On the Vibrations of an Axially Moving String With a Time-Dependent Velocity

2015 ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Axially Moving String ◽  
Fourier Sine Series ◽  
Axially Moving ◽  
Initial Boundary Value

The transverse vibrations of an axially moving string with a time-varying speed is studied in this paper. The governing equations of motion describing an axially moving string is analyzed using two different techniques. At first, the initial-boundary value problem is discretized using the Fourier sine series, and then the two timescales perturbation method is employed in search of infinite mode approximate solutions. Secondly, a new approach based on the two timescales perturbation method and the method of characteristics is used. It is found that there are infinitely many values of the velocity fluctuation frequency yielding infinitely many resonance conditions in the system. The response of the system with harmonically varying velocity function is computed for particular harmonic initial conditions.

scholarly journals On (Non) Applicability of a Mode-Truncation of a Damped Traveling String

2019 ◽  
Vol 38 (2) ◽  
pp. 471-478 ◽  
Author(s):  
Rajab Ali Malookani ◽  
Sajid Hussain Sandilo ◽  
Abdul Hanan
Keyword(s):  
Initial Conditions ◽  
Numerical Technique ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analytic Solutions ◽  
Amplitude Response ◽  
Lateral Vibrations ◽  
Axially Moving ◽  
Initial Boundary Value ◽  
Mode Truncation

This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximate analytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order ε -1

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

scholarly journals Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-31 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Difference Schemes ◽  
Parabolic Partial Differential Equation ◽  
Analysis Methods ◽  
Homotopy Analysis ◽  
Initial Boundary Value

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

Global weak entropy solutions to the Euler–Poisson system with spherical symmetry

2016 ◽  
Vol 26 (09) ◽  
pp. 1689-1734
Author(s):  
Jingjing Xiao
Keyword(s):  
Spherical Symmetry ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Poisson System ◽  
Godunov Scheme ◽  
Initial Value ◽  
Weak Entropy Solution ◽  
Initial Boundary Value

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform [Formula: see text] estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

scholarly journals APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

2020 ◽  
pp. 81-98
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Mixed Type ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Second Order ◽  
Initial Boundary Value

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

scholarly journals Inverse problems for the heat equation

2017 ◽  
Vol 21 (6) ◽  
pp. 62-75
Author(s):  
A.R. Zaynullov
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Initial Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Of Solution ◽  
Spatial Coordinates ◽  
Criterion Of Uniqueness ◽  
Initial Boundary Value ◽  
The Right

The inverse problem of finding initial conditions and the right-hand side had been studied for the inhomogeneous heat equation on the basis of formulas for the solution of the first initial-boundary value problem. A criterion of uniqueness of solution of the inverse problem for finding the initial condition was found with Spectral analysis. The right side of the heat equation is represented as a product of two functions, one of which depends on the spatial coordinates and the other from time. In one task, along with an unknown solution is sought factor on the right side, depending on the time, and in another - a factor that depends on the spatial coordinates. For these tasks, we prove uniqueness theorems, the existence and stability of solution.

scholarly journals The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
B. Deconinck ◽  
A. S. Fokas ◽  
J. Lenells
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Initial Value ◽  
Integrable Evolution ◽  
Initial Boundary Value ◽  
The Given

AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Sign in / Sign up

Export Citation Format

Share Document