Mechanical Engineering Journal (Nov 2021)
A modified perturbation method in which the applicable range of a small parameter for the solution is extended much larger than that by the conventional perturbation method
Abstract
A modified perturbation method, in which the applicable range of a small parameter ε for the solution is extended larger than that by the conventional perturbation method, is applied to two simple heat transfer problems with temperature-dependent thermal properties. The main procedure of the modified perturbation method is: (1) A perturbation parameter ε is assumed to be included in the nonlinear term of the differential equation. The solution θ is expressed by θ = φ + θf, where θf is an initial approximation of the solution and φ is θ − θf. (In the example problems of this paper, we assume that θf is a constant.) (2) θ = φ + θf is substituted into the differential equation and the nonlinear term is split into linear and nonlinear terms. (3) ε which is not in the nonlinear term is replaced by a newly introduced variable ε’. (4) An asymptotic expansion of φ in powers of ε is assumed for the solution of the differential equation, from which we obtain the perturbation solution of φ including ε’ and ε’. (5) ε in the perturbation solution of φ is replaced by ε. Then we obtain the perturbation solution of θ. By solving the two example problems, it is made clear that the solution by the modified perturbation method is more accurate than that by the conventional perturbation method. It is also made clear that the modified perturbation method extends drastically the applicable range of the perturbation parameter in comparison with the conventional perturbation method. The modifications of the perturbation method help reduce the contribution of the nonlinear term, which drastically improves the convergence characteristics of the solution. The reason for the good convergence characteristics of the modified perturbation solution is discussed.
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