Boundary Value Problems - Latest Articles
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The latest research articles published by Boundary Value Problems2015-04-17T00:00:00Z Global well-posedness of compressible magnetohydrodynamics with density-dependent viscosity and resistivity In this paper, we study the initial-boundary value problem for one-dimensional compressible magnetohydrodynamics (MHD) flows. Using the local estimates of strong solutions to three-dimensional compressible MHD (obtained by Fan and Yu in Nonlinear Anal. 69(10):3637-3660, 2008) and Sobolev’s inequalities, we get the unique global classical solution
(
ρ
,
u
,
b
)
, where
ρ
∈
C
1
(
[
0
,
T
]
;
H
1
(
[
0
,
1
]
)
)
,
u
∈
H
1
(
[
0
,
T
]
;
H
1
(
[
0
,
1
]
)
)
, and
b
∈
C
1
(
[
0
,
T
]
;
H
1
(
[
0
,
1
]
)
)
for any
T
>
0
. Here, we emphasize that the initial density
ρ
0
is permitted to contain vacuum states and the initial velocity
u
0
and the magnetic field
b
0
can be arbitrarily large. Also, both the viscosity coefficient μ and the resistivity coefficient ν depend on the density ρ.
http://www.boundaryvalueproblems.com/content/2015/1/62
Menglong SuYingzhao LiuWenzhuang ZhuJian WangBoundary Value Problems 2015, null:622015-04-17T00:00:00Zdoi:10.1186/s13661-015-0324-6/content/figures/s13661-015-0324-6-toc.gifBoundary Value Problems1687-2770${item.volume}622015-04-17T00:00:00ZXML Blow-up for a parabolic system with nonlocal sources and nonlocal boundary conditions This paper deals with blow-up properties of solutions to a nonlocal parabolic system with nonlocal boundary conditions. The global existence and finite time blow-up criteria are obtained. Moreover, for some special cases, we establish the precise blow-up rate estimates.
http://www.boundaryvalueproblems.com/content/2015/1/61
Guangsheng ZhongLixin TianBoundary Value Problems 2015, null:612015-04-16T00:00:00Zdoi:10.1186/s13661-015-0325-5/content/figures/s13661-015-0325-5-toc.gifBoundary Value Problems1687-2770${item.volume}612015-04-16T00:00:00ZXML Local well-posedness for periodic Benjamin equation with small initial data The local well-posedness of the periodic Benjamin equation with small initial value in
H
s
(
T
)
,
s
≥
−
1
/
2
, is given. It is here shown that
−
1
/
2
is the lower endpoint to obtain the bilinear estimates which are the crucial steps to obtain the local well-posedness by the Picard iteration.MSC: 35Q53, 35Q55.
http://www.boundaryvalueproblems.com/content/2015/1/60
Shaoguang ShiJunfeng LiBoundary Value Problems 2015, null:602015-04-04T12:00:00Zdoi:10.1186/s13661-015-0322-8/content/figures/s13661-015-0322-8-toc.gifBoundary Value Problems1687-2770${item.volume}602015-04-04T12:00:00ZXML A POD-based reduced-order TSCFE extrapolation iterative format for two-dimensional heat equations In this article, a proper orthogonal decomposition (POD) technique is employed to establish a POD-based reduced-order time-space continuous finite element (TSCFE) extrapolation iterative format for two-dimensional (2D) heat equations, which includes very few degrees of freedom but holds sufficiently high accuracy. The error estimates of the POD-based reduced-order TSCFE solutions and the algorithm implementation of the POD-based reduced-order TSCFE extrapolation iterative format are provided. A numerical example is used to illustrate that the results of the numerical computation are consistent with the theoretical conclusions. Moreover, it is shown that the POD-based reduced-order TSCFE extrapolation iterative format is feasible and efficient for solving 2D heat equations.MSC: 74S10, 65M15, 35Q35.
http://www.boundaryvalueproblems.com/content/2015/1/59
Zhendong LuoBoundary Value Problems 2015, null:592015-04-04T00:00:00Zdoi:10.1186/s13661-015-0320-x/content/figures/s13661-015-0320-x-toc.gifBoundary Value Problems1687-2770${item.volume}592015-04-04T00:00:00ZXML Optimal control problem for a generalized sixth order Cahn-Hilliard type equation with nonlinear diffusion In this paper, we study the initial-boundary-value problem for a generalized sixth order Cahn-Hilliard type equation, which describes the separation properties of oil-water mixtures when a substance enforcing the mixing of the phases is added. The optimal control under boundary condition is given and the existence of optimal solution is proved.MSC: 49J20, 35K35, 35K55.
http://www.boundaryvalueproblems.com/content/2015/1/58
Gongcao ShiChangchun LiuZhao WangBoundary Value Problems 2015, null:582015-04-03T12:00:00Zdoi:10.1186/s13661-015-0321-9/content/figures/s13661-015-0321-9-toc.gifBoundary Value Problems1687-2770${item.volume}582015-04-03T12:00:00ZXML The inverse scattering problem of some Schrödinger type equation with turning point In this paper the inverse scattering problem is considered for a version of the one-dimensional Schrödinger equation with turning point on the half-line
(
0
,
∞
)
. The scattering data of the problem is defined and the fundamental equation is derived. With the help of the derived fundamental equation, in terms of the scattering data, the potential is recovered uniquely.MSC: 58C40, 34L25, 34B05, 47A40.
http://www.boundaryvalueproblems.com/content/2015/1/57
Zaki El-RaheemFarouk SalamaBoundary Value Problems 2015, null:572015-04-03T00:00:00Zdoi:10.1186/s13661-015-0316-6/content/figures/s13661-015-0316-6-toc.gifBoundary Value Problems1687-2770${item.volume}572015-04-03T00:00:00ZXML Multiplicity of solutions for fractional Schrödinger equations with perturbation In this paper, we investigate a class of fractional Schrödinger equations with perturbation. By using the mountain pass theorem and Ekeland’s variational principle, we see that such equations possess two solutions. Recent results in the literature are generalized and significantly improved.MSC: 34B15.
http://www.boundaryvalueproblems.com/content/2015/1/56
Liu YangBoundary Value Problems 2015, null:562015-03-31T00:00:00Zdoi:10.1186/s13661-015-0317-5/content/figures/s13661-015-0317-5-toc.gifBoundary Value Problems1687-2770${item.volume}562015-03-31T00:00:00ZXML An iterative regularization method for an abstract ill-posed biparabolic problem In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a regularizing strategy based on the Kozlov-Maz’ya iteration method. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.MSC: 47A52, 65J22.
http://www.boundaryvalueproblems.com/content/2015/1/55
Abdelghani LakhdariNadjib BoussetilaBoundary Value Problems 2015, null:552015-03-28T00:00:00Zdoi:10.1186/s13661-015-0318-4/content/figures/s13661-015-0318-4-toc.gifBoundary Value Problems1687-2770${item.volume}552015-03-28T00:00:00ZXML Exact and asymptotic analysis of waves generated by sea-floor disturbances on a sloping beach A two-dimensional problem of small amplitude waves generated by some sea-bottom disturbance is studied on a sloping beach. The exact analytical solution for the wave height is provided, and with a periodic ground motion the asymptotic analysis of the waves is shown to exist for all time even in the vicinity of the shoreline. The novelty of this work lies in solving the corresponding Fredholm integral equation of the first kind and also to provide a uniform asymptotic estimate of the wave integral in the unsteady state involving both pole and saddle points for which Van der Waerden’s method is used. Within the framework of linear irrotational theory, explicit integral solutions of waves and their asymptotic and/or numerical computations presented here aim to provide an equivalent mathematical understanding to all such wave propagation which can be modelled in two dimensions at the open sea.
http://www.boundaryvalueproblems.com/content/2015/1/54
Arghya BandyopadhyayMaria Otero-EspinarBoundary Value Problems 2015, null:542015-03-26T12:00:00Zdoi:10.1186/s13661-015-0315-7/content/figures/s13661-015-0315-7-toc.gifBoundary Value Problems1687-2770${item.volume}542015-03-26T12:00:00ZXML Infinite first order differential systems with nonlocal initial conditions We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable ‘sub-linear’ growth then the system has at least one solution. The approach relies on the application, in a suitable Fréchet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of an application is also provided.MSC: 34A12, 34A34, 47H30.
http://www.boundaryvalueproblems.com/content/2015/1/53
Gennaro InfantePetru JebeleanFadila MadjidiBoundary Value Problems 2015, null:532015-03-21T12:00:00Zdoi:10.1186/s13661-015-0314-8/content/figures/s13661-015-0314-8-toc.gifBoundary Value Problems1687-2770${item.volume}532015-03-21T12:00:00ZXML