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
Related Articles
- EP ID EP558800
- DOI 10.18524/2519-206x.2018.2(32).149705
- Views 99
- Downloads 0
The block diagonalization of the linear homogeneous differential system with coefficients of oscillating type in resonance case
For the linear homogeneous system of the differential equations, coefficients of which are represented by an absolutely and uniformly convergent Fourier series with slowly varying coefficients and frequency, conditions o...
Approximation of solutions to the optimal control problem for the impulsive system with maximum
This paper presents the averaging method for two problems: impulsive system with maximum and for the optimal control problem of this kind of system. For the first problem the Krylov–Bogolyubov’s theorem is generalized. F...
BOUNDARY-VALUE PROBLEM FOR DIFFERENTIAL EQUATION WITH MIXED RIEMANN—LIOUVILLE DERIVATIVE OF FRACTIONAL ORDER
In this paper, we find a sufficient condition for solvability of boundary-value problem for differential equation with mixed Riemann-Liouville derivative of fractional order.
On exponential sums involving the divisor function over Z[𝑖]
We apply the van der Corput transform to investigate the sums of view Σ︀ 𝑟(𝑛)𝑔(𝑛)𝑒(𝑓(𝑛))), where 𝑟(𝑛) is the number of representations of 𝑛 as the sum of two squares of integer numbers. Such sums have been studied by M....
Step average linear differential inclusions of variable dimension
The theory of differential inclusions began its development in the early thirties of the 20th century with the publication A. Marsh and S. Zaremba. However, the rapid development of this theory began with the 60s of the...