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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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##### References:
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