TWO-SIDED APPROXIMATIONS METHOD AND ROTHE METHOD FOR SOLVING PROBLEMS FOR THE ONE-DIMENSIONAL SEMILINEAR HEAT EQUATION
Journal Title: Дослідження в математиці і механіці - Year 2018, Vol 23, Issue 2
Abstract
We consider the first initial-boundary problem for the one-dimensional semilinear heat equation. Based on the modified Rothe method at each time layer the original non-stationary problem is replaced by a nonlinear boundary-value problem for an ordinary differential equation. Using the Green’s functions method of nonlinear boundary value problems for an ordinary differential equation, a transition to an equivalent Hammerstein integral equation is considered, which is investigated as a nonlinear operator equation with a heterotone operator in the space of continuous functions that is semiordered by a cone of non-negative functions. To find a positive solution of the integral equation (and hence a generalized solution of the corresponding boundary value problem), a method of successive approximations with a two-sided character of convergence is constructed on each time layer. Thus, in the work for the first initial-boundary value problem for the one-dimensional semilinear heat equation with a variable heat conduction coefficient, a semi-discrete method for its solution was first built, based on the combined use of the modified Rothe lines method and the two-sided approximation method. A computational experiment was carried out for a heterotone power nonlinearity problem with exponential coefficient of thermal conductivity and parabolic initial temperature distribution.
Authors and Affiliations
M. V. Sidorov
Keywords
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- EP ID EP558800
- DOI 10.18524/2519-206x.2018.2(32).149705
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