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Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

Cabada Alberto, Dimitrov Nikolay D., Jonnalagadda Jagan Mohan

Opuscula Mathematica

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2024

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Vol. 44, no. 2

167--195

EN

This article is devoted to deduce the expression of the Green’s function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green’s function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green’s function is constructed as finite sums, the Green’s function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green’s function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green’s function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

2

On solvability of elliptic boundary value problems via global invertibility

Bełdziński Michał, Galewski Marek

Opuscula Mathematica

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2020

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Vol. 40, no. 1

37--47

EN

In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.

3

The impact of the Dirichlet boundary conditions on the convergence of the discretized system of nonlinear equations for potential problems

Kucwaj J.

Computer Assisted Methods in Engineering and Science

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2016

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Vol. 23, no. 1

69--81

EN

The purpose of this paper is the analysis of numerical approaches obtained by describing the Dirichlet boundary conditions on different connected components of the computational domain boundary for potential flow, provided that the domain is a rectangle. The considered problem is a potential flow around an airfoil profile. It is shown that in the case of a rectangular computational domain with two sides perpendicular to the speed direction, the potential function is constant on the connected components of these sides. This allows to state the Dirichlet conditions on the considered parts of the boundary instead of the potential jump on the slice connecting the trail edge with the external boundary. Furthermore, the adaptive remeshing method is applied to the solution of the considered problem.

4

Numerical solutions to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator

Ciesielski M., Leszczyński J.

Journal of Theoretical and Applied Mechanics

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2006

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Vol. 44 nr 2

393-403

EN

In this paper, we present a numerical solution to an ordinary differential equation of a fractional order in one-dimensional space. The solution to this equation can describe a steady state of the process of anomalous diffusion. The process arises from interactions within complex and non-homogeneous background. We present a numerical method which is based on the finite differences method. We consider a boundary value problem (Dirichlet conditions) for an equation with the Riesz-Feller fractional derivative. In the final part of this paper, some simulation results are shown. We present an example of non-linear temperature profiles in nanotubes which can be approximated by a solution to the fractional differential equation.

PL

W pracy zaprezentowano numeryczne rozwiązanie jednowymiarowego równania różniczkowego zwyczajnego niecałkowitego rzędu. Rozwiązanie tego równania może opisywać stan ustalony procesu anomalnej dyfuzji. Proces ten wynika z oddziaływań zachodzących w złożonych i niejednorodnych systemach. Zaprezentowana metoda numeryczna oparta jest na metodzie różnic skończonych. Rozważane było zagadnieriie brzegowe z warunkami Dirichleta dla tego równania z pochodną frakcjalną RieszaFellera. W końcowej części przedstawiono wyniki symulacji. Jako przykład zaprezentowano nieliniowy profil temperatury w nanorurkach, który może być przybliżony przez rozwiązanie frakcjalnego równania różniczkowego.