Abduragimov G.E. On the Existence and Uniqueness of a Positive Solution to the Dirichlet Problem for One Nonlinear Differential Equation
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https://doi.org/10.15688/mpcm.jvolsu.2026.1.1
Gusen E. Abduragimov
Candidate of Sciences (Physics and Mathematics), Associate Professor, Department of Applied Mathematics, Dagestan State University
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https://orcid.org/0000-0001-7095-932X
Dzerzhinsky St, 12, 367025 Makhachkala, Russian Federation
Abstract. The article deals with the boundary value problem △u + f(|x|, u) = 0, x ∈ S, U|Г = 0, where Г is the bound of S, and f a non-negative continuous function that satisfies one of the conditions:
(H1) : f(λ1v, λ2w) = λ1m1 λ2m2 f(v, w), ∀λ1, λ2 > 0, где m1 ≥ 0, m2 > 1 or
(H2) : f(λv, λw) = λm1+m2 f(v, w), ∀λ > 0, где m1 ≥ 0, m2 > 1.
In addition, it is assumed that
limu→+∞ min r∈[0,1] f(r, u)/u = +∞.
Under the assumption of non-negativity, continuity and homogeneity of the nonlinear component of the equation, using the linear group of transformations of C. Na, the existence of a unique positive radially symmetric solution to the problem under consideration was proven. Furthermore, a constructive non-iterative numerical algorithm for constructing such a solution is proposed. An example is provided to illustrate the results obtained.
Key words: Dirichlet problem, Cauchy problem, positive solution, radially symmetric solution, differential equation.
Numerical study of laser radiation estimation in the focal plane by Abduragimov G.E. On the Existence and Uniqueness of a Positive Solution to the Dirichlet Problem for One Nonlinear Differential Equation is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 29 No. 1 2026, pp. 5-12
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