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Auxiliary functions in the study of Stefan-like problems with variable thermal properties. (English) Zbl 1439.80009
The authors prove the existence and uniqueness of an auxiliary function solution to the following nonlinear differential system (called modified error function): \[((1+\delta y)y^{\prime })^{\prime }+2x(1+\gamma y)y^{\prime }=0\] for \(x\in (0,+\infty )\) and with the conditions \( y(0)=0\), \(y(+\infty )=1\), where \(\delta, \gamma \in (-1,+\infty )\). This problem is associated to phase-change processes where some thermal coefficients are assumed to vary with the material temperature. The main result of the paper is a proof of the existence of a unique bounded analytic solution \(\Phi _{\delta \gamma }\) which satisfies \(0\leq \Phi _{\delta \gamma }(x)\leq 1\) for all \(x\geq 0\) if the coefficients \(\delta \), \(\gamma \) satisfy a boundedness hypothesis. The proof uses Banach’s fixed point theory in an appropriate functional setting. The authors finally prove that the unique solution \(\Phi _{\delta \gamma }\) is increasing and if \(\delta \) is non-negative \(\Phi _{\delta \gamma }\) is concave.
MSC:
80A22 Stefan problems, phase changes, etc.
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Software:
COLSYS
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