共查询到20条相似文献，搜索用时 156 毫秒
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本文利用双曲函数展开法,在行波条件下,对五阶KdV方程,Fisher-Kolmogorov方程等两个非线性波动方程求解,并借助于计算机代数系统Maple,获得了这类偏微分方程的若干精确孤波解。 相似文献
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以非线性波动微分方程作为研究对象,运用李群分支算法对其进行变量分离及精确解分析。首先,利用不变子空间法通过线性常微分方程存在解的子空间中构建适合非线性波动微分方程和方程组的不变子空间,将子空间应用至方程算子中并进行降价和化简处理,推导出不变子空间的未知函数,从而得到等价转换的简化方程;其次,采用李群分支法将扩散方程的解空间分划为多个小轨道,选取相应无线维对称群的分支,每个解空间由自同构系统决定,获取方程解需选择对称群并由其构造新方程,再将符号不变量运用至方程组中,使它成为初始给定方程的求解条件,进而实现非线性波动微分方程的变量分离,求出其精确解。实验证明,所提方法可实现变量分离,得到精确解,为当代数学提供理论支持。 相似文献
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非线性微分方程算子的敏感域对于非线性微分方程的求解,非线性微分方程的实际应用分析具有重要意义。非线性微分方程敏感域分析的难点在于如何精确的对敏感域进行详细的建模,通过对方程各种影响因素的详细区分,构建对于整个非线性方程应用的分析模型。提出了一种非线性微分方程算子敏感域应用分析模型,采用每个独特解的解散布特性提取整体解特征,通过融合方法实现敏感域的有效分析。通过推到论证,结果证明,敏感域分析对于非线性微分方程分析具有很好的指导意义。 相似文献
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非线性科学已经被广泛应用于数学、物理、化学、经济等领域。许多非线性现象都可以用非线性偏微分方程来很好地描述,所以得到非线性偏微分方程的解具有重要的意义。在研究非线性科学的同时,出现了一些带有扰动项的非线性偏微分方程。为了研究这种扰动偏微分方程,一些以对称理论为基础的扰动方法相继产生。本文主要研究对称扰动理论在偏微分方程中的应用,寻求偏微分方程的近似对称约化和无穷级数解。 相似文献
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The design problem of collocated feedback controllers is addressed in this paper for a class of semi-linear distributed parameter systems described by parabolic partial differential equation (PDE), where a finite number of local actuators and sensors are intermittently distributed in space. A Lyapunov direct method for the exponential stability analysis of the resulting closed-loop system is first presented for the system, in which the first mean value theorem for integration and the Wirtinger's inequality are employed. The corresponding stabilization condition is then derived through the analysis result. Finally, the proposed design method is implemented on the feedback control of a fisher equation and its effectiveness is evaluated through simulation results. 相似文献
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Control of PDE-ODE cascades with Neumann interconnections 总被引：1，自引：0，他引：1
We extend several recent results on full-state feedback stabilization and state estimation of PDE-ODE cascades, where the PDEs are either of heat type or of wave type, from the previously considered cases where the interconnections are of Dirichlet type, to interconnections of Neumann type. The Neumann type interconnections constrain the PDE state to be subject to a Dirichlet boundary condition at the PDE-ODE interface, and employ the boundary value of the first spatial derivative of the PDE state to be the input to the ODE. In addition to considering heat-ODE and wave-ODE cascades, we also consider a cascade of a diffusion-convection PDE with an ODE, where the convection direction is “away” from the ODE. We refer to this case as a PDE-ODE cascade with “counter-convection.” This case is not only interesting because the PDE subsystem is unstable, but because the control signal is subject to competing effects of diffusion, which is in both directions in the one-dimensional domain, and counter-convection, which is in the direction that is opposite from the propagation direction of the standard delay (transport PDE) process. We rely on the diffusion process to propagate the control signal through the PDE towards the ODE, to stabilize the ODE. 相似文献
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Zhigang Ren Chao Xu Zhongcheng Zhou Zongze Wu Tehuan Chen 《Journal of The Franklin Institute》2019,356(1):173-195
In this paper, the boundary stabilization problem of a class of unstable reaction–advection–diffusion (RAD) systems described by a scalar parabolic partial differential equation (PDE) is considered. Different the previous research, we present a new gradient-based optimization framework for designing the optimal feedback kernel for stabilizing the unstable PDE system. Our new method does not require solving non-standard Riccati-type or Klein–Gorden-type PDEs. Instead, the feedback kernel is parameterized as a second-order polynomial whose coefficients are decision variables to be tuned via gradient-based dynamic optimization, where the gradients of the system cost functional (which penalizes both kernel and output magnitude) with respect to the decision parameters are computed by solving a so-called “costate” PDE in standard form. Special constraints are imposed on the kernel coefficients to ensure that the optimized kernel yields closed-loop stability. Finally, three numerical examples are illustrated to verify the effectiveness of the proposed approach. 相似文献
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A control system of an ODE and a diffusion PDE is discussed in this paper. The novelty lies in that the system is coupled. The method of PDE backstepping as well as some special skills is resorted in stabilizing the coupled PDE–ODE control system, which is transformed into an exponentially stable PDE–ODE cascade with an invertible integral transformation. And a state feedback boundary controller is designed. Moreover, an exponentially convergent observer for anti-collocated setup is proposed, and the output feedback boundary control problem is solved. For both the state and output feedback boundary controllers, exponential stability analyses in the sense of the corresponding norms for the resulting closed-loop systems are given through rigid proofs. 相似文献
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Amine Laghrib Aissam Hadri Abdelilah Hakim 《Journal of The Franklin Institute》2019,356(11):5834-5857
The multiframe image super-resolution (SR) has appeal great care due to its relevant practical applications. Since the SR methods are always considered as ill-posed problems, great interest is therefore investigated to reconstruct an image with less artefacts. In this paper, we propose a high-order partial differential equation (PDE) that preserve singularities and edge continuities while smoothing. This PDE is based on anisotropic diffusion behavior by adding some constraints to the Weickert coherence enhancing diffusion tensor in order to control the diffusion process near edges and avoids tiny corners destruction. In fact, this PDE combine between the diffusion process of Perona–Malik in the flat regions and the anisotropic diffusion of the Weickert model near strong edges. Moreover, the introduction of the high-order derivatives reduces unwanted artifacts such as staircasing. The given experimental results show the effectiveness of the proposed PDE compared to some competitive methods in both visually and quantitatively. 相似文献
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《Journal of The Franklin Institute》2023,360(2):841-861
This article proposes a sliding mode control strategy for hyperbolic PDE systems under the requirement of finite-time boundedness. First, the singular perturbation theory is introduced to model multi-time scales phenomena, and a quantized measurement method is employed to save the communication resources in network. In addition, by considering the effect of the singular perturbation phenomenon in PDE systems, a sliding surface dependent on spatial position and singular perturbation parameter is constructed, then a sliding mode control law is developed to drive state trajectories to the designed sliding surface in finite time. Moreover, a partitioning strategy is introduced to ensure that the system is finite-time bounded in the reaching phase and the sliding motion phase, respectively. Finally, some sufficient conditions are given to ensure that the system is finite-time bounded in both reaching phase and sliding motion phase, and a simulation example of the chemical tubular reactor demonstrates the effectiveness of the proposed method. 相似文献
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Jun-Wei Wang 《Journal of The Franklin Institute》2019,356(17):10405-10420
This paper deals with the problem of boundary control for a class of semi-linear parabolic partial differential equations (PDEs) with non-collocated distributed event-triggered observation. A semi-linear Luenberger PDE observer with an output error based event-triggering condition is constructed by using the event-triggered observation to exponentially track the PDE state. By the estimated state, a feedback controller is proposed. It has been shown by the Lyapunov technique, and a variant of Poincaré–Wirtinger inequality that the resulting closed-loop coupled PDEs is exponentially stable if a sufficient condition presented in terms of standard linear matrix inequality (LMI) is satisfied. Moreover, a rigorous proof is provided for existence of a minimal dwell-time between two triggering times. Finally, numerical simulation results are given to show the effectiveness of the proposed design method. 相似文献
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《Journal of The Franklin Institute》2017,354(12):5067-5090
In this paper, the issue of exponential synchronization for coupled systems on networks with mixed time-varying delays is concerned. An approach combining Kirchhoff’s matrix tree theorem in graph theory with Lyapunov method and periodically intermittent control is taken to investigate the problem. This method is different from the corresponding previous works. Two different kinds of synchronization conditions in the form of Lyapunov-type theorem and coefficients-type criterion are derived. They both reveal synchronization has a close relation with the topology structure of the network. Finally, the feasibility and effectiveness of the proposed method are illustrated by several numerical simulation figures. 相似文献
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以“相似形”一章教学为例,浅谈在定理证明教学中,如何提高学生的解题能力。首先抓住定理证明的方法教学,其中浅谈以下几种方法：参数证题法,公比过渡法,间接证题法,构造三角形证题法。其次是注意定理的结语教学。还要重视定理应用数学中的能力培养。通过以上三种做法,就会对教学的重难点有所突破。 相似文献