# Enforcing Mathematical Continuity for a System Containing Discontinuity

I am attempting to numerically solve a BVP for a solid core solenoid field. I am starting with:

$$\vec{\nabla{}^2}\vec{A_r}=0$$

$$\vec{\nabla{}^2}\vec{A_\phi}=\frac{\mu}{\mu_0}\vec{J_\phi}$$

$$\vec{\nabla{}^2}\vec{A_z}=0$$

in cylindrical coordinates where the current $$J$$ is traveling along a cylindrical surface in the $$\phi$$ direction. I am assuming that the current has been turned on for a time $$t>>t_0$$ so we ignore any hysteresis affects of ferromagnetic materials that produce non-linearity. So focusing on just the $$\phi$$ component, the $$\mu$$ is a function of space since we assume that the solid core of the solenoid is a cylinder of radius $$R$$ and length $$L$$. I can imagine then that $$\mu(r,z)$$ can be written in one of two ways:

$$\mu(r,z) = \mu_sH(L-|z|)H(R-r) + \mu_0H(|z|-L)H(r-R)$$

$$\mu(r,z) = \left\{ \begin{array}{lr} \mu_s, & \text{if } r < R \text{ or } |z| R \text{ or } |z|>L \end{array} \right\}$$

Regardless of the way this can be written, although I would prefer to use the first method as that would be easier to implement within code, I notice that there is a discontinuity occurring. We know the magnetic field lines must be continuous across the boundary that describes the metal core and the vacuum around it. I saw on the wiki article for Heaviside that at $$H(0)$$ it looks like it's value becomes $$0.5$$. However, this still doesn't suffice continuity. Neither would implementing an extra parameter in the piecewise for $$r = R \text{ or } |z|=L$$. The only idea I had was using the analytic approximation for the Heaviside since that is continuous while taking $$k\rightarrow\infty$$. However, it still changes value over a finite domain and so it doesn't represent the reality of the system in which essentially the material's surface and vacuum are creating. How then can I create a $$\mu$$ function that forces continuity and is properly describing physical reality?

• It does not matter what value you assign to $H(t)$ at $t=0$ and the vector potential is continuous everywhere, even at the interface where $\mu$ is discontinuous. this may help Apr 12, 2023 at 1:13
• @hyportnex So essentially so long as my conditions include continuity that means that my $\mu$ function can be in either form I wrote. Why doesn't continuity matter in terms of the function that describes permeability? That would mean the BVPs trump any $\mu$ function I create, yet I would think that defining $\mu$ forces discontinuity since it's in the original Poison equation. I am using conditions 7, 8 and 10 from that post along with two more conditions for my specific problem. Apr 12, 2023 at 7:31
• take a simple diff eq involving say, the signum function $sgn(x)=\pm 1$: $\frac{d}{dx}(sgn(x)\frac{dy}{dx})=0$ and solve it to find $y(x)$. Solve it and ask your self what does it matter if $sgn(0) = 0, 0.1 or -109.072$ when you integrate the equation for $y(x)$? Does the value of an integral change if you vary a single value of the integrand? Apr 12, 2023 at 8:10
• I worry that your conceptualisation of how you want to solve this problem is itself misguided. However, I will ignore that and mention another issue instead. Your magnetic constant" $\mu$ is specified wrongly. And your cylinder is length $2L$ if you have been using $|z|<L$ as the condition. I would suggest that you have $\mu = \mu_0 + (\mu_s-\mu_0)H(R-r)H(L-|z|)$ instead, because the specification of the conditions becomes much easier for you. Apr 13, 2023 at 6:56
• I am doubting your starting equation. Was it derived from $\text{rot}\left(\mu(r,z)^{-1}\text{rot}\vec{A}\right)=\vec{J}\;\;$? Apr 13, 2023 at 9:04