scholarly journals Sinc-Galerkin method for solving hyperbolic partial differential equations

2018 ◽  
Vol 8 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Hyperbolic Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Numerical Examples

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

Boundary control design based on partial differential equation observer for vibration suppression and attitude control of flexible satellites with multi-section solar panels

2021 ◽  
pp. 107754632199015
Author(s):  
Mohammad Mahdi Ataei ◽  
Hassan Salarieh ◽  
Hossein Nejat Pishkenari ◽  
Hadi Jalili
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Attitude Control ◽  
Vibration Suppression ◽  
Equation System ◽  
Solar Panels ◽  
Attitude Dynamics ◽  
Partial Differential ◽  
Satellite Attitude ◽  
Element Technique

A novel partial differential equation observer is proposed to be used in boundary attitude and vibration control of flexible satellites. Solar panels’ vibrations and attitude dynamics form a coupled partial differential equation–ordinary differential equation system which is controlled directly without discretization. Few feedback signals from boundaries are required which are estimated via a partial differential equation observer. Consequently, just satellite attitude and angular velocity should be measured and still the control system benefits information from continuous part vibrations. The closed-loop system is proved to be asymptotically stable. Simulations with a finite element technique illustrate good performance of this observer-based boundary controller.

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

scholarly journals Geometric shape features extraction using a steady state partial differential equation system

2019 ◽  
Vol 6 (4) ◽  
pp. 647-656 ◽  
Author(s):  
Takayuki Yamada
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Analytical Solution ◽  
Geometric Shape ◽  
Geometrical Shape ◽  
Shape Features ◽  
Partial Differential ◽  
Topological Constraint ◽  
Numerical Examples

Abstract A unified method for extracting geometric shape features from binary image data using a steady-state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantage of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method. Highlights A steady state partial differential equation for extraction of geometrical shape features is formulated. The functions for geometrical shape features are formulated by the solution of the proposed PDE. Analytical solution is provided in one-dimension. Numerical examples are provided in two-dimension.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

scholarly journals On Exact Solutions of Phi-4 Partial Differential Equation Using the Enhanced Modified Simple Equation Method

2018 ◽  
Vol 6 (4) ◽  
Author(s):  
Ziad Salem Rached
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Simple Equation ◽  
Equation Method ◽  
Partial Differential ◽  
New Exact Solutions ◽  
Modified Simple Equation Method

Constructing exact solutions of nonlinear ordinary and partial differential equations is an important topic in various disciplines such as Mathematics, Physics, Engineering, Biology, Astronomy, Chemistry,… since many problems and experiments can be modeled using these equations. Various methods are available in the literature to obtain explicit exact solutions. In this correspondence, the enhanced modified simple equation method (EMSEM) is applied to the Phi-4 partial differential equation. New exact solutions are obtained.

Controlling of the Pollution Based on a Prediction Model

2014 ◽  
Vol 513-517 ◽  
pp. 4094-4097
Author(s):  
Jing Mei Yang ◽  
Xiang Hui Zhao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Prediction Model ◽  
Environmental Monitoring ◽  
Partial Differential ◽  
Numerical Examples ◽  
Proposed Model ◽  
Efficient Measures

Controlling of the pollution plays an important role in environmental monitoring. This paper proposed a prediction model for the evolution of pollution in a certain region. With the statistic data of the pollution in this region, several parameters can be determined. With solving the related partial differential equation, the evolution of the pollution in next days can be predicted. Then some efficient measures can be adopted for controlling. The efficiency and the accuracy of the proposed model are shown in presented numerical examples.

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