Abstract
We have built a constructive method of investigation and approximate solution for nonlinear Gursa’s problem with prehistory. We have established sufficient condition of subsistence, existence of unity and constant signs solution of the investigated problem. At mathematical description to different nature process (gas sorption, the spread of moisture in the porous substances, pipes heating by a stream of hot water, drying by the airflow, etc. [1]) we often come to boundary value problems for nonlinear differential equations in partial derivatives, when not all output data are known, that is some of them need to be found from auxiliary nonlinear problems, which are mathematical models of processes that proceeded the research. These problems should be named as problems with prehistory. One approach to investigation and approximate solution to such a problem has been proposed in the current paper.
Highlights
Будується конструктивний метод дослiдження та наближеного розв’язання нелiнiйної задачi Гурса з передiсторiєю
We have built a constructive method of investigation and approximate solution for nonlinear Gursa’s problem with prehistory
At mathematical description to different nature process we often come to boundary value problems for nonlinear differential equations in partial derivatives, when not all output data are known, that is some of them need to be found from auxiliary nonlinear problems, which are mathematical models of processes that proceeded the research
Summary
(b) для всяких неперевно диференцiйовних пар функцiй zi(x, y), vi(x, y) ∈ B1,i, якi задовольняють умови D(κ,s)zi(x, y) D(κ,s)Vi(x, y), (x, y) ∈ Di, i = 1, 2, κ = 0, 1, s = 0, 1, κ+s < 2, в областi B1,i виконуються нерiвностi. (x, y) ∈ Di, 3) неперервнi функцiї Hi[zi(x, y); vi(x, y)] в областi B1,i задовольняють умову Лiпшиця, тобто для всяких з простору C1(Di) функцiй zi,r(x, y), vi,r(x, y) ∈ B1,i, r = 1, 2, виконуються умови. Функцiї zi,0(x, y), vi,0(x, y) ∈ C∗Di, якi належать областi B1,i i задовольняють умови (12), називаються функцiями порiвняння крайової задачi (1)—(5). Якщо функцiї fi[Ui(x, y)] ∈ C1(Bi) та iснують функцiї порiвняння задачi (1)—(5), то множина функцiй qi,p(x, y), ci,p(x, y), якi задовольняють умови (10), (16) не порожня. Аналогiчно доводиться i друга iз нерiвностей (16), якщо c(i,κp,s)(x, y) вибирати згiдно (18) i Лема 2 доведена
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