cJ_0(\lambda r) e^{-a \lambda^2 t}

[/tex]

I belive that this is a standard result in meny text books. Standard procedure in these books is then to proceed by explaining how to determine what values on [tex] \lambda [/tex] are valid and how the coefficients [tex] c_k [/tex] in the Bessel function are identified. This is pretty straight forward in the case of a fixed temperature on the boundary. But how is it done with the BC stated above? If someone could show me the steps, or point me to relevant literature, I would be very thankful.

Best regards,

Johan Ekh

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# Heat equation with Neumann BC

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