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Journal articles on the topic 'Nonlinear parabolic and elliptic boundary values problems'

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Published: 10 March 2023
Last updated: 11 March 2023

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1

Hamamuki, Nao, and Qing Liu. "A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 13. http://dx.doi.org/10.1051/cocv/2019076.

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This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.
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2

Shangbin, Cui. "Some comparison and uniqueness theorems for nonlinear elliptic boundary value problems and nonlinear parabolic initial-boundary value problems." Nonlinear Analysis: Theory, Methods & Applications 29, no. 9 (November 1997): 1079–90. http://dx.doi.org/10.1016/s0362-546x(96)00097-1.

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3

Chow, S. S., and R. D. Lazarov. "Superconvergence analysis of flux computations for nonlinear problems." Bulletin of the Australian Mathematical Society 40, no. 3 (December 1989): 465–79. http://dx.doi.org/10.1017/s0004972700017536.

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In this paper we consider the error estimates for some boundary-flux calculation procedures applied to two-point semilinear and strongly nonlinear elliptic boundary value problems. The case of semilinear parabolic problems is also studied. We prove that the computed flux is superconvergent with second and third order of convergence for linear and quadratic elements respectively. Corresponding estimates for higher order elements may also be obtained by following the general line of argument.
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4

SHAKHMUROV, VELI B., and AIDA SAHMUROVA. "Mixed problems for degenerate abstract parabolic equations and applications." Carpathian Journal of Mathematics 34, no. 2 (2018): 247–54. http://dx.doi.org/10.37193/cjm.2018.02.13.

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Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.
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5

Gavrilyuk, I. P. "Approximation of the Operator Exponential and Applications." Computational Methods in Applied Mathematics 7, no. 4 (2007): 294–320. http://dx.doi.org/10.2478/cmam-2007-0019.

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AbstractA review of the exponentially convergent approximations to the operator exponential is given. The applications to inhomogeneous parabolic and elliptic equations, nonlinear parabolic equations, tensor-product approximations of multidimensional solution operators as well as to parabolic problems with time dependent coefficients and boundary conditions are discussed.
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6

Indrei, Emanuel, and Andreas Minne. "Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 33, no. 5 (September 2016): 1259–77. http://dx.doi.org/10.1016/j.anihpc.2015.03.009.

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7

I. Vishik, Mark, and Sergey Zelik. "Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit." Communications on Pure & Applied Analysis 13, no. 5 (2014): 2059–93. http://dx.doi.org/10.3934/cpaa.2014.13.2059.

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8

Zhang, Qi S. "A general blow-up result on nonlinear boundary-value problems on exterior domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 2 (April 2001): 451–75. http://dx.doi.org/10.1017/s0308210500000950.

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In the first part, we study several exterior boundary-value problems covering three types of semilinear equations: elliptic, parabolic and hyperbolic. By a unified approach, we show that these problems share a common critical behaviour. In the second part we prove a blow-up result for an inhomogeneous porous medium equation with the critical exponent, which was left open in a previous paper.
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9

Andreu, F., N. Igbida, J. M. Mazón, and J. Toledo. "Renormalized solutions for degenerate elliptic–parabolic problems with nonlinear dynamical boundary conditions and L1-data." Journal of Differential Equations 244, no. 11 (June 2008): 2764–803. http://dx.doi.org/10.1016/j.jde.2008.02.022.

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10

Galkowski, Jeffrey. "Pseudospectra of semiclassical boundary value problems." Journal of the Institute of Mathematics of Jussieu 14, no. 2 (March 14, 2014): 405–49. http://dx.doi.org/10.1017/s1474748014000061.

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AbstractWe consider operators$ - \Delta + X $, where$ X $is a constant vector field, in a bounded domain, and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter$h$, which should be thought of as the reciprocal of the Péclet constant. This instability is due to the presence of the boundary: just as in the case of$ - \Delta + X $, some of our operators are normal when considered on$\mathbb{R}^d$. We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated with these elliptic operators.
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11

Lipitakis, Anastasia-Dimitra. "The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning Approach." Modelling and Simulation in Engineering 2019 (January 2, 2019): 1–9. http://dx.doi.org/10.1155/2019/5157145.

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Critical comments on the complexity of computational systems and the basic singularly perturbed (SP) concepts are given. A class of several complex SP nonlinear elliptic equations arising in various branches of science, technology, and engineering is presented. A classification of complex SP nonlinear PDEs with characteristic boundary value problems is described. A modified explicit preconditioned conjugate gradient method based on explicit inverse preconditioners is presented. The numerical solution of a characteristic 3D SP nonlinear parabolic model is analytically given and numerical results for several model problems are presented demonstrating both applicability and efficiency of the new computational methods.
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12

Rihova-Skabrahova, Dana. "Linear Scheme for Finite Element Solution of Nonlinear Parabolic-Elliptic Problems with Nonhomogeneous Dirichlet Boundary Condition." Applications of Mathematics 46, no. 2 (April 2001): 103–44. http://dx.doi.org/10.1023/a:1013783722140.

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13

Lieberman, Gary A. "On The Hölder Gradient Estimate For Solutions Of Nonlinear Elliptic And Parabolic Oblique Boundary Value Problems." Communications in Partial Differential Equations 15, no. 4 (January 1990): 515–23. http://dx.doi.org/10.1080/03605309908820696.

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14

Nefedov, N. N. "Development of Methods of Asymptotic Analysis of Transition Layers in Reaction–Diffusion–Advection Equations: Theory and Applications." Computational Mathematics and Mathematical Physics 61, no. 12 (December 2021): 2068–87. http://dx.doi.org/10.1134/s0965542521120095.

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Abstract This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.
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15

Zhang, Junjie, Shenzhou Zheng, and Chunyan Zuo. "$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values." Discrete & Continuous Dynamical Systems - S 14, no. 9 (2021): 3305. http://dx.doi.org/10.3934/dcdss.2021080.

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<p style='text-indent:20px;'>We prove a global <inline-formula><tex-math id="M1">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear elliptic equations <inline-formula><tex-math id="M2">\begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M3">\begin{document}$ C^{2, \alpha} $\end{document}</tex-math></inline-formula>-domain for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\in (0, 1) $\end{document}</tex-math></inline-formula>. Here, the nonlinearities <inline-formula><tex-math id="M5">\begin{document}$ F $\end{document}</tex-math></inline-formula> is assumed to be asymptotically <inline-formula><tex-math id="M6">\begin{document}$ \delta $\end{document}</tex-math></inline-formula>-regular to an operator <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula> that is <inline-formula><tex-math id="M8">\begin{document}$ (\delta, R) $\end{document}</tex-math></inline-formula>-vanishing with respect to <inline-formula><tex-math id="M9">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global <inline-formula><tex-math id="M10">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear parabolic equations <inline-formula><tex-math id="M11">\begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M12">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>-domain.</p>
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16

Gwinner, J. "Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/108043.

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The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.
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17

Abdullayev, Akmaljon, Kholsaid Kholturayev, and Nigora Safarbayeva. "Exact method to solve of linear heat transfer problems." E3S Web of Conferences 264 (2021): 02059. http://dx.doi.org/10.1051/e3sconf/202126402059.

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When approximating multidimensional partial differential equations, the values of the grid functions from neighboring layers are taken from the previous time layer or approximation. As a result, along with the approximation discrepancy, an additional discrepancy of the numerical solution is formed. To reduce this discrepancy when solving a stationary elliptic equation, parabolization is carried out, and the resulting equation is solved by the method of successive approximations. This discrepancy is eliminated in the approximate analytical method proposed below for solving two-dimensional equations of parabolic and elliptic types, and an exact solution of the system of finite difference equations for a fixed time is obtained. To solve problems with a boundary condition of the first kind on the first coordinate and arbitrary combinations of the first, second and third kinds of boundary conditions on the second coordinate, it is proposed to use the method of straight lines on the first coordinate and ordinary sweep method on the second coordinate. Approximating the equations on the first coordinate, a matrix equation is built relative to the grid functions. Using eigenvalues and vectors of the three-diagonal transition matrix, linear combinations of grid functions are compiled, where the coefficients are the elements of the eigenvectors of the three-diagonal transition matrix. Boundary conditions, and for a parabolic equation, initial conditions are formed for a given combination of grid functions. The resulting one-dimensional differential-difference equations are solved by the ordinary sweep method. From the resulting solution, proceed to the initial grid functions. The method provides a second order of approximation accuracy on coordinates. And the approximation accuracy in time when solving the parabolic equation can be increased to the second order using the central difference in time. The method is used to solve heat transfer problems when the boundary conditions are expressed by smooth and discontinuous functions of a stationary and non-stationary nature, and the right-hand side of the equation represents a moving source or outflow of heat.
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18

KASCHENKO, S. A. "BIFURCATIONAL FEATURES IN SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS WITH WEAK DIFFUSION." International Journal of Bifurcation and Chaos 15, no. 11 (November 2005): 3595–606. http://dx.doi.org/10.1142/s0218127405014258.

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Asymptotic solutions of parabolic boundary value problems are studied in a neighborhood of both an equilibrium state and a cycle in near-critical cases which can be considered as infinite-dimensional due to small values of the diffusion coefficients. Algorithms are developed to construct normalized equations in such situations. Principle difference between bifurcations in two-dimensional and one-dimensional spatial systems is demonstrated.
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19

Da Silva, Edcarlos Domingos, and Francisco Odair De Paiva. "Landesman-lazer type conditions and multiplicity results for nonlinear elliptic problems with neumann boundary values." Acta Mathematica Sinica, English Series 30, no. 2 (January 15, 2014): 229–50. http://dx.doi.org/10.1007/s10114-014-2750-2.

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20

Koshanov, B. D., N. Kakharman, R. U. Segizbayeva, and Zh B. Sultangaziyeva. "Two theorems on estimates for solutions of one class of nonlinear equations in a finite-dimensional space." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 107, no. 3 (September 30, 2022): 70–84. http://dx.doi.org/10.31489/2022m3/70-84.

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The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory and quantum physics. In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items. In the first subsection, the notation used and the statement of the main results are given. In the second subsection, the main lemmas are given. The third section is devoted to the proof of Theorem 1. In the fourth section, Theorem 2 is proved. The conditions of the theorems are such that they can be used in studying a certain class of initial-boundary value problems to obtain strong a priori estimates in the presence of weak a priori estimates. This is the meaning of these theorems.
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21

Shan, Maria. "Keller-Osserman a priori estimates for doubly nonlinear anisotropic parabolic equations with absorption term." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 149–59. http://dx.doi.org/10.37069/1683-4720-2018-32-15.

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We are concerned with divergence type quasilinear parabolic equation with measurable coefficients and lower order terms model of which is a doubly nonlinear anisotropic parabolic equations with absorption term. This class of equations has numerous applications which appear in modeling of electrorheological fluids, image precessing, theory of elasticity, theory of non-Newtonian fluids with viscosity depending on the temperature. But the qualitative theory doesn't construct for these anisotropic equations. So, naturally, that during the last decade there has been growing substantial development in the qualitative theory of second order anisotropic elliptic and parabolic equations. The main purpose is to obtain the pointwise upper estimates in terms of distance to the boundary for nonnegative solutions of such equations. This type of estimates originate from the work of J. B. Keller, R. Osserman, who obtained a simple upper bound for any solution, in any number of variables for Laplace equation. These estimates play a crucial role in the theory of existence or nonexistence of so called large solutions of such equations, in the problems of removable singularities for solutions to elliptic and parabolic equations. Up to our knowledge all the known estimates for large solutions to elliptic and parabolic equations are related with equations for which some comparison properties hold. We refer to I.I. Skrypnik, A.E. Shishkov, M. Marcus , L. Veron, V.D. Radulescu for an account of these results and references therein. Such equations have been the object of very few works because in general such properties do not hold. The main ones concern equations only in the precise choice of absorption term \(f(u)=u^q\). Among the people who published significative results in this direction are I.I. Skrypnik, J. Vetois, F.C. Cirstea, J. Garcia-Melian, J.D. Rossi, J.C. Sabina de Lis. The main result of the paper is a priori estimates of Keller-Osserman type for nonnegative solutions of a doubly nonlinear anisotropic parabolic equations with absorption term that have been proven despite of the lack of comparison principle. To obtain these estimates we exploit the method of energy estimations and De Giorgy iteration techniques.
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22

Nkashama, M. N., and S. B. Robinson. "Resonance and non-resonance in terms of average values. II." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 5 (October 2001): 1217–35. http://dx.doi.org/10.1017/s0308210500001359.

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We prove existence results for semilinear elliptic boundary-value problems in both the resonance and non-resonance cases. What sets our results apart is that we impose sufficient conditions for solvability in terms of the (asymptotic) average values of the nonlinearities, thus allowing the nonlinear term to have significant oscillations outside the given spectral gap as long as it remains within the interval on the average in some sense. This work generalizes the results of a previous paper, which dealt exclusively with the ordinary differential equation (ODE) case and relied on ODE techniques.
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23

Bokalo, Mykola, Oleh Buhrii, and Nikolyetta Hryadil. "Initial–boundary value problems for nonlinear elliptic–parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity." Nonlinear Analysis 192 (March 2020): 111700. http://dx.doi.org/10.1016/j.na.2019.111700.

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24

Warma, Mahamadi. "Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains." Nonlinear Analysis: Theory, Methods & Applications 75, no. 14 (September 2012): 5561–88. http://dx.doi.org/10.1016/j.na.2012.05.004.

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25

Turkyilmazoglu, Mustafa. "Solution of Initial and Boundary Value Problems by an Effective Accurate Method." International Journal of Computational Methods 14, no. 06 (August 2017): 1750069. http://dx.doi.org/10.1142/s0219876217500694.

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The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M. [2013] “Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type,” Appl. Math. Mod. 37, 7539–7548; Turkyilmazoglu, M. [2014] “An effective approach for numerical solutions of high-order Fredholm integro-differential equations,” Appl. Math. Comput. 227, 384–398; Turkyilmazoglu, M. [2015] “Parabolic partial differential equations with nonlocal initial and boundary values,” Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.
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26

Devaud, Denis. "Petrov–Galerkin space-time hp-approximation of parabolic equations in H1/2." IMA Journal of Numerical Analysis 40, no. 4 (October 16, 2019): 2717–45. http://dx.doi.org/10.1093/imanum/drz036.

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Abstract We analyse a class of variational space-time discretizations for a broad class of initial boundary value problems for linear, parabolic evolution equations. The space-time variational formulation is based on fractional Sobolev spaces of order $1/2$ and the Riemann–Liouville derivative of order $1/2$ with respect to the temporal variable. It accommodates general, conforming space discretizations and naturally accommodates discretization of infinite horizon evolution problems. We prove an inf-sup condition for $hp$-time semidiscretizations with an explicit expression of stable test functions given in terms of Hilbert transforms of the corresponding trial functions; inf-sup constants are independent of temporal order and the time-step sequences, allowing quasi-optimal, high-order discretizations on graded time-step sequences, and also $hp$-time discretizations. For solutions exhibiting Gevrey regularity in time and taking values in certain weighted Bochner spaces, we establish novel exponential convergence estimates in terms of $N_t$, the number of (elliptic) spatial problems to be solved. The space-time variational setting allows general space discretizations and, in particular, for spatial $hp$-FEM discretizations. We report numerical tests of the method for model problems in one space dimension with typical singular solutions in the spatial and temporal variable. $hp$-discretizations in both spatial and temporal variables are used without any loss of stability, resulting in overall exponential convergence of the space-time discretization.
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27

ANDREU, FUENSANTA, JOSÉ M. MAZÓN, and MIRCEA SOFONEA. "ENTROPY SOLUTIONS IN THE STUDY OF ANTIPLANE SHEAR DEFORMATIONS FOR ELASTIC SOLIDS." Mathematical Models and Methods in Applied Sciences 10, no. 01 (February 2000): 99–126. http://dx.doi.org/10.1142/s0218202500000082.

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The concept of entropy solution was recently introduced in the study of Dirichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the method of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal antiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic constitutive law is physically nonlinear and the friction is described by a static law. We present a variational formulation of the model and prove the existence and the uniqueness of a weak solution in the case when the body forces and the prescribed surface tractions have the regularity L∞. The proof is based on classical results for elliptic variational inequalities and measure theory arguments. We also define the concept of entropy solution and we prove an existence and uniqueness result in the case when the body forces and the surface tractions have the regularity L1. The proof is based on properties of the trace operators for functions which are not in Sobolev spaces. Finally, we present a regularity result for the entropy solution and we give some concrete examples and mechanical interpretation.
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28

Vyas, Prashant Dineshbhai, Harish C. Thakur, and Veera P. Darji. "Nonlinear analysis of convective-radiative longitudinal fin of various profiles." International Journal of Numerical Methods for Heat & Fluid Flow 30, no. 6 (May 29, 2019): 3065–82. http://dx.doi.org/10.1108/hff-08-2018-0444.

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Purpose This paper aims to study nonlinear heat transfer through a longitudinal fin of three different profiles. Design/methodology/approach A truly meshfree method is used to undertake a nonlinear analysis to predict temperature distribution and heat-transfer rate. Findings A longitudinal fin of three different profiles, such as rectangular, triangular and concave parabolic, are analyzed. Temperature variation, along with the fin length and rate of heat transfer in steady state, under convective and convective-radiative environments has been demonstrated and explained. Moving least square (MLS) approximants are used to approximate the unknown function of temperature T(x) with Th(x). Essential boundary conditions are imposed using the penalty method. An iterative predictor–corrector scheme is used to handle nonlinearity. Research limitations/implications Modelling fin in a convective-radiative environment removes the assumption of no radiation condition. It also allows to vary convective heat-transfer coefficient and predict the closer values to the real problems for the corresponding fin surfaces. Originality/value The meshless local Petrov–Galerkin method can solve nonlinear fin problems and predict an accurate solution.
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29

Teirumnieka, Ērika, Ilmārs Kangro, Edmunds Teirumnieks, and Harijs Kalis. "The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers." Environment. Technology. Resources. Proceedings of the International Scientific and Practical Conference 3 (June 16, 2015): 186. http://dx.doi.org/10.17770/etr2015vol3.618.

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<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>
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30

Rotko, V. V. "INVERSE PROBLEMS FOR MATHEMATICAL MODELS WITH THE POINTWISE OVERDETERMINATION." Yugra State University Bulletin 14, no. 3 (December 15, 2018): 57–66. http://dx.doi.org/10.17816/byusu2018057-66.

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In the article we examine well-posedness questions in the Sobolev spaces of an inverse source problem in the case of a quasilinear parabolic system of the second order. These problem arise when describing heat and mass transfer, diffusion, filtration, and in many other fields. The main part of the operator is linear. The unknowns occur in the nonlinear right-hand side. In particular, this class of problems includes the coefficient inverse problems on determinations of the lower order coefficients in a parabolic equation or a system. The overdetermination conditions are the values of a solution at some collection of points lying inside the spacial domain. The Dirichlet and oblique derivative problems under consideration. The problems are studied in a bounded domain with smooth boundary. However, the results can be generalized to the case of unbounded domains as well for which the corresponding solvability theorems hold. The conditions ensuring local (in time) well-posedness of the problem in the Sobolev classes are exposed. The conditions on the data are minimal. The results are sharp. The problem is reduced to an operator equation whose solvability is proven with the use of a priori bounds and the fixed point theorem. A solution possesses all generalize derivatives occurring in the system which belong to the space with and some additional necessary smoothness in some neighborhood about the overdetermination points.
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31

Crochet, Marcel J. "Numerical Simulation of Viscoelastic Flow: A Review." Rubber Chemistry and Technology 62, no. 3 (July 1, 1989): 426–55. http://dx.doi.org/10.5254/1.3536253.

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Abstract It is evident that a major effort has been accomplished over the last ten years toward the development of numerical methods for solving viscoelatic flow. The problem was clearly much harder than expected. Several extensive reviews have been devoted to a detailed account of the difficulties encountered in reaching moderate values of the Weissenberg number. The numerical and analytical work undertaken in parallel by several research groups has led to some important conclusions which paved the way for recent promising development. First, numerical algorithms for solving highly nonlinear advective problems must be used with the greatest carefulness. When spurious solutions or unexpected effects such as limit points arise in numerical simulations, we have learned to question the validity of the numerical method as well as that of the constitutive equation. Typically, successive failures of numerical calculations with the Maxwell model at low values of We have often been attributed to its singular behavior in uniaxial elongational flow but, in the meantime, better adapted algorithms have led to solutions at ever increasing values of We. Secondly, the mathematical analysis of the partial differential equations governing the flow of viscoelastic fluids has revealed the possibility of changes of type of the vorticity equation under some circumstances, i.e., when the velocity of the fluid becomes comparable with the velocity of shear waves. The coexistence of hyperbolic and elliptic regions in a steady flow may be of great importance in explaining a number of experimental observations. Simultaneously, the analysis has led to the identification of artificial changes of type which partly explain some numerical failures, or at least give a pertinent diagnosis of numerical inaccuracy. Thirdly, it has been found that numerical algorithms must take into account the specific features of viscoelastic flow; among these, stress boundary layers, stress singularities, and advective (or memory) terms in the constitutive equations are prominent.
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32

Sumin, Vladimir I. "Volterra funktional equations in the stability problem for the existence of global solutions of distributed controlled systems." Russian Universities Reports. Mathematics, no. 132 (2020): 422–40. http://dx.doi.org/10.20310/2686-9667-2020-25-132-422-440.

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Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z(t)=f(t,A[z](t),v(t) ), t≡{t^1,⋯,t^n }∈Π⊂R^n, z∈L_p^m≡(L_p (Π) )^m, where f(.,.,.):Π×R^l×R^s→R^m; v(.)∈D⊂L_k^s – control function; A:L_p^m (Π)→L_q^l (Π)- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H∈T, the restriction A├ [z]┤|_H does not depend on the values of ├ z┤|_(Π\H); (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H∈T). This continuation is realized with the help of the chain {H_1⊂H_2⊂⋯⊂H_(k-1)⊂H_k≡Π}, where H_i∈T, i=¯(1,k.) A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p=q=k=∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F_v [z(.) ](t)≡f(t,A[z](t),v(t)) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L_∞^m (H) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p,q,k ≤∞, (this case covers a much wider class of CIBVP), the operator F_v; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L_p^m (H), for which the operator F_v is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.
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33

Berselli, Luigi C., and Michael Růžička. "Space–time discretization for nonlinear parabolic systems with p-structure." IMA Journal of Numerical Analysis, December 23, 2020. http://dx.doi.org/10.1093/imanum/draa079.

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Abstract In this paper we consider nonlinear parabolic systems with elliptic part, depending only on the symmetric gradient, which can be also degenerate. We prove optimal error estimates for solutions with natural regularity. The main novelty, with respect to previous results, is that we obtain the estimates directly without introducing intermediate semidiscrete problems, which enables the treatment of homogeneous Dirichlet boundary conditions. In addition, we prove the existence of solutions of the continuous problem with the requested regularity, if the data of the problem are smooth enough.
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34

Dancer, E. N. "Stable and Finite Morse Index Solutions for Dirichlet Problems with Small Diffusion in a Degenerate Case and Problems with Infinite Boundary Values." Advanced Nonlinear Studies 9, no. 4 (January 1, 2009). http://dx.doi.org/10.1515/ans-2009-0405.

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AbstractWe consider weakly nonlinear elliptic equations with small diffusion in the case where the nonlinearity has a non-nodal zero. We show that there is an unexpected connection with problems with infinite boundary values.
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35

Yang, Guangchong, and Yanqiu Chen. "Nonnegative Solutions of a Nonlinear System and Applications to Elliptic BVPs*." Journal of Applied Mathematics & Bioinformatics, August 31, 2021, 15–26. http://dx.doi.org/10.47260/jamb/1122.

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Abstract In this communication, we study the existence of nonnegative solutions of a nonlinear system in Banach spaces. These maps involved in the system defined on cone do not necessarily take values in the cone. Using fixed point theorems just established for this type of mappings, nonnegative solutions of the system are obtained and used to investigate elliptic boundary value problems (BVPs). MSC(2010): 47H10, 35J57. Keywords: Nonlinear system, Nonnegative solutions, Nowhere normal-outward maps, Fixed point, Elliptic BVPs.
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36

Balázsová, Monika, Miloslav Feistauer, and Anna-Margarete Sändig. "Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder." Journal of Numerical Mathematics, June 25, 2022. http://dx.doi.org/10.1515/jnma-2021-0113.

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Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.
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37

Baraket, Sami, and Vicenţiu D. Rădulescu. "Combined Effects of Concave-Convex Nonlinearities in a Fourth-Order Problem with Variable Exponent." Advanced Nonlinear Studies 16, no. 3 (January 1, 2016). http://dx.doi.org/10.1515/ans-2015-5032.

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AbstractWe study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes the existence of a nontrivial weak solution in the case of a small perturbation of the right-hand side. The proof combines variational methods, including the Ekeland variational principle and the mountain pass theorem of Ambrosetti and Rabinowitz. Next we consider a very related eigenvalue problem and we prove the existence of nontrivial weak solutions for large values of the parameter. The direct method of the calculus of variations, estimates of the levels of the associated energy functional and basic properties of the Lebesgue and Sobolev spaces with variable exponent have an important role in our arguments.
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38

Baber, Muhammad Z., Aly R. Seadway, Nauman Ahmed, Muhammad S. Iqbal, and Muhammad W. Yasin. "Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross–Pitaevskii equation in dispersive media." International Journal of Modern Physics B, December 21, 2022. http://dx.doi.org/10.1142/s0217979223501916.

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In this study, the Gross–Pitaevskii equation perturbed with multiplicative time noise is under consideration numerically and analytically. The NLSE is a universal governing model that helps in evolution of complex fields that are used in dispersive media. For the numerical solution, we used the stochastic forward Euler (SFE) scheme. To find the exact solutions, we chose the techniques namely [Formula: see text]-model expansion. For the analysis of the proposed scheme, we checked the stability of the scheme with the help of Von-Neumann criteria and the consistency of the scheme with the mean of Ito’s sense. The exact solutions of the model are constructed successfully in the Jacobi elliptic function in the form of trigonometric and hyperbolic functions. Last, we compared the graphical behavior of the proposed scheme with some exact solutions by using the unique selection of initial and boundary conditions. The plots are constructed in the form of 3D, line, and contour representation by choosing the different values of parameters.
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39

Baber, Muhammad Z., Aly R. Seadway, Nauman Ahmed, Muhammad S. Iqbal, and Muhammad W. Yasin. "Selection of solitons coinciding the numerical solutions for uniquely solvable physical problems: A comparative study for the nonlinear stochastic Gross–Pitaevskii equation in dispersive media." International Journal of Modern Physics B, December 21, 2022. http://dx.doi.org/10.1142/s0217979223501919.

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In this study, the Gross–Pitaevskii equation perturbed with multiplicative time noise is under consideration numerically and analytically. The NLSE is a universal governing model that helps in evolution of complex fields that are used in dispersive media. For the numerical solution, we used the stochastic forward Euler (SFE) scheme. To find the exact solutions, we chose the techniques namely [Formula: see text]-model expansion. For the analysis of the proposed scheme, we checked the stability of the scheme with the help of Von-Neumann criteria and the consistency of the scheme with the mean of Ito’s sense. The exact solutions of the model are constructed successfully in the Jacobi elliptic function in the form of trigonometric and hyperbolic functions. Last, we compared the graphical behavior of the proposed scheme with some exact solutions by using the unique selection of initial and boundary conditions. The plots are constructed in the form of 3D, line, and contour representation by choosing the different values of parameters.
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