Numerical solution of nonlocal contact problems for elliptic equations

Nonlocal contact problem for two-dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in the case of Poisson’s equation. Then, the more general problem with nonlocal multipoint contact conditions for elliptic equation with variable coefficients is considered, and the iterative method to solve the problem numerically is constructed and investigated. The uniqueness and existence of the regular solution are proved. The iterative method allows reducing the solution of a nonlocal contact problem to the solution of a sequence of classical boundary value problems.

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“…Φ k is defined from (15), and I − , I + -from ( 11) and ( 12), respectively. Thus, the following theorem is true.…”

Section: Considering the Boundary Conditions Amentioning

confidence: 99%

On One Generalization of the Multipoint Nonlocal Contact Problem for Elliptic Equation in Rectangular Area

Davitashvili

Meladze

Criado-Aldeanueva

et al. 2022

Journal of Mathematics

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“…Pokornyi [21] studied the eigenfunctions of a particular problem on graphs with Dirichlet conditions at border nodes, as well as the spectrum and the impact of eigenvalue multiplicity. Using the double-sweep method, Gordeziani et al [22] provided a numerical way to solve ordinary differential equations on graphs, as well as an investigation of the existence and uniqueness results of these kinds of problems. Currie and Watson [23] provided asymptotic approximations for eigenvalues and examined the spectral structure of second-order boundary value problems on graphs.…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Nieto,

Yadav,

Mathur

et al. 2024

Symmetry

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Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

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“…In 1989, Zavgorodnij explored differential equations using a geometric net (see [26]), with the recommended solutions to boundary value problems placed at the inner vertices of the system. However, in [27], the authors used the double-sweep technique, which they discovered to be more effective on graphs, to obtain numerical solutions for differential equations.…”

Section: Introductionmentioning

confidence: 99%

Study of Fractional Differential Equations Emerging in the Theory of Chemical Graphs: A Robust Approach

Turab

Rosli

2022

Mathematics

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The study of the interconnections between chemical systems is known as chemical graph theory. Through the use of star graphs, a limited group of researchers has examined the space of possible solutions for boundary-value problems. They recognized that for their strategy to function, they needed a core node related to other nodes but not to itself; as a result, they opted to use star graphs. In this sense, the graphs of neopentane will be helpful in extending the scope of our technique. It has the CAS number 463-82-1 and the chemical formula C5H12, and it is a component of a petrochemical precursor. In order to determine whether or not the suggested boundary-value problems on these graphs have any known solutions, we use the theorems developed by Schaefer and Krasnoselskii on fixed points. In addition, we illustrate our preliminary results with the help of an example that we present.

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Numerical solution of nonlocal contact problems for elliptic equations

Cited by 4 publications

References 3 publications

On One Generalization of the Multipoint Nonlocal Contact Problem for Elliptic Equation in Rectangular Area

On One Generalization of the Multipoint Nonlocal Contact Problem for Elliptic Equation in Rectangular Area

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Study of Fractional Differential Equations Emerging in the Theory of Chemical Graphs: A Robust Approach

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