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Journal articles on the topic 'Fully nonlinear equation'

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Author: Grafiati

Published: 11 March 2023

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1

Trudinger, Neil S. "On degenerate fully nonlinear elliptic equations in balls." Bulletin of the Australian Mathematical Society 35, no. 2 (April 1987): 299–307. http://dx.doi.org/10.1017/s0004972700013253.

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We establish derivative estimates and existence theorems for the Dirichlet and Neumann problems for nonlinear, degenerate elliptic equations of the form F (D2u) = g in balls. The degeneracy arises through the possible vanishing of the function g and the degenerate Monge-Ampère equation is covered as a special case.

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2

Zhang, Hong-sheng, Hua-wei Zhou, Guang-wen Hong, and Jian-min Yang. "A FULLY NONLINEAR BOUSSINESQ MODEL FOR WATER WAVE PROPAGATION." Coastal Engineering Proceedings 1, no. 32 (January 31, 2011): 12. http://dx.doi.org/10.9753/icce.v32.waves.12.

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A set of high-order fully nonlinear Boussinesq-type equations is derived from the Laplace equation and the nonlinear boundary conditions. The derived equations include the dissipation terms and fully satisfy the sea bed boundary condition. The equations with the linear dispersion accurate up to [2,2] padé approximation is qualitatively and quantitatively studied in details. A numerical model for wave propagation is developed with the use of iterative Crank-Nicolson scheme, and the two-dimensional fourth-order filter formula is also derived. With two test cases numerically simulated, the modeled results of the fully nonlinear version of the numerical model are compared to those of the weakly nonlinear version.

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3

Ivanov, S. K., and A. M. Kamchatnov. "WAVE PULSE EVOLUTION FOR FULLY NONLINEAR SERRE EQUATION." XXII workshop of the Council of nonlinear dynamics of the Russian Academy of Sciences 47, no. 1 (April 30, 2019): 58–60. http://dx.doi.org/10.29006/1564-2291.jor-2019.47(1).15.

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Although the shallow-water theory is a classical subject of investigations with a huge number of papers devoted to it, it still remains very active field of research with many important applications. When one neglects dissipation effects and non-uniformity of the basin’s bottom, the interplay of nonlinearity and dispersion effects leads to quite complicated wave patterns which form depends crucially on the initial profile of the pulse. If the nonlinearity and dispersion effects are taken into account in the lowest approximation and one considers a one-directional propagation of the wave, then its dynamics is governed by the famous Korteweg-de Vries (KdV) equation. Comparison with experiments shows that the KdV approximation is not good enough and one needs to go beyond it. Therefore considerable efforts were directed to the derivation of the corresponding wave equation that was able to better describe the system. One of the most popular models was first suggested and studied in much detail by Serre (Serre, 1953). For such a model, in which evolution is described by the Serre (Su-Gardner, Green- Naghdi) equation, El made an important study of the law of conservation of the “number of waves” and its soliton analogue (El, 2006). Using El’s method one can find the laws of motion of the edges of the dispersive shock waves (DSW) in problems related with self-similar evolution of step-like initial discontinuities. In (Kamchatnov, 2018) these methods were shown that allow one to go beyond such an initial profile. In this report, we will show the application of the methods of this work to study of simple wave initial pulses evolution in the theory of the Serre equations and give an analytical solution for the laws of motion of edges of DSW formed in the process of evolution of the initial pulses. Analytical results are confirmed by numerical calculations. The reported study was funded by RFBR according to the research project №19- 01-00178 А.

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4

Dunphy, M., C. Subich, and M. Stastna. "Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves." Nonlinear Processes in Geophysics 18, no. 3 (June 14, 2011): 351–58. http://dx.doi.org/10.5194/npg-18-351-2011.

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Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

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5

Trudinger, Neil S. "Hölder gradient estimates for fully nonlinear elliptic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 108, no. 1-2 (1988): 57–65. http://dx.doi.org/10.1017/s0308210500026512.

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SynopsisIn this paper we prove interior and global Hölder estimates for Lipschitz viscosity solutions of second order, nonlinear, uniformly elliptic equations. The smoothness hypotheses on the operators are more general than previously considered for classical solutions, so that our estimates are also new in this case and readily extend to embrace obstacle problems. In particular Isaac's equations of stochastic differential game theory constitute a special case of our results, and moreover our techniques, in combination with recent existence theorems of Ishii, lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic Isaac's equation.

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6

CHOI, WOOYOUNG, and ROBERTO CAMASSA. "Fully nonlinear internal waves in a two-fluid system." Journal of Fluid Mechanics 396 (October 10, 1999): 1–36. http://dx.doi.org/10.1017/s0022112099005820.

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Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

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7

Akagi, Goro. "Local solvability of a fully nonlinear parabolic equation." Kodai Mathematical Journal 37, no. 3 (October 2014): 702–27. http://dx.doi.org/10.2996/kmj/1414674617.

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8

Lee, H. Y. "Fully discrete methods for the nonlinear Schrödinger equation." Computers & Mathematics with Applications 28, no. 6 (September 1994): 9–24. http://dx.doi.org/10.1016/0898-1221(94)00148-0.

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9

Tam, Luen-Fai, and Tom Yau-Heng Wan. "A fully nonlinear equation in relativistic Teichmüller theory." International Journal of Mathematics 30, no. 13 (December 2019): 1940004. http://dx.doi.org/10.1142/s0129167x19400044.

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We obtain some basic estimates for a Monge–Ampère type equation introduced by Moncrief in the study of the Relativistic Teichmüller Theory. We then give another proof of the parametrization of the Teichmüller space obtained by Moncrief. Our approach provides yet another proof of the classical Teichmüller theorem that the Teichmüller space of a compact oriented surface of genus [Formula: see text] is diffeomorphic to the disk of dimension [Formula: see text]. We also give another proof of properness of a certain energy function on the Teichmüller space.

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10

Chernitskii, Alexander A. "Born-infeld electrodynamics: Clifford number and spinor representations." International Journal of Mathematics and Mathematical Sciences 31, no. 2 (2002): 77–84. http://dx.doi.org/10.1155/s016117120210620x.

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The Clifford number formalism for Maxwell equations is considered. The Clifford imaginary unit for space-time is introduced as coordinate independent form of fully antisymmetric fourth-rank tensor. The representation of Maxwell equations in massless Dirac equation form is considered; we also consider two approaches to the invariance of Dirac equation with respect to the Lorentz transformations. According to the first approach, the unknown column is invariant and according to the second approach it has the transformation properties known as spinorial ones. The Clifford number representation for nonlinear electrodynamics equations is obtained. From this representation, we obtain the nonlinear like Dirac equation which is the form of nonlinear electrodynamics equations. As a special case we have the appropriate representations for Born-Infeld nonlinear electrodynamics.

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11

Zakaria, La, Wahyu Megarani, Ahmad Faisol, Aang Nuryaman, and Ulfah Muharramah. "Computational Mathematics: Solving Dual Fully Fuzzy Nonlinear Matrix Equations Numerically using Broyden’s Method." International Journal of Mathematical, Engineering and Management Sciences 8, no. 1 (February 1, 2023): 60–77. http://dx.doi.org/10.33889/ijmems.2023.8.1.004.

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Fuzzy numbers have many applications in various mathematical models in both linear and nonlinear forms. In the form of a nonlinear system, fuzzy nonlinear equations can be constructed in the form of matrix equations. Unfortunately, the matrix equations used to solve the problem of dual fully fuzzy nonlinear systems are still relatively few found in the publication of research results. This article attempts to solve a dual fully fuzzy nonlinear equation system involving triangular fuzzy numbers using Broyden’s method. This article provides the pseudocode algorithm and the implementation of the algorithm into the MATLAB program for the iteration process to be carried out quickly and easily. The performance of the given algorithm is the fastest in finding system solutions and provides a minimum error value.

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12

Kachulin, Dmitry, Alexander Dyachenko, and Vladimir Zakharov. "Soliton Turbulence in Approximate and Exact Models for Deep Water Waves." Fluids 5, no. 2 (May 10, 2020): 67. http://dx.doi.org/10.3390/fluids5020067.

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We investigate and compare soliton turbulence appearing as a result of modulational instability of the homogeneous wave train in three nonlinear models for surface gravity waves: the nonlinear Schrödinger equation, the super compact Zakharov equation, and the fully nonlinear equations written in conformal variables. We show that even at a low level of energy and average wave steepness, the wave dynamics in the nonlinear Schrödinger equation fundamentally differ from the dynamics in more accurate models. We study energy losses of wind waves due to their breaking for large values of total energy in the super compact Zakharov equation and in the exact equations and show that in both models, the wave system loses 50% of energy very slowly, during few days.

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13

STOCKER, J. R., and D. H. PEREGRINE. "The current-modified nonlinear Schrödinger equation." Journal of Fluid Mechanics 399 (November 25, 1999): 335–53. http://dx.doi.org/10.1017/s0022112099006618.

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By comparison with both experimental and numerical data, Dysthe's (1979) O(ε4) modified nonlinear Schrödinger; equation has been shown to model the evolution of a slowly varying wavetrain well (here ε is the wave steepness). In this work, we extend the equation to include a prescribed, large-scale, O(ε2) surface current which varies about a mean value. As an introduction, a heuristic derivation of the O(ε3) current-modified equation, used by Bakhanov et al. (1996), is given, before a more formal approach is used to derive the O(ε4) equation. Numerical solutions of the new equations are compared in one horizontal dimension with those from a fully nonlinear solver for velocity potential in the specific case of a sinusoidal surface current, such as may be due to an underlying internal wave. The comparisons are encouraging, especially for the O(ε4) equation.

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14

Gao, T., Z. Wang, and P. A. Milewski. "Nonlinear hydroelastic waves on a linear shear current at finite depth." Journal of Fluid Mechanics 876 (July 31, 2019): 55–86. http://dx.doi.org/10.1017/jfm.2019.528.

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This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

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15

Emmrich, Etienne, and David Šiška. "Full discretisation of second-order nonlinear evolution equations: strong convergence and applications." Computational Methods in Applied Mathematics 11, no. 4 (2011): 441–59. http://dx.doi.org/10.2478/cmam-2011-0025.

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Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

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16

Wu, Ruili, and Junyan Li. "Boundary value problem for a fully nonlinear elliptic equation." Journal of Physics: Conference Series 1978, no. 1 (July 1, 2021): 012028. http://dx.doi.org/10.1088/1742-6596/1978/1/012028.

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17

Huang, Nanjing. "Existence of periodic solutions for fully nonlinear wave equation." Applicable Analysis 60, no. 3-4 (April 1996): 321–26. http://dx.doi.org/10.1080/00036819608840435.

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18

Broadbridge, Philip, and Joanna M. Goard. "Exact solution of a degenerate fully nonlinear diffusion equation." Zeitschrift für angewandte Mathematik und Physik 55, no. 3 (May 2004): 534–38. http://dx.doi.org/10.1007/s00033-004-3015-1.

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19

DEBSARMA, SUMA, K. P. DAS, and JAMES T. KIRBY. "Fully nonlinear higher-order model equations for long internal waves in a two-fluid system." Journal of Fluid Mechanics 654 (May 11, 2010): 281–303. http://dx.doi.org/10.1017/s0022112010000601.

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Fully nonlinear model equations, including dispersive effects at one-order higher approximation than in the model of Choi & Camassa (J. Fluid Mech., vol. 396, 1999, pp. 1–36), are derived for long internal waves propagating in two spatial horizontal dimensions in a two-fluid system, where the lower layer is of infinite depth. The model equations consist of two coupled equations for the displacement of the interface and the horizontal velocity of the upper layer at an arbitrary elevation, and they are correct to O(μ2) terms, where μ is the ratio of thickness of the upper-layer fluid to a typical wavelength. For solitary waves propagating in one horizontal direction, the two coupled equations reduce to a single equation for the elevation of the interface. Solitary wave profiles obtained numerically from this equation for different wave speeds are in good agreement with computational results based on Euler's equations. A numerical approach for the propagation of solitary waves is provided in the weakly nonlinear case.

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20

Huang, Rongli, and Yunhua Ye. "On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I." International Mathematics Research Notices 2019, no. 18 (November 13, 2017): 5539–76. http://dx.doi.org/10.1093/imrn/rnx278.

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Abstract In this article, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solutions of the fully nonlinear parabolic equations and prove the long time existence and convergence of the flow. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs in Euclidean and pseudo-Euclidean space.

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21

Antontsev, Stanislav, and Sergey Shmarev. "On a class of fully nonlinear parabolic equations." Advances in Nonlinear Analysis 8, no. 1 (November 23, 2016): 79–100. http://dx.doi.org/10.1515/anona-2016-0055.

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Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points where {\Delta u=0} , the equation degenerates if {m>2} , or becomes singular if {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as {t\to\infty} . Sufficient conditions for exponential or power decay of {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

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22

Castro, Angel, and David Lannes. "Fully nonlinear long-wave models in the presence of vorticity." Journal of Fluid Mechanics 759 (October 27, 2014): 642–75. http://dx.doi.org/10.1017/jfm.2014.593.

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AbstractWe study here Green–Naghdi type equations (also called fully nonlinear Boussinesq, or Serre equations) modelling the propagation of large-amplitude waves in shallow water without a smallness assumption on the amplitude of the waves. The novelty here is that we allow for a general vorticity, thereby allowing complex interactions between surface waves and currents. We show that the a priori ($2+1$)-dimensional dynamics of the vorticity can be reduced to a finite cascade of two-dimensional equations. With a mechanism reminiscent of turbulence theory, vorticity effects contribute to the averaged momentum equation through a Reynolds-like tensor that can be determined by a cascade of equations. Closure is obtained at the precision of the model at the second order of this cascade. We also show how to reconstruct the velocity field in the ($2+1$)-dimensional fluid domain from this set of two-dimensional equations and exhibit transfer mechanisms between the horizontal and vertical components of the vorticity, thus opening perspectives for the study of rip currents, for instance.

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23

Fadimba, Koffi B. "Error Analysis for a Galerkin Finite Element Method Applied to a Coupled Nonlinear Degenerate System of Advection-diffusion Equations." Computational Methods in Applied Mathematics 6, no. 1 (2006): 3–30. http://dx.doi.org/10.2478/cmam-2006-0001.

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AbstractWe consider a standard Galerkin Method applied to both the pressure equation and the saturation equation of a coupled nonlinear system of degenerate advection-diffusion equations modeling a two-phase immiscible flow through porous media. After regularizing the problem and establishing some regularity results, we derive error estimates for a semi-discretized Galerkin Method. A decoupled nonlinear scheme is then proposed for a fully discretized (backward in time) Galerkin Method, and error estimates are derived for that method. We also prove the existence and uniqueness for the nonlinear operator intervening in the backward time discretization.

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24

PĂRĂU, E. I., J. M. VANDEN-BROECK, and M. J. COOKER. "Nonlinear three-dimensional interfacial flows with a free surface." Journal of Fluid Mechanics 591 (October 30, 2007): 481–94. http://dx.doi.org/10.1017/s0022112007008452.

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A configuration consisting of two superposed fluids bounded above by a free surface is considered. Steady three-dimensional potential solutions generated by a moving pressure distribution are computed. The pressure can be applied either on the interface or on the free surface. Solutions of the fully nonlinear equations are calculated by boundary-integral equation methods. The results generalize previous linear and weakly nonlinear results. Fully localized gravity–capillary interfacial solitary waves are also computed, when the free surface is replaced by a rigid lid.

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25

Van Gorder, Robert A. "Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He." Journal of Fluid Mechanics 707 (July 24, 2012): 585–94. http://dx.doi.org/10.1017/jfm.2012.308.

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AbstractWe obtain the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. As the relevant friction parameters are small, we linearize terms involving such parameters, while keeping the remaining nonlinearities, which accurately describe the curvature of the vortex filament, intact. The resulting equation is a type of nonlinear Schrödinger equation, and, under an appropriate change of variables, this equation is shown to have a first integral. This is in direct analogy with the simpler equation studied previously in the literature; indeed, in the limit where the superfluid parameters are taken to zero, we recover the results of Van Gorder. While this first integral is mathematically interesting, it is not particularly useful for computing solutions to the nonlinear partial differential equation which governs the vortex filament. As such, we introduce a new change of dependent variable, which results in a nonlinear four-dimensional system that can be numerically integrated. Integrating this system, we recover solutions to the fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He. We find that the qualitative features of the solutions depend not only on the superfluid friction parameters, but also strongly on the initial conditions taken, the curvature and the normal fluid velocity.

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26

Sabi’u, Jamilu, Hadi Rezazadeh, Rodica Cimpoiasu, and Radu Constantinescu. "Traveling wave solutions of the generalized Rosenau–Kawahara-RLW equation via the sine–cosine method and a generalized auxiliary equation method." International Journal of Nonlinear Sciences and Numerical Simulation 23, no. 3-4 (November 29, 2021): 539–51. http://dx.doi.org/10.1515/ijnsns-2019-0206.

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Abstract In this paper, we have approached a complicated nonlinear wave equation which links the Rosenau–Kawahara equation to the regularized long wave (RLW) equation. Taking advantages from the sine–cosine method as well as from the generalized auxiliary equation method, we have successfully reached to three types of traveling wave solutions: periodic, hyperbolic and exponential ones. Our results do constitute themselves as a challenge to apply the mentioned techniques in order to solve other generalized dynamical models, for example, the ones which involve phenomena such as a fully nonlinear dispersion and a fully nonlinear convection.

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27

Hamid Sharif, Nahidh, and Nils‐Erik Wiberg. "Interface‐capturing finite element technique for transient two‐phase flow." Engineering Computations 20, no. 5/6 (August 1, 2003): 725–40. http://dx.doi.org/10.1108/02644400310488835.

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A numerical model is presented for the computation of unsteady two‐fluid interfaces in nonlinear porous media flow. The nonlinear Forchheimer equation is included in the Navier‐Stokes equations for porous media flow. The model is based on capturing the interface on a fixed mesh domain. The zero level set of a pseudo‐concentration function, which defines the interface between the two fluids, is governed by a time‐dependent advection equation. The time‐dependent Navier‐Stokes equations and the advection equation are spatially discretized by the finite element (FE) method. The fully coupled implicit time integration scheme and the explicit forward Eulerian scheme are implemented for the advancement in time. The trapezoidal rule is applied to the fully implicit scheme, while the operator‐splitting algorithm is used for the velocity‐pressure segregation in the explicit scheme. The spatial and time discretizations are stabilized using FE stabilization techniques. Numerical examples of unsteady flow of two‐fluid interfaces in an earth dam are investigated.

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LIU, PHILIP L. F., and JIANGANG WEN. "Nonlinear diffusive surface waves in porous media." Journal of Fluid Mechanics 347 (September 25, 1997): 119–39. http://dx.doi.org/10.1017/s0022112097006472.

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A fully nonlinear, diffusive, and weakly dispersive wave equation is derived for describing gravity surface wave propagation in a shallow porous medium. Darcy's flow is assumed in a homogeneous and isotropic porous medium. In deriving the general equation, the depth of the porous medium is assumed to be small in comparison with the horizontal length scale, i.e. O(μ2) =O(h0/L)2[Lt ]1. The order of magnitude of accuracy of the general equation is O(μ4). Simplified governing equations are also obtained for the situation where the magnitude of the free-surface fluctuations is also small, O(ε)=O(a/h0)[Lt ]1, and is of the same order of magnitude as O(μ2). The resulting equation is of O(μ4, ε2) and is equivalent to the Boussinesq equations for water waves. Because of the dissipative nature of the porous medium flow, the damping rate of the surface wave is of the same order magnitude as the wavenumber. The tide-induced groundwater fluctuations are investigated by using the newly derived equation. Perturbation solutions as well as numerical solutions are obtained. These solutions compare very well with experimental data. The interactions between a solitary wave and a rectangular porous breakwater are then examined by solving the Boussinesq equations and the newly derived equations together. Numerical solutions for transmitted waves for different porous breakwaters are obtained and compared with experimental data. Excellent agreement is observed.

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29

Yu, Xiaohui. "Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros." Communications on Pure and Applied Analysis 12, no. 1 (September 2012): 451–59. http://dx.doi.org/10.3934/cpaa.2013.12.451.

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30

Guan, Bo, and Qun Li. "A Monge-Ampère type fully nonlinear equation on Hermitian manifolds." Discrete and Continuous Dynamical Systems - Series B 17, no. 6 (May 2012): 1991–99. http://dx.doi.org/10.3934/dcdsb.2012.17.1991.

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31

Wu, Feng, Zheng Yao, and Wanxie Zhong. "Fully nonlinear (2+1)-dimensional displacement shallow water wave equation." Chinese Physics B 26, no. 5 (May 2017): 054501. http://dx.doi.org/10.1088/1674-1056/26/5/054501.

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32

Minhós, F., T. Gyulov, and A. I. Santos. "Lower and upper solutions for a fully nonlinear beam equation." Nonlinear Analysis: Theory, Methods & Applications 71, no. 1-2 (July 2009): 281–92. http://dx.doi.org/10.1016/j.na.2008.10.073.

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33

Bernoff, Andrew J. "Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation." Physica D: Nonlinear Phenomena 30, no. 3 (April 1988): 363–81. http://dx.doi.org/10.1016/0167-2789(88)90026-7.

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34

Geng, Weihua, and Shan Zhao. "Fully implicit ADI schemes for solving the nonlinear Poisson-Boltzmann equation." Computational and Mathematical Biophysics 1 (April 24, 2013): 109–23. http://dx.doi.org/10.2478/mlbmb-2013-0006.

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AbstractThe Poisson-Boltzmann (PB) model is an effective approach for the electrostatics analysis of solvated biomolecules. The nonlinearity associated with the PB equation is critical when the underlying electrostatic potential is strong, but is extremely difficult to solve numerically. In this paper, we construct two operator splitting alternating direction implicit (ADI) schemes to efficiently and stably solve the nonlinear PB equation in a pseudo-transient continuation approach. The operator splitting framework enables an analytical integration of the nonlinear term that suppresses the nonlinear instability. A standard finite difference scheme weighted by piecewise dielectric constants varying across the molecular surface is employed to discretize the nonhomogeneous diffusion term of the nonlinear PB equation, and yields tridiagonal matrices in the Douglas and Douglas-Rachford type ADI schemes. The proposed time splitting ADI schemes are different from all existing pseudo-transient continuation approaches for solving the classical nonlinear PB equation in the sense that they are fully implicit. In a numerical benchmark example, the steady state solutions of the fully-implicit ADI schemes based on different initial values all converge to the time invariant analytical solution, while those of the explicit Euler and semi-implicit ADI schemes blow up when the magnitude of the initial solution is large. For the solvation analysis in applications to real biomolecules with various sizes, the time stability of the proposed ADI schemes can be maintained even using very large time increments, demonstrating the efficiency and stability of the present methods for biomolecular simulation.

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35

CLAMOND, DIDIER, and JOHN GRUE. "A fast method for fully nonlinear water-wave computations." Journal of Fluid Mechanics 447 (October 30, 2001): 337–55. http://dx.doi.org/10.1017/s0022112001006000.

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A fast computational method for fully nonlinear non-overturning water waves is derived in two and three dimensions. A corresponding time-stepping scheme is developed in the two-dimensional case. The essential part of the method is a fast converging iterative solution procedure of the Laplace equation. One part of the solution is obtained by fast Fourier transform, while another part is highly nonlinear and consists of integrals with kernels that decay quickly in space. The number of operations required is asymptotically O(N log N), where N is the number of nodes at the free surface. While any accuracy of the computations is achieved by a continued iteration of the equations, one iteration is found to be sufficient for practical computations, while maintaining high accuracy. The resulting explicit approximation of the scheme is tested in two versions. Simulations of nonlinear wave fields with wave slope even up to about unity compare very well with reference computations. The numerical scheme is formulated in such a way that aliasing terms are partially or completely avoided.

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36

Tsai, Ching-Piao, Hong-Bin Chen, and John R. C. Hsu. "Second-Order Time-Dependent Mild-Slope Equation for Wave Transformation." Mathematical Problems in Engineering 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/341385.

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This study is to propose a wave model with both wave dispersivity and nonlinearity for the wave field without water depth restriction. A narrow-banded sea state centred around a certain dominant wave frequency is considered for applications in coastal engineering. A system of fully nonlinear governing equations is first derived by depth integration of the incompressible Navier-Stokes equation in conservative form. A set of second-order nonlinear time-dependent mild-slope equations is then developed by a perturbation scheme. The present nonlinear equations can be simplified to the linear time-dependent mild-slope equation, nonlinear long wave equation, and traditional Boussinesq wave equation, respectively. A finite volume method with the fourth-order Adams-Moulton predictor-corrector numerical scheme is adopted to directly compute the wave transformation. The validity of the present model is demonstrated by the simulation of the Stokes wave, cnoidal wave, and solitary wave on uniform depth, nonlinear wave shoaling on a sloping beach, and wave propagation over an elliptic shoal. The nearshore wave transformation across the surf zone is simulated for 1D wave on a uniform slope and on a composite bar profile and 2D wave field around a jetty. These computed wave height distributions show very good agreement with the experimental results available.

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Lu, Dian Chen, and Ruo Yu Zhu. "Exponential Stability Estimate of Fully Nonlinear Aceive Equation by Boundary Control." Key Engineering Materials 467-469 (February 2011): 1078–83. http://dx.doi.org/10.4028/www.scientific.net/kem.467-469.1078.

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The well-posed problem for the fully nonlinear Aceive diffusion and dispersion equation on the domain [0, 1] is investigated by using boundary control. The existence and uniqueness of the solutions with the help of the Banach fixed point theorem and the theory of operator semigroups are verified. By using some inequalities and integration by parts, the exponential stability of the fully nonlinear Aceive diffusion and dispersion equation with the designed boundary feedback is also proved.

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Hong, CC. "Advanced frequency analysis of thick FGM plates using third-order shear deformation theory with a nonlinear shear correction coefficient." Journal of Structural Engineering & Applied Mechanics 5, no. 3 (September 30, 2022): 143–60. http://dx.doi.org/10.31462/jseam.2022.03143160.

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The effects of third-order shear deformation theory (TSDT) displacements and advanced nonlinear varied shear correction coefficient on the free vibration frequency of thick functionally graded material (FGM) plates under environment temperature are studied. The nonlinear coefficient term of TSDT displacements is included to derive the advanced equation of nonlinear varied shear correction coefficient for the thick FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under free vibration can be represented into fully homogeneous equation and the natural frequency can be found. The parametric effects of nonlinear coefficient term of TSDT, environment temperature and FGM power law index on the natural frequency of thick FGM plates are investigated.

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39

Bokanowski, Olivier, Athena Picarelli, and Christoph Reisinger. "Stability and convergence of second order backward differentiation schemes for parabolic Hamilton–Jacobi–Bellman equations." Numerische Mathematik 148, no. 1 (May 2021): 187–222. http://dx.doi.org/10.1007/s00211-021-01202-x.

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AbstractWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.

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Kong, Tao, Weidong Zhao, and Tao Zhou. "Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs." Communications in Computational Physics 18, no. 5 (November 2015): 1482–503. http://dx.doi.org/10.4208/cicp.240515.280815a.

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AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

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41

Henderson, K. L., D. H. Peregrine, and J. W. Dold. "Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation." Wave Motion 29, no. 4 (May 1999): 341–61. http://dx.doi.org/10.1016/s0165-2125(98)00045-6.

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IGNAT, LIVIU I. "FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES." Mathematical Models and Methods in Applied Sciences 17, no. 04 (April 2007): 567–91. http://dx.doi.org/10.1142/s0218202507002029.

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We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

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Berezhiani, V. I., L. N. Tsintsadze, and P. K. Shukla. "Nonlinear interaction of an intense electromagnetic wave with an unmagnetized electron—positron plasma." Journal of Plasma Physics 48, no. 1 (August 1992): 139–43. http://dx.doi.org/10.1017/s0022377800016421.

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The nonlinear interaction of an arbitrarily large-amplitude circularly polarized electromagnetic wave with an unmagnetized electron-positron plasma is considered, taking into account relativistic particle-mass variation as well as large-scale density perturbations created by radiation pressure. It is found that the interaction is governed by an equation for the electromagnetic wave envelope, which is coupled with a pair of equations describing fully nonlinear longitudinal plasma motions. The dynamics of the nonlinear electromagnetic wave packet is studied.

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Mace, R. L., M. A. Hellberg, R. Bharuthram, and S. Baboolal. "Electron-acoustic solitons in a weakly relativistic plasma." Journal of Plasma Physics 47, no. 1 (February 1992): 61–74. http://dx.doi.org/10.1017/s0022377800024089.

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Weakly relativistic electron-acoustic solitons are investigated in a two-electron-component plasma whose cool electrons form a relativistic beam. A general Korteweg-de Vries (KdV) equation is derived, in the small-|ø| domain, for a plasma consisting of an arbitrary number of relativistically streaming fluid components and a hot Boltzmann component. This equation is then applied to the specific case of electron-acoustic waves. In addition, the fully nonlinear system of fluid and Poisson equations is integrated to yield electron-acoustic solitons of arbitrary amplitude. It is shown that relativistic beam effects on electron-acoustic solitons significantly increase the soliton amplitude beyond its non-relativistic value. For intermediate- to large-amplitude solitons, a finite cool-electron temperature is found to destroy the balance between nonlinearity and dispersion, yielding soliton break-up. Also, only rarefactive electronacoustic soliton solutions of our equations are found, even though the relativistic beam provides a positive contribution to the nonlinear coefficient of the KdV equation, describing relativistic, nonlinear electron-acoustic waves.

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Coutino, Aaron, and Marek Stastna. "The fully nonlinear stratified geostrophic adjustment problem." Nonlinear Processes in Geophysics 24, no. 1 (January 30, 2017): 61–75. http://dx.doi.org/10.5194/npg-24-61-2017.

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Abstract. The study of the adjustment to equilibrium by a stratified fluid in a rotating reference frame is a classical problem in geophysical fluid dynamics. We consider the fully nonlinear, stratified adjustment problem from a numerical point of view. We present results of smoothed dam break simulations based on experiments in the published literature, with a focus on both the wave trains that propagate away from the nascent geostrophic state and the geostrophic state itself. We demonstrate that for Rossby numbers in excess of roughly 2 the wave train cannot be interpreted in terms of linear theory. This wave train consists of a leading solitary-like packet and a trailing tail of dispersive waves. However, it is found that the leading wave packet never completely separates from the trailing tail. Somewhat surprisingly, the inertial oscillations associated with the geostrophic state exhibit evidence of nonlinearity even when the Rossby number falls below 1. We vary the width of the initial disturbance and the rotation rate so as to keep the Rossby number fixed, and find that while the qualitative response remains consistent, the Froude number varies, and these variations are manifested in the form of the emanating wave train. For wider initial disturbances we find clear evidence of a wave train that initially propagates toward the near wall, reflects, and propagates away from the geostrophic state behind the leading wave train. We compare kinetic energy inside and outside of the geostrophic state, finding that for long times a Rossby number of around one-quarter yields an equal split between the two, with lower (higher) Rossby numbers yielding more energy in the geostrophic state (wave train). Finally we compare the energetics of the geostrophic state as the Rossby number varies, finding long-lived inertial oscillations in the majority of the cases and a general agreement with the past literature that employed either hydrostatic, shallow-water equation-based theory or stratified Navier–Stokes equations with a linear stratification.

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Barbu, Luminiţa, and Gheorghe Moroşanu. "Elliptic-like regularization of a fully nonlinear evolution inclusion and applications." Communications in Contemporary Mathematics 19, no. 05 (May 13, 2016): 1650037. http://dx.doi.org/10.1142/s0219199716500371.

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Consider in a Hilbert space [Formula: see text] the Cauchy problem [Formula: see text]: [Formula: see text], and associate with it the second-order problem [Formula: see text]: [Formula: see text], where [Formula: see text] is a (possibly set-valued) maximal monotone operator, [Formula: see text] is a Lipschitz operator, and [Formula: see text] is a positive small parameter. Note that [Formula: see text] is an elliptic-like regularization of [Formula: see text] in the sense suggested by Lions in his book on singular perturbations. We prove that the solution [Formula: see text] of [Formula: see text] approximates the solution [Formula: see text] of [Formula: see text]: [Formula: see text]. Applications to the nonlinear heat equation as well as to the nonlinear telegraph system and the nonlinear wave equation are presented.

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Omrani, K. "ON FULLY DISCRETE GALERKIN APPROXIMATIONS FOR THE CAHN‐HILLIARD EQUATION." Mathematical Modelling and Analysis 9, no. 4 (December 31, 2004): 313–26. http://dx.doi.org/10.3846/13926292.2004.9637262.

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Standard Galerkin approximations, using smooth splines to solutions of the nonlinear evolutionary Cahn‐Hilliard equation are analysed. The existence, uniqueness and convergence of the fully discrete Crank‐Nicolson scheme are discussed. At last a linearized Galerkin approximation is presented, which is also second order accurate in time fully discrete scheme.

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Zhang, Yunfei, and Minghe Pei. "Existence of Periodic Solutions for Nonlinear Fully Third-Order Differential Equations." Journal of Function Spaces 2020 (April 6, 2020): 1–7. http://dx.doi.org/10.1155/2020/6793721.

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In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.

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Qu, Meng, Ping Li, and Liu Yang. "Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation." Communications on Pure & Applied Analysis 19, no. 3 (2020): 1337–49. http://dx.doi.org/10.3934/cpaa.2020065.

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He, Xiaoming, Xin Zhao, and Wenming Zou. "Maximum principles for a fully nonlinear nonlocal equation on unbounded domains." Communications on Pure & Applied Analysis 19, no. 9 (2020): 4387–99. http://dx.doi.org/10.3934/cpaa.2020200.

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