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Search Results (6)

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Keywords = nonlinear parabolic elliptic system

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22 pages, 317 KiB  
Article
Two-Phase Incompressible Flow with Dynamic Capillary Pressure in a Heterogeneous Porous Media
by Mohamed Lamine Mostefai, Abdelbaki Choucha, Salah Boulaaras and Mufda Alrawashdeh
Mathematics 2024, 12(19), 3038; https://doi.org/10.3390/math12193038 - 28 Sep 2024
Viewed by 700
Abstract
We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. [...] Read more.
We prove the existence of weak solutions of a two-incompressible immiscible wetting and non-wetting fluids phase flow model in porous media with dynamic capillary pressure. This model is a coupled system which includes a nonlinear parabolic saturation equation and an elliptic pressure–velocity equation. In the regularized case, the existence and uniqueness of the weak solution are obtained. We let the regularization parameter be η→0 to prove the existence of weak solutions. Full article
17 pages, 363 KiB  
Article
Control Problem Related to a 2D Parabolic–Elliptic Chemo-Repulsion System with Nonlinear Production
by Exequiel Mallea-Zepeda and Luis Medina
Symmetry 2023, 15(10), 1949; https://doi.org/10.3390/sym15101949 - 21 Oct 2023
Viewed by 1206
Abstract
In this work, we analyze a bilinear optimal control problem related to a 2D parabolic–elliptic chemo-repulsion system with a nonlinear chemical signal production term. We prove the existence of global optimal solutions with bilinear control, and applying a generic result on the existence [...] Read more.
In this work, we analyze a bilinear optimal control problem related to a 2D parabolic–elliptic chemo-repulsion system with a nonlinear chemical signal production term. We prove the existence of global optimal solutions with bilinear control, and applying a generic result on the existence of Lagrange multipliers in Banach spaces, we obtain first-order necessary optimality conditions and derive an optimality system for a local optimal solution. Full article
(This article belongs to the Special Issue Optimal Control and Symmetry)
24 pages, 708 KiB  
Article
Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation
by Dossan Baigereyev, Dinara Omariyeva, Nurlan Temirbekov, Yerlan Yergaliyev and Kulzhamila Boranbek
Mathematics 2022, 10(8), 1319; https://doi.org/10.3390/math10081319 - 15 Apr 2022
Cited by 8 | Viewed by 2233
Abstract
In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important [...] Read more.
In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper. Full article
(This article belongs to the Section E: Applied Mathematics)
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15 pages, 2779 KiB  
Article
Convergence and Numerical Solution of a Model for Tumor Growth
by Juan J. Benito, Ángel García, María Lucía Gavete, Mihaela Negreanu, Francisco Ureña and Antonio M. Vargas
Mathematics 2021, 9(12), 1355; https://doi.org/10.3390/math9121355 - 11 Jun 2021
Cited by 6 | Viewed by 2307
Abstract
In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the [...] Read more.
In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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11 pages, 2252 KiB  
Communication
Propagation-Invariant Off-Axis Elliptic Gaussian Beams with the Orbital Angular Momentum
by Alexey A. Kovalev, Victor V. Kotlyar and Darya S. Kalinkina
Photonics 2021, 8(6), 190; https://doi.org/10.3390/photonics8060190 - 28 May 2021
Cited by 4 | Viewed by 3666
Abstract
We studied paraxial light beams, obtained by a continuous superposition of off-axis Gaussian beams with their phases chosen so that the whole superposition is invariant to free-space propagation, i.e., does not change its transverse intensity shape. Solving a system of five nonlinear equations [...] Read more.
We studied paraxial light beams, obtained by a continuous superposition of off-axis Gaussian beams with their phases chosen so that the whole superposition is invariant to free-space propagation, i.e., does not change its transverse intensity shape. Solving a system of five nonlinear equations for such superpositions, we obtained an analytical expression for a propagation-invariant off-axis elliptic Gaussian beam. For such an elliptic beam, an analytical expression was derived for the orbital angular momentum, which was shown to consist of two terms. The first one is intrinsic and describes the momentum with respect to the beam center and is shown to grow with the beam ellipticity. The second term depends parabolically on the distance between the beam center and the optical axis (similar to the Steiner theorem in mechanics). It is shown that the ellipse orientation in the transverse plane does not affect the normalized orbital angular momentum. Such elliptic beams can be used in wireless optical communications, since their superpositions do not interfere in space, if they do not interfere in the initial plane. Full article
(This article belongs to the Special Issue Singular Optics)
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21 pages, 4034 KiB  
Article
Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model
by Roman Cherniha, Vasyl’ Davydovych and John R. King
Symmetry 2018, 10(5), 171; https://doi.org/10.3390/sym10050171 - 17 May 2018
Cited by 5 | Viewed by 3535
Abstract
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete [...] Read more.
A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other is not. A complete Lie symmetry classification, including a number of the cases characterised as being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications (tumour growth). Graphical representations of the solutions are provided and a biological interpretation is briefly addressed. The results are generalised on multi-dimensional case under the assumption of the radially symmetrical shape of the tumour. Full article
(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models, II)
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