On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1

2018 ◽  
Vol 25 (3) ◽  
pp. 427-439
Author(s):  
Arezki Kheloufi
Keyword(s):  
Dimensional Space ◽  
Boundary Value ◽  
Smooth Functions ◽  
Geometrical Properties ◽  
Regular Type ◽  
Noncylindrical Domain ◽  
Significant Case ◽  
Type Boundary ◽  
The One ◽  
Type Boundary Condition

AbstractThis paper is devoted to the analysis of the boundary value problem {\partial_{t}u-\Delta u=f}, with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q=\{(t,x_{1})\in\mathbb{R}^{2}:0<t<T,\varphi_{1}(t)<x_{1}<\varphi_{2}(t)\}% \times\prod_{i=1}^{N-1}{]0,b_{i}[}, where {\varphi_{1}} and {\varphi_{2}} are smooth functions. One of the main issues of the paper is that the domain can possibly be non-regular; for instance, the significant case when {\varphi_{1}(0)=\varphi_{2}(0)} is allowed. We prove well-posedness results for the problem in a number of different settings and under natural assumptions on the coefficients and on the geometrical properties of the domain. This work is an extension of the one-dimensional case studied in [4].

scholarly journals Advective balance in pipe-formed vortex rings

2017 ◽  
Vol 836 ◽  
pp. 773-796
Author(s):  
Karim Shariff ◽  
Paul S. Krueger
Keyword(s):  
Reynolds Numbers ◽  
Vortex Rings ◽  
Quadratic Term ◽  
Axisymmetric Vortex ◽  
Order Of Magnitude ◽  
Cylindrical Diffusion ◽  
Type Boundary ◽  
The One ◽  
Type Boundary Condition ◽  
Edge Layer

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

scholarly journals The existence of three positive solutions to integral type BVPs for singular second ODEs with one-dimensional p-Laplacian

2011 ◽  
Vol 27 (2) ◽  
pp. 239-248
Author(s):  
YUJI LIU ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Integral Type ◽  
Singular Differential Equations ◽  
One Dimensional ◽  
Type Boundary ◽  
The One

This paper is concerned with the integral type boundary value problems of the second order singular differential equations with one-dimensional p-Laplacian. Sufficient conditions to guarantee the existence of at least three positive solutions are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensional p-Laplacian term [ρ(t)Φ(x 0 (t))]0 involved with the function ρ, which makes the solutions un-concave. Furthermore, f, g, h and ρ may be singular at t = 0 or t = 1.

Fractional advection–diffusion equation with memory and Robin-type boundary condition

2019 ◽  
Vol 14 (3) ◽  
pp. 306 ◽  
Author(s):  
Itrat Abbas Mirza ◽  
Dumitru Vieru ◽  
Najma Ahmed
Keyword(s):  
Diffusion Equation ◽  
Fourier Transforms ◽  
Solute Concentration ◽  
Minimum Concentration ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Type Boundary ◽  
The One ◽  
Type Boundary Condition ◽  
With Memory

The one-dimensional fractional advection–diffusion equation with Robin-type boundary conditions is studied by using the Laplace and finite sine-cosine Fourier transforms. The mathematical model with memory is developed by employing the generalized Fick’s law with time-fractional Caputo derivative. The influence of the fractional parameter (the non-local effects) on the solute concentration is studied. It is found that solute concentration can be minimized by decreasing the memory parameter. Also, it is found that, at small values of time the ordinary model leads to minimum concentration, while at large values of the time the fractional model is recommended.

scholarly journals SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR SINGULAR QUASI-LAPLACIAN DIFFERENTIAL EQUATIONS ON THE WHOLE LINE

2012 ◽  
Vol 17 (3) ◽  
pp. 423-446 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Continuous Operator ◽  
Completely Continuous ◽  
One Dimensional ◽  
Type Boundary ◽  
The One

This paper is concerned with some integral type boundary value problems associated to second order singular differential equations with quasi-Laplacian on the whole line. The emphasis is put on the one-dimensional p-Laplacian term involving a nonnegative function ρ that may be singular at t = 0 and such that . A Banach space and a nonlinear completely continuous operator are defined in this paper. By using the Schauder's fixed point theorem, sufficient conditions to guarantee the existence of at least one solution are established. An example is presented to illustrate the main theorem.

Diffraction contrast of dislocations in icosahedral quasicrystals

1990 ◽  
Vol 48 (4) ◽  
pp. 454-455
Author(s):  
K. Urban ◽  
Z. Zhang ◽  
M. Wollgarten ◽  
D. Gratias
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Characterization ◽  
Extinction Condition ◽  
Burgers Vector ◽  
Diffraction Contrast ◽  
Lattice Geometry ◽  
Reference Space ◽  
Icosahedral Quasicrystals ◽  
The One

Recently dislocations have been observed by electron microscopy in the icosahedral quasicrystalline (IQ) phase of Al65Cu20Fe15. These dislocations exhibit diffraction contrast similar to that known for dislocations in conventional crystals. The contrast becomes extinct for certain diffraction vectors g. In the following the basis of electron diffraction contrast of dislocations in the IQ phase is described. Taking account of the six-dimensional nature of the Burgers vector a “strong” and a “weak” extinction condition are found.Dislocations in quasicrystals canot be described on the basis of simple shear or insertion of a lattice plane only. In order to achieve a complete characterization of these dislocations it is advantageous to make use of the one to one correspondence of the lattice geometry in our three-dimensional space (R3) and that in the six-dimensional reference space (R6) where full periodicity is recovered . Therefore the contrast extinction condition has to be written as gpbp + gobo = 0 (1). The diffraction vector g and the Burgers vector b decompose into two vectors gp, bp and go, bo in, respectively, the physical and the orthogonal three-dimensional sub-spaces of R6.

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