Some symmetric boundary value problems and non-symmetric solutions

Non-symmetric low-index solutions for a symmetric boundary value problem

The (k,ψ)-fractional derivative based on the k-gamma function is a more general version of the Hilfer fractional derivative. It is widely used in differential equations to describe physical phenomena, population dynamics, and biological genetic memory problems. In this article, we mainly study the 4m+2-point symmetric integral boundary value problem of nonlinear (k,ψ)-fractional differential coupled Laplacian equations. The existence and uniqueness of solutions are obtained by the Krasnosel’skii fixed-point theorem and Banach’s contraction mapping principle. Furthermore, we also apply the calculus inequality techniques to discuss the stability of this system. Finally, three interesting examples and numerical simulations are given to further verify the correctness and effectiveness of the conclusions.

It is known that even entirely symmetric boundary value problems can admit solutions in which the inherent symmetry of the governing equations gets spontaneously broken. When this happens, two non-symmetric “twin solutions” occur so that the symmetry of the boundary value problem broken in the two individual solutions becomes restored by the twins. The present paper shows that, as a combined effect of buoyancy, viscous dissipation and pressure work, in a mixed convection flow in a vertical porous channel with isothermal walls, in addition to symmetric solutions, precisely this kind of twin solutions with broken symmetry can occur, although the walls are kept at the same temperature. The existence of another remarkable solution branch which describes symmetric adiabatic flows corresponding to nonlinear eigenvalues of the problem is also reported. The heat–work balance of all these steady-flow states is discussed in detail. The analysis is supported by an intuitive point-mechanical analogy. The equivalence of this mechanical energy analysis to the standard phase space method is also discussed shortly.

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

In this paper we discuss the uniqueness of the large solutions and metasolutions in a general class of radially symmetric singular boundary value problems.

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.

Penny-shaped crack in an elastic layer bonded to dissimilar half spaces

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280].

Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

ABSTRACTIn this paper, we study the initial-boundary value problem of the generalized Burgers equation posed on a finite interval with non-homogeneous boundary conditions. The boundary conditions are given in a general form, which covers the usual Dirichlet, Neumann or Robin boundary conditions. For the generalized Burgers equation, we establish the local well-posedness for the weak solution in when the Sobolev index is negative. Besides, for the classical Burgers equation with Dirichlet boundary conditions, we obtain the global well-posedness.

One-dimensional thermal wave transport in multilayered systems with an interface thermal resistance is studied under the framework of the Cattaneo–Vernotte hyperbolic heat conduction model, considering modulated heat excitation under Dirichlet and Neumann boundary conditions. For a single semi-infinite layer, analytical formulas useful in the measurement of its thermal relaxation time as well as additional thermal properties are presented. For a composite-layered system, in the thermally thin regime, with the Dirichlet boundary condition, the well known effective thermal resistance formula is obtained, while for the Neumann problem, only the heat capacity identity is found. In contrast, in the thermally thick case, an analytical expression for both Dirichlet and Neumann conditions is obtained for the effective thermal diffusivity of the whole system in terms of the thermal properties of the individual layers and their interface thermal resistance. The limits of applicability of this equation, in the thermally thick regime, are shown to provide useful and simple results in the characterization of layered systems and that they can be reduced to the results obtained using the Fourier approach. The role of the thermal relaxation time, the interface thermal resistance, and the implications of these results in the possibility of enhancement in heat transport are discussed.

This paper proposes a global approximation method to solve elliptic boundary value Poisson problems in arbitrary shaped 2‐D domains. Using transfinite interpolation, a symmetric finite element formulation is derived for degrees of freedom arranged mostly along the boundary of the domain. In cases where both Dirichlet and Neumann boundary conditions occur, the numerical solution is based on bivariate Coons interpolation using the boundary only. Furthermore, in case of only Dirichlet boundary conditions and no existing axes of symmetry, it is proposed to use at least one internal point and apply transfinite interpolation. The theory is sustained by five numerical examples applied to domains of square, circular and elliptic shape. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.

For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆.

We consider the one‐dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non‐Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.