scholarly journals FRACTIONAL APPROXIMATIONS OF THERMOELASTIC PLATE SYSTEMS

2021 ◽  
Author(s):  
Flank Bezerra ◽  
CÁSSIO FEITOSA
Keyword(s):  
Initial Conditions ◽  
Integral Formula ◽  
Geometric Theory ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Systems ◽  
Thermoelastic Plate ◽  
Initial Boundary Value ◽  
Plate System

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partial\Omega$ is assumed to be a $\mathcal{C}^4$-hypersurface. In this paper we consider the initial-boundary value problem associated with the following thermoelastic plate system \[ \begin{cases} \partial_t^2u +\Delta^2 u+\Delta\theta=f(u),\ & x\in\Omega,\ t>0, \\ \partial_t\theta-\Delta \theta-\Delta \partial_tu=0,\ & x\in\Omega,\ t>0, \end{cases} \] subject to boundary conditions \[ \begin{cases} u=\Delta u=0,\ & x\in\partial\Omega,\ t>0,\\ \theta=0,\ & x\in\partial\Omega,\ t>0, \end{cases} \] and initial conditions \[ u(x,0)=u_0(x),\ \partial_tu(x,0)=v_0(x)\ \mbox{and}\ \theta(x,0)=\theta_0(x),\ x\in\Omega. \] We calculate explicit the fractional powers of the thermoelastic plate operator associated with this system via Balakrishnan integral formula and we present a fractional approximated system. We obtain a result of local well-posedness of the thermoelastic plate system and of its fractional approximations via geometric theory of semilinear parabolic systems.

Positivity for elliptic and parabolic systems

1978 ◽  
Vol 79 (3-4) ◽  
pp. 183-191 ◽  
Author(s):  
J. F. G. Auchmuty
Keyword(s):  
Initial Conditions ◽  
Elliptic Systems ◽  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Biological Phenomena ◽  
Weakly Coupled ◽  
Coupling Terms ◽  
Initial Boundary Value

SynopsisThe positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

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 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.

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