Journal articles: 'Differential equations, Partial – Asymptotic theory'

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Bandle, Catherine. "SOME ASYMPTOTIC PROBLEMS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (Lezioni Lincee)." Bulletin of the London Mathematical Society 30, no. 3 (May 1998): 326–27. http://dx.doi.org/10.1112/s0024609397274096.

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S., L. R., Hans G. Kaper, and Marc Garbey. "Asymptotic Analysis and the Numerical Solution of Partial Differential Equations." Mathematics of Computation 59, no. 199 (July 1992): 303. http://dx.doi.org/10.2307/2153003.

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Khodja, Farid Ammar, Assia Benabdallah, and Djamel Teniou. "Stability of coupled systems." Abstract and Applied Analysis 1, no. 3 (1996): 327–40. http://dx.doi.org/10.1155/s1085337596000176.

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The exponential and asymptotic stability are studied for certain coupled systems involving unbounded linear operators and linear infinitesimal semigroup generators. Examples demonstrating the theory are also given from the field of partial differential equations.

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RUDNICKI, RYSZARD, and KATARZYNA PICHÓR. "MARKOV SEMIGROUPS AND STABILITY OF THE CELL MATURITY DISTRIBUTION." Journal of Biological Systems 08, no. 01 (March 2000): 69–94. http://dx.doi.org/10.1142/s0218339000000067.

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A model of the maturity-structured cell population is considered. This model is described by a partial differential equation with a transformed argument. Using the theory of Markov semigroups we establish a new criterion for asymptotic stability of such equations.

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Hibino, Masaki. "Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type, II." Publications of the Research Institute for Mathematical Sciences 37, no. 4 (2001): 579–614. http://dx.doi.org/10.2977/prims/1145477330.

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Bouatta, Mohamed A., Sergey A. Vasilyev, and Sergey I. Vinitsky. "The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation." Discrete and Continuous Models and Applied Computational Science 29, no. 2 (December 15, 2021): 126–45. http://dx.doi.org/10.22363/2658-4670-2021-29-2-126-145.

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The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

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Rogacheva, Nelly. "THE REFINED THEORY OF ELASTIC PLATES AS ASYMPTOTIC APPROACH OF 3D PROBLEM." MATEC Web of Conferences 196 (2018): 02037. http://dx.doi.org/10.1051/matecconf/201819602037.

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The refined theory of elastic thin and thick plates is constructed by the asymptotic method for reducing three-dimensional (3D) equations of linear elasticity to two-dimensional ones without the use of any assumptions. The resulting refined theory is much more complicated than the known classical Kirchhoff theory: the required values of the refined theory vary in thickness of the plate by more complex laws; the system of partial differential equations of the refined theory has a higher order than the system of equations of the classical theory. A comparison of the obtained theory with the popular refined theory of Timoshenko and E. Reissner, taking into account the transverse shear deformation is made. It is shown that the inclusion only of the transverse shear deformation is insufficient. In addition to the transverse shear deformation, many additional terms having the same order as the transverse shear deformation must be taken into account.

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Hayashi, Nakao, and Elena I. Kaikina. "Benjamin-Ono Equation on a Half-Line." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–38. http://dx.doi.org/10.1155/2010/714534.

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We consider the initial-boundary value problem for Benjamin-Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

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Hibino, Masaki. "Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —." Communications on Pure & Applied Analysis 2, no. 2 (2003): 211–31. http://dx.doi.org/10.3934/cpaa.2003.2.211.

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Allognissode, Fulbert Kuessi, Mamadou Abdoul Diop, Khalil Ezzinbi, and Carlos Ogouyandjou. "Stochastic partial functional integrodifferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior." Random Operators and Stochastic Equations 27, no. 2 (June 1, 2019): 107–22. http://dx.doi.org/10.1515/rose-2019-2009.

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Abstract This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion {S_{Q}^{H}(t)} , with Hurst parameter {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufficient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory.

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Anikin, Anatoly Yu, Sergey Yu Dobrokhotov, Alexander I. Klevin, and Brunello Tirozzi. "Short-Wave Asymptotics for Gaussian Beams and Packets and Scalarization of Equations in Plasma Physics." Physics 1, no. 2 (August 31, 2019): 301–20. http://dx.doi.org/10.3390/physics1020023.

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We study Gaussian wave beam and wave packet types of solutions to the linearized cold plasma system in a toroidal domain (tokamak). Such solutions are constructed with help of Maslov’s complex germ theory (short-wave or semi-classical asymptotics with complex phases). The term “semi-classical” asymptotics is understood in a broad sense: asymptotic solutions of evolutionary and stationary partial differential equations from wave or quantum mechanics are expressed through solutions of the corresponding equations of classical mechanics. This, in particular, allows one to use useful geometric considerations. The small parameter of the expansion is h = λ / 2 π L where λ is the wavelength and L the dimension of the system. In order to apply the asymptotic algorithm, we need this parameter to be small, so we deal only with high-frequency waves, which are in the range of lower hybrid waves used to heat the plasma. The asymptotic solution appears to be a Gaussian wave packet divided by the square root of the determinant of an appropriate Jacobi matrix (“complex divergence”). When this determinant is zero, focal points appear. Our approach allows one to write out asymptotics near focal points. We also claim that this approach is very practical and leads to formulas that can be used for numerical simulations in software like Wolfram Mathematica, Maple, etc. For the particular case of high-frequency beams, we present a recipe for constructing beams and packets and show the results of their numerical implementation. We also propose ideas to treat the more difficult general case of arbitrary frequency. We also explain the main ideas of asymptotic theory used to obtain such formulas.

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Kaikina, Elena I. "Forced cubic Schrödinger equation with Robin boundary data: large-time asymptotics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2159 (November 8, 2013): 20130341. http://dx.doi.org/10.1098/rspa.2013.0341.

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We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.

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Vorotnikov, V. I. "On Problem of Partial Stability for Functional Differential Systems with Holdover." Mekhatronika, Avtomatizatsiya, Upravlenie 20, no. 7 (July 4, 2019): 398–404. http://dx.doi.org/10.17587/mau.20.398-404.

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The theory of systems of functional differential equations is a significant and rapidly developing sphere of modern mathematics which finds extensive application in complex systems of automatic control and also in economic, modern technical, ecological, and biological models. Naturally, the problems arises of stability and partial stability of the processes described by the class of the equation. The article studies the problem of partial stability which arise in applications either from the requirement of proper performance of a system or in assessing system capability. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of partial stability. We suppose that the system have the zero equilibrium position. A conditions are obtained under which the uniform stability (uniform asymptotic stability) of the zero equilibrium position with respect to the part of the variables implies the uniform stability (uniform asymptotic stability) of this equilibrium position with respect to the other, larger part of the variables, which include an additional group of coordinates of the phase vector. These conditions include: 1) the condition for uniform asymptotic stability of the zero equilibrium position of the "reduced" subsystem of the original system with respect to the additional group of variables; 2) the restriction on the coupling between the "reduced" subsystem and the rest parts of the system. Application of the obtained results to a problem of stabilization with respect to a part of the variables for nonlinear controlled systems is discussed.

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Fiza, Mehreen, Hakeem Ullah, Saeed Islam, Qayum Shah, Farkhanda Inayat Chohan, and Mustafa Bin Mamat. "Modifications of the Multistep Optimal Homotopy Asymptotic Method to Some Nonlinear KdV-Equations." European Journal of Pure and Applied Mathematics 11, no. 2 (April 27, 2018): 537–52. http://dx.doi.org/10.29020/nybg.ejpam.v11i2.3194.

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In this article we have introduced the mathematical theory of multistep optimal homotopy asymptotic method (MOHAM). The proposed method is implemented to different models having system of partial differential equations (PDEs). The results obtained by proposed method are compared with Homotopy Analysis Method (HAM) and closed form solutions. The comparisons of these results show that MOHAM is simpler in applicability, effective, explicit, control the convergence through optimal constants, involve less computational work. The MOHAM is independent of the assumption of initial conditions and small parameters like Homotopy Perturbation Method (HPM), HAM, Variational Iteration Method (VIM), Adomian Decomposition Method (ADM) and Perturbation Method (PM).

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Saha Ray, S. "The comparison of two reliable methods for the accurate solution of fractional Fisher type equation." Engineering Computations 34, no. 8 (November 6, 2017): 2598–613. http://dx.doi.org/10.1108/ec-03-2017-0074.

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Purpose The purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method. Design/methodology/approach Comparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems. Findings The comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method. Originality/value In this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency.

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Kao, Yonggui, and Hamid Reza Karimi. "Stability in Mean of Partial Variables for Coupled Stochastic Reaction-Diffusion Systems on Networks: A Graph Approach." Abstract and Applied Analysis 2014 (2014): 1–13. http://dx.doi.org/10.1155/2014/597502.

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This paper is devoted to investigating stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks (CSRDSNs). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations (SODE) and using Itô formula, we establish some novel stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. These stability principles have a close relation with the topology property of the network. We also provide a systematic method for constructing global Lyapunov function for these CSRDSNs by using graph theory. The new method can help to analyze the dynamics of complex networks. An example is presented to illustrate the effectiveness and efficiency of the obtained results.

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BUCKWAR, E., M. G. RIEDLER, and P. E. KLOEDEN. "THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE." Stochastics and Dynamics 11, no. 02n03 (September 2011): 265–81. http://dx.doi.org/10.1142/s0219493711003279.

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An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

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Yao, Minghui, Li Ma, and Wei Zhang. "Nonlinear Dynamics of the High-Speed Rotating Plate." International Journal of Aerospace Engineering 2018 (2018): 1–23. http://dx.doi.org/10.1155/2018/5610915.

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High speed rotating blades are crucial components of modern large aircraft engines. The rotating blades under working condition frequently suffer from the aerodynamic, elastic and inertia loads, which may lead to large amplitude nonlinear oscillations. This paper investigates nonlinear dynamic responses of the blade with varying rotating speed in supersonic airflow. The blade is simplified as a pre-twist and presetting cantilever composite plate. Warping effect of the rectangular cross-section of the plate is considered. Based on the first-order shear deformation theory and von-Karman nonlinear geometric relationship, nonlinear partial differential dynamic equations of motion for the plate are derived by using Hamilton’s principle. Galerkin approach is applied to discretize the partial differential governing equations of motion to ordinary differential equations. Asymptotic perturbation method is exploited to derive four-degree-of-freedom averaged equation for the case of 1 : 3 internal resonance-1/2 sub-harmonic resonance. Based on the averaged equation, numerical simulation is used to analyze the influence of the perturbation rotating speed on nonlinear dynamic responses of the blade. Bifurcation diagram, phase portraits, waveforms and power spectrum prove that periodic motion and chaotic motion exist in nonlinear vibration of the rotating cantilever composite plate.

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Vassalos, Paris. "Asymptotic results on the condition number of FD matrices approximating semi-elliptic PDEs." Electronic Journal of Linear Algebra 34 (February 21, 2018): 566–81. http://dx.doi.org/10.13001/1081-3810.3852.

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This work studies the asymptotic behavior of the spectral condition number of the matrices $A_{nn}$ arising from the discretization of semi-elliptic partial differential equations of the form \bdm -\left( a(x,y)u_{xx}+b(x,y)u_{yy}\right)=f(x,y), \edm on the square $\Omega=(0,1)^2,$ with Dirichlet boundary conditions, where the smooth enough variable coefficients $a(x,y), b(x,y)$ are nonnegative functions on $\overline{\Omega}$ with zeros. In the case of coefficient functions with a single and common zero, it is discovered that apart from the minimum order of the zero also the direction that it occurs is of great importance for the characterization of the growth of the condition number of $A_{nn}$. On the contrary, when the coefficient functions have non intersecting zeros, it is proved that independently of the order their zeros, and their positions, the condition number of $A_{nn}$ behaves asymptotically exactly as in the case of strictly elliptic differential equations, i.e., it grows asymptotically as $n^2$. Finally, the more complicated case of coefficient functions having curves of roots is considered, and conjectures for future work are given. In conclusion, several experiments are presented that numerically confirm the developed theoretical analysis.

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Meng, Xin-You, and Jiao-Guo Wang. "Analysis of a delayed diffusive model with Beddington–DeAngelis functional response." International Journal of Biomathematics 12, no. 04 (May 2019): 1950047. http://dx.doi.org/10.1142/s1793524519500475.

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In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington–DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis.

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DIXIT, SHIVSAI AJIT, and O. N. RAMESH. "Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers." Journal of Fluid Mechanics 615 (November 25, 2008): 445–75. http://dx.doi.org/10.1017/s0022112008004047.

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Experiments were done on sink flow turbulent boundary layers over a wide range of streamwise pressure gradients in order to investigate the effects on the mean velocity profiles. Measurements revealed the existence of non-universal logarithmic laws, in both inner and defect coordinates, even when the mean velocity descriptions departed strongly from the universal logarithmic law (with universal values of the Kármán constant and the inner law intercept). Systematic dependences of slope and intercepts for inner and outer logarithmic laws on the strength of the pressure gradient were observed. A theory based on the method of matched asymptotic expansions was developed in order to explain the experimentally observed variations of log-law constants with the non-dimensional pressure gradient parameter (Δp=(ν/ρU3τ)dp/dx). Towards this end, the system of partial differential equations governing the mean flow was reduced to inner and outer ordinary differential equations in self-preserving form, valid for sink flow conditions. Asymptotic matching of the inner and outer mean velocity expansions, extended to higher orders, clearly revealed the dependence of slope and intercepts on pressure gradient in the logarithmic laws.

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Liu, Jia, and Xuebing Zhang. "Stability and Hopf bifurcation of a delayed reaction–diffusion predator–prey model with anti-predator behaviour." Nonlinear Analysis: Modelling and Control 24, no. 3 (April 23, 2019): 387–406. http://dx.doi.org/10.15388/na.2019.3.5.

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In this paper, we study the dynamics of a delayed reaction–diffusion predator–prey model with anti-predator behaviour. By using the theory of partial functional differential equations, Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the model, and a phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. By utilizing upperlower solution method, the global asymptotic stability of the interior equilibrium is studied. Finally, numerical simulation results are presented to validate the theoretical analysis.

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Zhang, Jia-Fang. "Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects." Abstract and Applied Analysis 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/856725.

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This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

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Durastante, Fabio, and Isabella Furci. "Spectral analysis of saddle-point matrices from optimization problems with elliptic PDE constraints." Electronic Journal of Linear Algebra 36, no. 36 (December 12, 2020): 773–98. http://dx.doi.org/10.13001/ela.2020.5151.

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The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. They uncover the existence of an hidden structure in these matrix sequences, namely, they show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. They show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that they deal with saddle--point matrices. Finally, they exploit it to propose an optimal preconditioner strategy for the GMRES, and Flexible--GMRES methods.

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Jallouli, Aymen, Najib Kacem, and Joseph Lardies. "Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers." Micromachines 9, no. 11 (November 5, 2018): 575. http://dx.doi.org/10.3390/mi9110575.

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In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

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Agiasofitou, Eleni, and Markus Lazar. "Phason dynamics of quasicrystals." Acta Crystallographica Section A Foundations and Advances 70, a1 (August 5, 2014): C86. http://dx.doi.org/10.1107/s2053273314099136.

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Phason dynamics constitutes a challenging and interesting subject in the study of quasicrystals, since there is not a unique model in the literature for the description of the dynamics of the phason fields. Here, we introduce the elastodynamic model of wave-telegraph type for the description of dynamics of quasicrystals [1, 2]. Phonons are represented by waves, and phasons by waves damped in time and propagating with finite velocity; that means the equations of motion for the phonons are partial differential equations of wave type, and for the phasons partial differential equations of telegraph type. The proposed model constitutes a unified theory in the sense that already established models in the literature can be recovered as asymptotic cases of it. Several noteworthy features characterize the proposed model. The influence of the damping in the dynamic behavior of the phasons is expressed by the tensor of phason friction coefficients, which gives the possibility to take into account that the phason waves can be damped anisotropically. In terms of the phason friction coefficient and the average mass density of the material an important quantity, the characteristic time of damping, can been defined. Another important advantage of the model is that it provides a theory valid in the whole regime of possible wavelengths for the phasons. In addition, with the telegraph type equation there is no longer the drawback of the infinite propagation velocity that exists with the equation of diffusion type.

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BARAVIERA, ALEXANDRE TAVARES, and MARCELO MENDES DISCONZI. "ASYMPTOTIC STATES IN COUPLED MAP LATTICES." International Journal of Bifurcation and Chaos 18, no. 02 (February 2008): 285–311. http://dx.doi.org/10.1142/s0218127408020318.

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Coupled Map Lattices (CML) are a kind of dynamical systems that appear naturally in some contexts, like the discretization of partial differential equations, and as a simple model of coupling between nonlinear systems. The coupling creates new and rich properties, that has been the object of intense investigation during the last decades. In this work we have two goals: first we give a nontechnical introduction to the theory of invariant measures and equilibrium in dynamics (with analogies with equilibrium in statistical mechanics) because we believe that sometimes a lot of interesting problems on the interface between physics and mathematics are not being developed simply due to the lack of a common language. Our second goal is to make a small contribution to the theory of equilibrium states for CML. More specifically, we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. The goal is to show that we start in a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the dynamics of most orbits coupling. Associating a symbolic dynamics with symbols +1, 0 and -1 to the system we describe a transition characterized by an asymptotic state composed only of symbols +1 or only of symbols -1. We rigorously prove our assertions and provide numerical experiments with two goals: first, as illustration of our rigorous results and second, to motivate some conjectures concerning the problem and some of its possible variations.

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Rega, Giuseppe. "Nonlinear vibrations of suspended cables—Part I: Modeling and analysis." Applied Mechanics Reviews 57, no. 6 (November 1, 2004): 443–78. http://dx.doi.org/10.1115/1.1777224.

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This review article is the first of three parts of a Special Issue dealing with finite-amplitude oscillations of elastic suspended cables. This part is concerned with system modeling and methods of analysis. After shortly reporting on cable historical literature and identifying the topic and scope of the review, the article begins with a presentation of the mechanical system and of the ensuing mathematical models. Continuum equations of cable finite motion are formulated, their linearized version is reported, and nonlinear discretized models for the analysis of 2D or 3D vibration problems are discussed. Approximate methods for asymptotic analysis of either single or multi-degree-of-freedom models of small-sag cables are addressed, as well as asymptotic models operating directly on the original partial differential equations. Numerical tools and geometrical techniques from dynamical systems theory are illustrated with reference to the single-degree-of-freedom model of cable, reporting on measures for diagnosis of nonlinear and chaotic response, as well as on techniques for local and global bifurcation analysis. The paper ends with a discussion on the main features and problems encountered in nonlinear experimental analysis of vibrating suspended cables. This review article cites 226 references.

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Carvalho, Alexandre N. "Contracting sets and dissipation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 6 (1995): 1305–29. http://dx.doi.org/10.1017/s0308210500030523.

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In this work we study reaction–diffusion systems in fractional power spaces Xα which are embedded in L∞. We prove that the solution operators T(t) to these problems are globally defined, point dissipative, locally bounded and compact. That ensures the existence of global attractors. We also find a set containing the range of every function in the attractor, providing good estimates on asymptotic concentrations. This is done under very few hypotheses on the reaction term. These hypotheses are natural and easy to verify in many applications. The tools employed are the theory of invariant regions for systems of parabolic partial differential equations, the notion of contracting sets and the variation of constants formula. Several examples are considered to emphasise the applicability of these techniques.

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Martel, Yvan, and Frank Merle. "Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg—de Vries equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 2 (April 2011): 287–317. http://dx.doi.org/10.1017/s030821051000003x.

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We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

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Tuckwell, Henry C., Roger Rodriguez, and Frederic Y. M. Wan. "Determination of Firing Times for the Stochastic Fitzhugh-Nagumo Neuronal Model." Neural Computation 15, no. 1 (January 1, 2003): 143–59. http://dx.doi.org/10.1162/089976603321043739.

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We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.

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Afsar, Mohammed Z., Adrian Sescu, and Stewart J. Leib. "Modelling and prediction of the peak-radiated sound in subsonic axisymmetric air jets using acoustic analogy-based asymptotic analysis." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2159 (October 14, 2019): 20190073. http://dx.doi.org/10.1098/rsta.2019.0073.

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This paper uses asymptotic analysis within the generalized acoustic analogy formulation (Goldstein 2003 JFM 488 , 315–333. ( doi:10.1017/S0022112003004890 )) to develop a noise prediction model for the peak sound of axisymmetric round jets at subsonic acoustic Mach numbers (Ma). The analogy shows that the exact formula for the acoustic pressure is given by a convolution product of a propagator tensor (determined by the vector Green's function of the adjoint linearized Euler equations for a given jet mean flow) and a generalized source term representing the jet turbulence field. Using a low-frequency/small spread rate asymptotic expansion of the propagator, mean flow non-parallelism enters the lowest order Green's function solution via the streamwise component of the mean flow advection vector in a hyperbolic partial differential equation. We then address the predictive capability of the solution to this partial differential equation when used in the analogy through first-of-its-kind numerical calculations when an experimentally verified model of the turbulence source structure is used together with Reynolds-averaged Navier–Stokes solutions for the jet mean flow. Our noise predictions show a reasonable level of accuracy in the peak noise direction at Ma = 0.9, for Strouhal numbers up to about 0.6, and at Ma = 0.5 using modified source coefficients. Possible reasons for this are discussed. Moreover, the prediction range can be extended beyond unity Strouhal number by using an approximate composite asymptotic formula for the vector Green's function that reduces to the locally parallel flow limit at high frequencies. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

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Ma, Li, Minghui Yao, Wei Zhang, and Dongxing Cao. "Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow." Applied Mathematics and Mechanics 41, no. 12 (October 28, 2020): 1861–80. http://dx.doi.org/10.1007/s10483-020-2668-8.

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AbstractTurbo-machineries, as key components, have a wide utilization in fields of civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Kármán nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton’s principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.

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Choi, Daniel. "On Geometrical Rigidity of Surfaces." Mathematical Models and Methods in Applied Sciences 07, no. 04 (June 1997): 507–55. http://dx.doi.org/10.1142/s0218202597000281.

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In this paper we first present a panorama about geometrical rigidity and inextensional displacements (also called infinitesimal bendings) for surfaces with kinematic boundary conditions and for surfaces with edges (in the sense of folds or junctions). This theory is fundamental for thin linear elastic shells, as it rules their asymptotic behavior when the thickness tends to zero. This behavior enlights some difficulties encountered in numerical studies of very thin elastic shells. Our approach is based on the introduction of a nonclassical space denoted by R(S) and related to inextensional displacements. It permits us to obtain new results concerning developable surfaces and hyperbolic surfaces, with one or two edges (most of them assumed to keep constant angle), including a theorem of rigid edge when the edge is an asymptotic line of the surface. By applying these results, we are able to exhibit a new example of sensitive problem for a shell with hyperbolic mean surface and with two edges keeping constant angle. In the Appendix, we give a nonclassical variant of Goursat problem for hyperbolic linear partial differential equations system, used in the proof of a rigidity result.

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O'Brien, S. B. G. M. "On Marangoni drying: nonlinear kinematic waves in a thin film." Journal of Fluid Mechanics 254 (September 1993): 649–70. http://dx.doi.org/10.1017/s0022112093002290.

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In the field of industrial drying, a recent innovation has exploited the occurrence of Marangoni effects in such a way that the resultant free-surface flow enhances the drying process. To this end, alcohol vapour, soluble in water, is introduced above a drying film and as a result of diffusion through the air and water phases a favourable concentration gradient gives rise to the required shear flow. We consider here a simple process driven by this mechanism, and by means of asymptotic simplification and the concepts of singular perturbation theory a leading-order approximation is obtained in which the alcohol concentration in the water is a specified function of space and time. The evolution of the free surface thus reduces to a single nonlinear partial differential equation of a similar form to the Korteweg–de Vries and Burgers equations, higher-derivative terms corresponding to surface tension and gravity effects. Numerical solutions of this equation are obtained and are compared to the application of first order nonlinear kinematic wave theory with corresponding shock solutions.

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Paninski, Liam. "The Spike-Triggered Average of the Integrate-and-Fire Cell Driven by Gaussian White Noise." Neural Computation 18, no. 11 (November 2006): 2592–616. http://dx.doi.org/10.1162/neco.2006.18.11.2592.

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We compute the exact spike-triggered average (STA) of the voltage for the nonleaky integrate-and-fire (IF) cell in continuous time, driven by gaussian white noise. The computation is based on techniques from the theory of renewal processes and continuous-time hidden Markov processes (e.g., the backward and forward Fokker-Planck partial differential equations associated with first-passage time densities). From the STA voltage, it is straightforward to derive the STA input current. The theory also gives an explicit asymptotic approximation for the STA of the leaky IF cell, valid in the low-noise regime σ → 0. We consider both the STA and the conditional average voltage given an observed spike “doublet” event, that is, two spikes separated by some fixed period of silence. In each case, we find that the STA as a function of time-preceding-spike, τ, has a square root singularity as τ approaches zero from below and scales linearly with the scale of injected noise current. We close by briefly examining the discrete-time case, where similar phenomena are observed.

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YAN, XIANG-PING, and WAN-TONG LI. "STABILITY AND HOPF BIFURCATION FOR A DELAYED COOPERATIVE SYSTEM WITH DIFFUSION EFFECTS." International Journal of Bifurcation and Chaos 18, no. 02 (February 2008): 441–53. http://dx.doi.org/10.1142/s0218127408020434.

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The main purpose of this paper is to investigate the stability and Hopf bifurcation for a delayed two-species cooperative diffusion system with Neumann boundary conditions. By linearizing the system at the positive equilibrium and analyzing the corresponding characteristic equation, the asymptotic stability of positive equilibrium and the existence of Hopf oscillations are demonstrated. It is shown that, under certain conditions, the system undergoes only a spatially homogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through a sequence of critical values; under the other conditions, except for the previous spatially homogeneous Hopf bifurcations, the system also undergoes a spatially inhomogeneous Hopf bifurcation at the positive equilibrium when the delay crosses through another sequence of critical values. In particular, in order to determine the direction and stability of periodic solutions bifurcating from spatially homogeneous Hopf bifurcations, the explicit formulas are given by using the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs). Finally, to verify our theoretical predictions, some numerical simulations are also included.

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Bertozzi, Andrea L., Shane D. Johnson, and Michael J. Ward. "Mathematical modelling of crime and security: Special Issue of EJAM." European Journal of Applied Mathematics 27, no. 3 (April 28, 2016): 311–16. http://dx.doi.org/10.1017/s0956792516000176.

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This special issue of the European journal of applied mathematics features research articles that involve the application of mathematical methodologies to the modelling of a broad range of problems related to crime and security. Some specific topics in this issue include recent developments in mathematical models of residential burglary, a dynamical model for the spatial spread of riots initiated by some triggering event, the analysis and development of game-theoretic models of crime and conflict, the study of statistically based models of insurgent activity and terrorism using real-world data sets, models for the optimal strategy of police deployment under realistic constraints, and a model of cyber crime as related to the study of spiking behaviour in social network cyberspace communications. Overall, the mathematical and computational methodologies employed in these studies are as diverse as the specific applications themselves and the scales (spatial or otherwise) to which they are applied. These methodologies range from statistical and stochastic methods based on maximum likelihood methods, Bayesian equilibria, regression analysis, self-excited Hawkes point processes, agent-based random walk models on networks, to more traditional applied mathematical methods such as dynamical systems and stability theory, the theory of Nash equilibria, rigorous methods in partial differential equations and travelling wave theory, and asymptotic methods that exploit disparate space and time scales.

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Mossaiby, Farshid, Majid Bazrpach, and Arman Shojaei. "Extending the method of exponential basis functions to problems with singularities." Engineering Computations 32, no. 2 (April 20, 2015): 406–23. http://dx.doi.org/10.1108/ec-01-2014-0019.

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Purpose – The purpose of this paper is to aim at extending the method of exponential basis functions (EBF) to solve a class of problems with singularities. Design/methodology/approach – In the procedure of EBF a summation of EBF satisfying the governing differential equation with unknown constant coefficients is considered for the solution. These coefficients are determined by the satisfaction of prescribed boundary conditions through a collocation approach. The applied basis functions are available in the case of linear partial differential equations (PDEs) with constant coefficients. Moreover, the method contributes to yield highly accurate results with ultra convergence rates for problems with smooth solution. This leads EBF to offer many advantages for a variety of engineering problems. However, owing to the global and smooth nature of the bases, the performance of EBF deteriorates in problems with singularities. In the present study, some exponential-like influence functions are developed, and a few of them are added to original bases. Findings – The new bases are capable of forming the constitutive terms of the asymptotic solution near the singularity points and alleviate the aforementioned limitation. The appealing feature of this method is that all the advantages of EBF such as its simplicity and efficiency are completely preserved. Research limitations/implications – In its current form, EBF can only solve PDEs with constant coefficients. Originality/value – Application of the method to some benchmark problems demonstrates its robustness over some other boundary approximation methods. This research may pave the road for future investigations corresponding to a wide range of practical engineering problems.

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Roos, H. G. "Oleinik, O.: Some Asymptotic Problems in the Theory of Partial Differential Equations. Cambridge, Cambridge University Press 1996. X, 203 pp., £ 35.00 (hardback)/£ 12.95 (paperback). ISBN 0-521-48083-3 (hardback)/0-521-48537-1 (paperback) (Lezioni Lincee)." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 77, no. 7 (1997): 542. http://dx.doi.org/10.1002/zamm.19970770714.

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Zuev, A. L. "Partial asymptotic stability of abstract differential equations." Ukrainian Mathematical Journal 58, no. 5 (May 2006): 709–17. http://dx.doi.org/10.1007/s11253-006-0096-3.

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TZOU, J. C., Y. NEC, and M. J. WARD. "The stability of localized spikes for the 1-D Brusselator reaction–diffusion model." European Journal of Applied Mathematics 24, no. 4 (April 10, 2013): 515–64. http://dx.doi.org/10.1017/s0956792513000089.

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In a one-dimensional domain, the stability of localized spike patterns is analysed for two closely related singularly perturbed reaction–diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a non-local eigenvalue problem (NLEP), whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer–Meinhardt and Gray–Scott models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous, rather than synchronous, oscillatory instability of the spike amplitudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike amplitudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike amplitudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full partial differential equations system.

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CRIMINALE, W. O., T. L. JACKSON, D. G. LASSEIGNE, and R. D. JOSLIN. "Perturbation dynamics in viscous channel flows." Journal of Fluid Mechanics 339 (May 25, 1997): 55–75. http://dx.doi.org/10.1017/s0022112097005235.

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Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.

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Al-Hammadi, A. S. A. "Asymptotic theory for third‐order differential equations." Mathematika 35, no. 2 (December 1988): 225–32. http://dx.doi.org/10.1112/s0025579300015229.

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Ignatyev, Oleksiy. "Partial asymptotic stability in probability of stochastic differential equations." Statistics & Probability Letters 79, no. 5 (March 2009): 597–601. http://dx.doi.org/10.1016/j.spl.2008.10.005.

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Badiale, Marino, and Martino Bardi. "Asymptotic symmetry of solutions of nonlinear partial differential equations." Communications on Pure and Applied Mathematics 45, no. 7 (August 1992): 899–921. http://dx.doi.org/10.1002/cpa.3160450705.

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Berti, Massimiliano. "KAM Theory for Partial Differential Equations." Analysis in Theory and Applications 35, no. 3 (June 2019): 235–67. http://dx.doi.org/10.4208/ata.oa-0013.

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Maheswari, R., and S. Karunanithi. "Asymptotic Stability of Stochastic Impulsive Neutral Partial Functional Differential Equations." International Journal of Computer Applications 85, no. 18 (January 16, 2014): 22–26. http://dx.doi.org/10.5120/14941-3423.

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Nagajothi, N., and V. Sadhasivam. "ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF FORCED FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS." Advances in Mathematics: Scientific Journal 9, no. 8 (August 19, 2020): 6177–93. http://dx.doi.org/10.37418/amsj.9.8.85.

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Caraballo, TomáS. "Asymptotic exponential stability of stochastic partial differential equations with delay." Stochastics and Stochastic Reports 33, no. 1-2 (November 1990): 27–47. http://dx.doi.org/10.1080/17442509008833662.

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Journal articles on the topic 'Differential equations, Partial – Asymptotic theory'

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