scholarly journals Cubic Spline Method for 1D Wave Equation in Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Polar Coordinates ◽  
Spline Method ◽  
Implicit Difference Scheme ◽  
Time Direction ◽  
Numerical Examples ◽  
Theoretical Results

Using nonpolynomial cubic spline approximation in space and finite difference in time direction, we discuss three-level implicit difference scheme of O(k2+h4) for the numerical solution of 1D wave equations in polar coordinates, where k>0 and h>0 are grid sizes in time and space coordinates, respectively. The proposed method is applicable to problems with singularity. Stability theory of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

scholarly journals Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Iterative Method ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Poisson’S Equation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Poisson's Equation ◽  
Polar Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Diffusion Convection

Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

scholarly journals Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 597 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Wave Equation ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Wave Equations ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
High Rate ◽  
Numerical Examples ◽  
Linear And Nonlinear

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

scholarly journals Local Convexity-PreservingC2Rational Cubic Spline for Convex Data

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd Majid ◽  
Jamaludin Md. Ali
Keyword(s):  
Shape Parameter ◽  
Cubic Spline ◽  
Convex Curve ◽  
Shape Feature ◽  
Local Convexity ◽  
Shape Parameters ◽  
Numerical Examples ◽  
2D Data ◽  
Industrial Demand ◽  
Made In

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data usingC2rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

On the existence of compressional MHD oscillations in an inhomogeneous magnetoplasma

1990 ◽  
Vol 44 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Andrew N. Wright
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Density Distribution ◽  
Cold Plasma ◽  
Wave Equations ◽  
Perturbation Field ◽  
Plasma Density Distribution ◽  
Background Magnetic Field ◽  
Transverse Fields ◽  
The Magnetic Field

In a cold plasma the wave equation for solely compressional magnetic field perturbations appears to decouple in any surface orthogonal to the background magnetic field. However, the compressional fields in any two of these surfaces are related to each other by the condition that the perturbation field b be divergence-free. Hence the wave equations in these surfaces are not truly decoupled from one another. If the two solutions happen to be ‘matched’ (i.e. V.b = 0) then the medium may execute a solely compressional oscillation. If the two solutions are unmatched then transverse fields must evolve. We consider two classes of compressional solutions and derive a set of criteria for when the medium will be able to support pure compressional field oscillations. These criteria relate to the geometry of the magnetic field and the plasma density distribution. We present the conditions in such a manner that it is easy to see if a given magnetoplasma is able to executive either of the compressional solutions we investigate.

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals Stability Analysis of Impulsive Control Systems with Finite and Infinite Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xuling Wang ◽  
Xiaodi Li ◽  
Gani Tr. Stamov
Keyword(s):  
Control Systems ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Infinite Delays ◽  
Impulsive Control Systems ◽  
Stable Matrix ◽  
Theoretical Results

This paper studies impulsive control systems with finite and infinite delays. Several stability criteria are established by employing the largest and smallest eigenvalue of matrix. Our sufficient conditions are less restrictive than the ones in the earlier literature. Moreover, it is shown that by using impulsive control, the delay systems can be stabilized even if it contains no stable matrix. Finally, some numerical examples are discussed to illustrate the theoretical results.

A Solution Method for an Optimal Controlled Vibrating Circle Shell by Measure and Classical Trajectory

2011 ◽  
Vol 52-54 ◽  
pp. 1855-1860
Author(s):  
J.A. Fakharzadeh ◽  
F.N. Jafarpoor
Keyword(s):  
Wave Equations ◽  
Classical Trajectory ◽  
Polar Coordinates ◽  
Approximation Schemes ◽  
Solution Technique ◽  
Underlying Space ◽  
The Mean ◽  
Positive Coefficients ◽  
And Control ◽  
Finite Linear

The mean idea of this paper is to present a new combinatorial solution technique for the controlled vibrating circle shell systems. Based on the classical results of the wave equations on circle domains, the trajectory is considered as a finite trigonometric series with unknown coefficients in polar coordinates. Then, the problem is transferred to one in which its unknowns are a positive Radon measure and some positive coefficients. Extending the underlying space helps us to prove the existence of the solution. By using the density properties and some approximation schemes, the problem is deformed into a finite linear programming and the nearly optimal trajectory and control are identified simultaneously. A numerical example is also given.

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