An exact methodology for solving nonlinear diffusion equations based on integral transforms. (English) Zbl 0628.65108
Let T and k(T) denote the temperature and the thermal conductivity, respectively. Assuming a truncated Taylor-series expansion of the form \(k(T)=k_ 0\sum^{N}_{n=0}\beta_ n(T-T_ 0)^ n,\) where \(\beta_ 0\equiv 1\), \(T_ 0\) is the reference temperature, the authors derive the nonlinar temperature field equation at once for a slab, a cylinder and a sphere. Suitable initial and boundary conditions are derived as well. The case of the slab for \(N=1\) is examined in details. Applying an integral transform enables to obtain the solution in the form of an infinite series. To get terms of this series one has to solve a system of nonlinear Volterra integral equations of the second kind. To do this one a numerical procedure is proposed. Numerical calculations are performed and results are compared to those ones obtained by a standard finite difference method.
Reviewer: St.Burys
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65R10 | Numerical methods for integral transforms |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 45G10 | Other nonlinear integral equations |
| 35C15 | Integral representations of solutions to PDEs |
Keywords:
nonlinear diffusion equations; temperature-dependent thermal conductivity; comparison of methods; nonlinear heat equation; truncated Taylor-series expansion; system of nonlinear Volterra integral equations of the second kindReferences:
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