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A two level finite difference scheme for one dimensional Pennes’ bioheat equation. (English) Zbl 1086.65090
The following initial-boundary value problem is considered for one dimensional Pennes’ bioheat equation.
\[ \Theta_t+a*\Theta-b*(k(x)*\Theta_x)_x =0,\quad \Theta(x,0)=0,\, \Theta(0,t)=\Theta_0,\, \Theta_x(L,t)=0, \tag{1} \] where \(k(x)\) is the thermal conductivity of the tissue, \(a\) and \(b\) are positive parameters. The problem (1) is replaced by the two-layers difference scheme with second order of approximation with regard to the uniform space step size \( \Delta x\) and the uniform time step size \(\Delta t.\) The Neumann boundary condition in the point \(x=L\) is approximated with the first order by a left difference. The (known) results about the stability of the difference scheme with regard to the initial data and absolute convergence are established. Several numerical examples are given.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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