Steady-state solution of the PTC thermistor problem using a quadratic spline finite element method (Q1877378)

This paper deals with a dimensionless mathematical model of the PCT thermistor: \[ u_t=u_{xx}+\gamma\sigma(u)|\phi_x|^2, \;0<x<1, \;t>0 \] subject to the conditions \(u_x=0\) for \(x=0\), \(t>0\), \(u_x+\beta u=0\) for \(x=1\), \(t>0\), \(u(x,0)=0\) for \(0\leq x\leq 1\), where \(u\) is the temperature and \(\phi\) is the electrical potential. A finite element solution is constructed using quadratic spatial elements and finite difference approximation in time. The numerical and analytical steady-state solution of the problem is covered.