We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: {\it If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane}. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one-dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system.

The formulation and analysis of two operator Boundary-Domain Integral Equation systems for variable coefficient mixed BVP with in two dimensional domain is discussed. To analyse the two-operator approach, we applied one of its linear versions to the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE and reduced it to four different BDIE systems. %The two-operator BDIE systems are nonstandard systems of equations containing integral operators defined on the domain under consideration and potential type and pseudo-differential operators defined on open sub-manifolds of the boundary. Using the results as in CMN09 andAM11, a rigorous analysis of the two-operator BDIE systems is given and it is shown that they are equivalent to the mixed BVP and thus are uniquely solvable, while the corresponding boundary domain integral operators are invertible in the appropriate Sobolev-Slobodetski (Bessel potential) spaces.

In this paper, the Boundary-Domain Integral Equations (BDIEs) for the mixed boundary value problem(BVP) for a compressible Stokes system of partial differential equation (PDE) with variable coefficient in 2D is considered . An appropriate parametrix is used to reduce this BVP to the BDIEs. Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems in appropriate Sobolev spaces are proved.

In this paper, the Dirichlet and Neumann boundary value problems for the steady state Stokes system of partial differential equations for a compressible viscous fluid with variable viscosity coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to a system of direct segregated boundary-domain integral equations (BDIEs). The BDIEs in the two dimensional case have special properties in comparison with the three dimension because of the logarithmic term in the parametrix for the associated partial differential equations. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of corresponding parametrix-based hydrodaynamic single layer and hypersingular potentials and hence the unique solvability of BDIEs. Equivalence of the BDIE systems to the Dirichlet and Neumann BVPs and the invertibility of the corresponding boundary-domain integral operators in appropriate Sobolev spaces are shown.

The paper is devoted to the Stackelberg control of a linear parabolic equation with missing initial conditions. The strategy involves two controls called follower and leader. The objective of the follower is to bring the state to a desired state while the leader has to bring the system to rest at the final time. The results are obtained by means of Fenchel-Legendre transform and appropriate Carleman inequalities.

The aim of this work is to present new approach to study weighted pseudo almost periodic functions with infinite delay using the measure theory. We study the existence and uniqueness of (μ,ν)-pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed on ]-∞,0].

The newly generalized energy storage component namely memristor is a fundamental circuit element so called universal charge-controlled mem-element is proposed for controlling the analysis and coexisting attractors. The governing differential equations of memristor are highly non-linear for mathematical relationships. The mathematical model of memristor is established in terms of newly defined fractal-fractional differential operators so called Atangana-Baleanu, Caputo-Fabrizio and Caputo fractal-fractional differential operator. A novel numerical approach is developed for the governing differential equations of memristor on the basis of Atangana-Baleanu, Caputo-Fabrizio and Caputo fractal-fractional differential operator. We discussed chaotic behavior of memristor under three criteria as (i) varying fractal order, we fixed fractional order, (ii) varying fractional order, we fixed fractal order and (ii) varying fractal and fractional orders simultaneously. Our investigated graphical illustrations and simulated results via MATLAB for the chaotic behaviors of memristor suggest that newly presented Atangana-Baleanu, Caputo-Fabrizio and Caputo fractal-fractional differential operator has generates significant results as compared with classical approach.

An SIS epidemic reaction-diffusion model with saturated incidence rate and spontaneous infection is considered. We establish the existence of endemic equilibrium by using a fixed point theorem. We mainly investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium. Our analysis shows that the spontaneous infection can enhance persistence of infectious disease.

In this study, we investigate the sum-type singular nonlinear fractional q-integro-differential $m$-point boundary value problem. The existence of positive solutions is obtained by the properties of the Green function, standard Caputo $q$-derivative, Riemann-Liouville fractional $q$-integral and the means of a fixed point theorem on a real Banach space $(\mathcal{X}, \|.\|)$ which has a partially order by using a cone $P \subset \mathcal{X}$. The proofs are based on solving the operator equation $\mathcal{O}_1 x + \mathcal{O}_2 x = x $ such that the operator $\mathcal{O}_1$, $\mathcal{O}_2$ are $r$-convex, sub-homogeneous, respectively and define on cone $P$. As applications, we provide an example illustrating the primary effects.

We study properties of a piecewise deterministic Markov process modeling the changes in concentration of specific antibodies.The evolution of densities of the process is described by a stochastic semigroup. The long-time behaviour of this semigroup is studied. In particular we prove theorems on its asymptotic stability.

In this paper, we focus on designing a well-conditioned Glarkin spectral methods for solving a two-sided fractional diffusion equations with drift, in which the fractional operators are defined neither in Riemann-Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann-Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well-conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion.

In this paper, we deal with the wave equation with acoustic boundary conditions. The exponential stabilization is obtained by Lyapunov approach and Riemannian geometry method. We then apply our main theorem to the wave equations with memory type acoustic boundary conditions, which is not available in the literature and give an example in the end.

The Dirichlet, Neumann and mixed boundary value problems for the linear second-order scalar elliptic differential equation with variable coefficient in a bounded three-dimensional domain are considered. The PDE right-hand side belongs to $H^{-1}(\Omega)$ or $\widetilde{H}^{-1}(\Omega),$ when neither classical nor canonical co-normal derivatives are well defined. Using the two-operator approach and appropriate parametrix (Levi function) each problem is reduced to different systems of boundary domain integral equations (BDIEs). Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as invertibility of the BDIE operators are analysed in appropriate Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

We consider a linear RLC network, modeled by differential-algebraic equations, containing distributed semiconductor devices, modeled by multi-dimensional parabolic-elliptic equations. With appropriate coupling conditions, we obtain a system of multidimensional partial differential-algebraic equations. For the resulting system, we prove an existence and uniqueness result, and study the asymptotic behavior of the solutions.

A linear manifold ${\mathcal K}_2$ of evolutionary equations for a pseudovector field on ${\Bbb R}^3$ is described. An infinitisimal shift of each equation is determined by a second-order differential operator of divergent type. All operators are invariant with respect to space translations in ${\Bbb R}^3$, relative to time translations, and they are transformed by covariant way relative to rotations of ${\Bbb R}^3$. It is proved that the linear space ${\mathcal M}_2 \subset {\mathcal K}_2$ of differential operators preserving solenoidal property and unimodularity of the field is one-dimensional and an explicit form of such operators is found.