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We prove convergence of a sequence of weak solutions of the nonlocal Cahn-Hilliard equation to the strong solution of the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a W 1 , 1 -kernel. The proof is based on the relative entropy method. Additionally, we prove the strong L 2 -convergence of the nonlocal operator to the negative Laplacian together with a rate of convergence. [ABSTRACT FROM AUTHOR]

We develop a sparse spectral method for a class of fractional differential equations, posed on R , in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on [ - 1 , 1 ] whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping [ - 1 , 1 ] to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size O (n) × O (n) , with O (n) nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an O (n) complexity solve. Applications to fractional heat and wave equations are considered. [ABSTRACT FROM AUTHOR]

We prove a characterization of Hardy's inequality in Sobolev–Slobodeckiĭ spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods. [ABSTRACT FROM AUTHOR]

In this paper, Witte's conditions for the uniqueness solution of nonlinear differential equations with integer and non-integer order derivatives are investigated. We present a detailed analysis of the uniqueness solutions of four classes of nonlinear differential equations with nonlocal operators. These classes include classical and fractional ordinary differential equations in fractal calculus. For each case, theorems and lemmas and their proofs are presented in detail.

In this paper we prove existence of solutions to Schrödinger-Maxwell type systems involving mixed local-nonlocal operators. Two different models are considered: classical Schrödinger-Maxwell equations and Schrödinger-Maxwell equations with a coercive potential, and the main novelty is that the nonlocal part of the operator is allowed to be nonpositive definite according to a real parameter. We then provide a range of parameter values to ensure the existence of solitary standing waves, obtained as Mountain Pass critical points for the associated energy functionals. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide sufficient conditions implying that the effective domain $ D(A\phi) $ of an $ m $-accretive operator $ A\phi $ in $ L^1 $ is dense in $ L^1 $. Here, $ A\phi $ refers to the composition $ A\circ \phi $ in $ L^1 $ of the part $ A = (\partial\mathcal{E})_{\vert L^{1\cap \infty}} $ in $ L^{1\cap\infty} $ of the subgradient $ \partial\mathcal{E} $ in $ L^2 $ of a convex, proper, lower semicontinuous functional $ \mathcal{E} $ on $ L^2 $ and a continuous, strictly increasing function $ \phi $ on the real line $ \mathbb{R} $. To illustrate the role of the sufficient conditions, we apply our main result to the class of doubly nonlinear operators $ A\phi $, where $ A $ is a classical Leray-Lions operator. [ABSTRACT FROM AUTHOR]

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases. [ABSTRACT FROM AUTHOR]

In this survey we discuss some existence and asymptotic results, originally obtained in [4,3], for functions of least Ws,1-fractional seminorm. We present the connection between these functions and nonlocal minimal surfaces, leveraging this relation to build a function of least fractional seminorm. We further prove that a function of least fractional seminorm is the limit for p → 1 of the sequence of minimizers of the Ws,p-energy. Additionally, we consider the fractional 1-Laplace operator and study the equivalence between weak solutions and functions of least fractional seminorm.

We present some recent results on a class of degenerate operators which are modeled on the fractional Laplacian, converge to the truncated Laplacian, and are extremal among operators with fractional diffusion along subspaces of possibly different dimensions. In particular, we will recall basic properties of these operators, validity of maximum principles, and related phenomena.

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

We study a nonlinear inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated with a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis, we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction to the case of fractional elasticity. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity. [ABSTRACT FROM AUTHOR]

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings. [ABSTRACT FROM AUTHOR]

In this work we study nonlocal operators and corresponding spaces with a focus on operators of order near zero. We investigate the interior regularity of eigenfunctions and of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem and we prove that eigenfunctions are of class C ∞ if the kernel satisfies this property away from its singularity. Similarly in this case, if in the Poisson problem the right-hand is of class C ∞ , then also any weak solution is of class C ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal Cahn–Hilliard equation (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn–Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in L 1 of the Cahn–Hilliard Equation. We also show that the Cahn–Hilliard equation is the gradient flow of the Ginzburg–Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]

We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of p-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and R N with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allows us to cover the nonlocal counterparts of a large scope of local diffusion problems like, for example, Stefan problems, Hele–Shaw problems, diffusion in porous media problems and obstacle problems. Nonlinear semigroup theory is the basis for this study. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a simple nonlocal-in-time dynamic system proposed for the effective modeling of complex diffusive regimes in heterogeneous media. We present its solutions and their commonly studied statistics such as the mean square distance. This interesting model employs a nonlocal operator to replace the conventional first-order time-derivative. It introduces a finite memory effect of a constant length encoded through a kernel function. The nonlocal-in-time operator is related to fractional time derivatives that rely on the entire timehistory on one hand, while reduces to, on the other hand, the classical time derivative if the length of the memory window diminishes. This allows us to demonstrate the effectiveness of the nonlocal-in-time model in capturing the crossover widely observed in nature between the initial sub-diffusion and the long time normal diffusion. [ABSTRACT FROM AUTHOR]

We obtain new equitightness and $ C([0, T];L^p({\mathbb{R}}^N)) $-convergence results for finite-difference approximations of generalized porous medium equations of the form$ \partial_tu-\mathfrak{L}[\varphi(u)] = g\qquad\text{in }~~ {\mathbb{R}}^N\times(0, T) , $where $ \varphi: {\mathbb{R}}\to {\mathbb{R}} $ is continuous and nondecreasing, and $ \mathfrak{L} $ is a local or nonlocal diffusion operator. Our results include slow diffusions, strongly degenerate Stefan problems, and fast diffusions above a critical exponent. These results improve the previous $ C([0, T];L_ \text{loc}^p({\mathbb{R}}^N)) $-convergence obtained in a series of papers on the topic by the authors. To have equitightness and global $ L^p({\mathbb{R}}^N) $-convergence, some additional restrictions on $ \mathfrak{L} $ and $ \varphi $ are needed. Most commonly used symmetric operators $ \mathfrak{L} $ are still included: the Laplacian, fractional Laplacians, and other generators of symmetric Lévy processes with some fractional moment. We also discuss extensions to nonlinear possibly strongly degenerate convection-diffusion equations. [ABSTRACT FROM AUTHOR]

We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $ \partial_t u = - \mathcal{L} u^m $ posed on a bounded Euclidean domain $ \Omega\subset \mathbb{R}^N $ with smooth boundary and $ N\ge 1 $. The linear diffusion operator $ \mathcal{L} $ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $ u^m = |u|^{m-1}u $ with $ 0

We study the generalized eigenvalue problem on the whole space for a class of integro-differential elliptic operators. The nonlocal operator is over a finite measure, but this has no particular structure. Some of our results even hold for singular kernels. The first part of the paper presents results concerning the existence of a principal eigenfunction. Then we present various necessary and/or sufficient conditions for the maximum principle to hold, and use these to characterize the simplicity of the principal eigenvalue. [ABSTRACT FROM AUTHOR]

For the spectral fractional diffusion operator of order 2s, s ∈ (0 , 1) , in bounded, curvilinear polygonal domains Ω ⊂ R 2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm H s (Ω) . The first hp discretization is based on writing the solution as a co-normal derivative of a 2 + 1 -dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in Ω . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in Ω , exponential convergence rates for solutions u ∈ H s (Ω) of L s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards ∂ Ω . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L - s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in Ω . The present analysis for either approach extends to (polygonal subsets M ~ of) analytic, compact 2-manifolds M , parametrized by a global, analytic chart χ with polygonal Euclidean parameter domain Ω ⊂ R 2 . Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced. [ABSTRACT FROM AUTHOR]

In this article we consider direct and inverse problems for α -stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of directional antilocality as introduced by Y. Ishikawa in the 80s. We consider the Dirichlet problem for these operators on the respective "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the direct problem with infinite data and single measurement uniqueness results for the associated inverse problems. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise. [ABSTRACT FROM AUTHOR]

In this paper we investigate whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated. [ABSTRACT FROM AUTHOR]

We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Hölder estimate.

We study the validity of the comparison and maximum principles and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one-dimensional fractional diffusion. [ABSTRACT FROM AUTHOR]

Nonlocal models have recently had a major impact in nonlinear continuum mechanics and are used to describe physical systems/processes which cannot be accurately described by classical, calculus based "local" approaches. In part, this is due to their multiscale nature that enables aggregation of micro-level behavior to obtain a macro-level description of singular/irregular phenomena such as peridynamics, crack propagation, anomalous diffusion and transport phenomena. At the core of these models are nonlocal differential operators, including nonlocal analogs of the gradient/Hessian. This paper initiates the use of such nonlocal operators in the context of optimization and learning. We define and analyze the convergence properties of nonlocal analogs of (stochastic) gradient descent and Newton's method on Euclidean spaces. Our results indicate that as the nonlocal interactions become less noticeable, the optima corresponding to nonlocal optimization converge to the "usual" optima. At the same time, we argue that nonlocal learning is possible in situations where standard calculus fails. As a stylized numerical example of this, we consider the problem of non-differentiable parameter estimation on a non-smooth translation manifold and show that our nonlocal gradient descent recovers the unknown translation parameter from a non-differentiable objective function. [ABSTRACT FROM AUTHOR]

In this paper we study the (BV , L p) -decomposition, p = 1 , 2 , of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p = 1 we also study the associated geometric problem and the thresholding parameters describing the behavior of its solutions. [ABSTRACT FROM AUTHOR]

The optimal control of partial differential equations (PDEs) driven by nonlocal operators presents many numerical challenges. In contrast to the existing methods available in the literature, we propose a wavelet-based approach. This allows us to directly treat the nonlocal operators without the need to extend the underlying PDE into a higher spatial dimension. Due to their possessing vanishing moments, wavelets offer efficient compression strategies that lead to O(N)-algorithms for the forward equation, where N is the number of degrees of freedom. While wavelet schemes have been used very succesfully in the context of boundary element methods, their potential for the simulation of forward problems on domains involving nonlocal operators has yet to be fully exploited. These computational advantages carry over to the solution of the class of control problems under consideration. The latter are equivalent to a coupled system of nonsmooth operator equations with nonlocal operators. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any initial data in the Sobolev space H s (S) with s > 3 / 2. In this paper, we consider the Cauchy problem in the critical space H 1 / 2 (S). Two interesting new features, at this level of regularity, are that the equation can be written in the form ∂ t u + V ∂ x u + γ Λ u = 0 , where V is not bounded and γ is not bounded from below. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any initial data in H 1 / 2 (S). Our proof relies on sharp commutators estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm. [ABSTRACT FROM AUTHOR]

In this manuscript we deal with existence/uniqueness and regularity issues of suitable weak solutions to nonlocal problems driven by fractional Laplace type operators. Different from previous researches, in our approach we consider gradient non-linearity sources with subcritical growth, as well as appropriated measures as sources and boundary datum. We provide an in-depth discussion on the notions of solutions involved together with existence/uniqueness results in different regimes and for different boundary value problems. Finally, this work extends previous ones by dealing with more general nonlocal operators, source terms and boundary data. [ABSTRACT FROM AUTHOR]

A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided. [ABSTRACT FROM AUTHOR]

Abstract A comprehensive study about the spread of COVID-19 cases in Turkey and South Africa has been presented in this paper. An exhaustive statistical analysis was performed using data collected from Turkey and South Africa within the period of 11 March 2020 to 3 May 2020 and 05 March and 3 of May, respectively. It was observed that in the case of Turkey, a negative Spearman correlation for the number of infected class and a positive Spearman correlation for both the number of deaths and recoveries were obtained. This implied that the daily infections could decrease, while the daily deaths and number of recovered people could increase under current conditions. In the case of South Africa, a negative Spearman correlation for both daily deaths and daily infected people were obtained, indicating that these numbers may decrease if the current conditions are maintained. The utilization of a statistical technique predicted the daily number of infected, recovered, and dead people for each country; and three results were obtained for Turkey, namely an upper boundary, a prediction from current situation and lower boundary. The histograms of the daily number of newly infected, recovered and death showed a sign of lognormal and normal distribution, which is presented using the Bell curving method parameters estimation. A new mathematical model COVID-19 comprised of nine classes was suggested; of which a formula of the reproductive number, well-poseness of the solutions and the stability analysis were presented in detail. The suggested model was further extended to the scope of nonlocal operators for each case; whereby a numerical method was used to provide numerical solutions, and simulations were performed for different non-integer numbers. Additionally, sections devoted to control optimal and others dedicated to compare cases between Turkey and South Africa with the aim to comprehend why there are less numbers of deaths and infected people in South Africa than Turkey were presented in detail.

This paper deals with the qualitative analysis of solutions to the following (p, q)-fractional equation: (−∆)ps1u + (−∆)qs1u + V(x)(|u|p-2u+ |u| q−2u) = K(x) f(u)/|x|β in RN, where 1 < q < p, 0 < s2 ≤ s1 < 1, ps1 = N, β ∈ [0, N), and V, K : RN → R, f : R → R are continuous functions satisfying some natural hypotheses. We are concerned both with the case when f has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser–Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational and topological methods, we prove the existence of nonnegative solutions. [ABSTRACT FROM AUTHOR]

This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s). [ABSTRACT FROM AUTHOR]

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator ∂ t - K , where K is the following nonlocal kinetic operator: K f (x , v) = p. v. ∫ R d (f (x , v + w) - f (x , v)) κ (x , v , w) | w | d + α d w + v · ∇ x f (x , v) + b (x , v) · ∇ v f (x , v). Here κ 0 - 1 ⩽ κ (x , v , w) ⩽ κ 0 belongs to C b ∞ (R 3 d) and is symmetric in w, p.v. stands for the Cauchy principal value, and b ∈ C b ∞ (R 2 d ; R d) . [ABSTRACT FROM AUTHOR]

The paper deals with possible behaviour at infinity of solutions to the Cauchy problem for a parabolic type equation whose elliptic part is the generator of a Markov jump process , i.e. a nonlocal diffusion operator. The analysis of the behaviour of the solutions at infinity is based on the results on the asymptotics of the fundamental solutions of nonlocal parabolic problems. It is shown that such fundamental solutions might have different asymptotics and decay rates in the regions of moderate, large and super-large deviations. The asymptotic formulae for the said fundamental solutions are then used for describing classes of unbounded functions in which the studied Cauchy problem is well-posed. We also consider the question of uniqueness of a solution in these functional classes.

The main goal of this article is to study a new class of nonlocal operators and the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated with them. The fundamental solutions of these equations are transition functions of Markov processes on an n-dimensional vector space over the p-adic numbers. We also study some properties of these Markov processes, including the first passage time problem.

A comprehensive study about the spread of COVID-19 cases in Turkey and South Africa has been presented in this paper. An exhaustive statistical analysis was performed using data collected from Turkey and South Africa within the period of 11 March 2020 to 3 May 2020 and 05 March and 3 of May, respectively. It was observed that in the case of Turkey, a negative Spearman correlation for the number of infected class and a positive Spearman correlation for both the number of deaths and recoveries were obtained. This implied that the daily infections could decrease, while the daily deaths and number of recovered people could increase under current conditions. In the case of South Africa, a negative Spearman correlation for both daily deaths and daily infected people were obtained, indicating that these numbers may decrease if the current conditions are maintained. The utilization of a statistical technique predicted the daily number of infected, recovered, and dead people for each country; and three results were obtained for Turkey, namely an upper boundary, a prediction from current situation and lower boundary. The histograms of the daily number of newly infected, recovered and death showed a sign of lognormal and normal distribution, which is presented using the Bell curving method parameters estimation. A new mathematical model COVID-19 comprised of nine classes was suggested; of which a formula of the reproductive number, well-poseness of the solutions and the stability analysis were presented in detail. The suggested model was further extended to the scope of nonlocal operators for each case; whereby a numerical method was used to provide numerical solutions, and simulations were performed for different non-integer numbers. Additionally, sections devoted to control optimal and others dedicated to compare cases between Turkey and South Africa with the aim to comprehend why there are less numbers of deaths and infected people in South Africa than Turkey were presented in detail. [ABSTRACT FROM AUTHOR]

We consider parabolic nonlocal Venttsel’ problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution. The nonlocal term can be regarded as a regional fractional Laplacian on the boundary. The regularity results deeply rely on a priori estimates, obtained via the so-called Munchhausen trick, and sophisticated extension theorem for anisotropic weighted Sobolev spaces. MSC 2010: Primary 35K20, 35B65; Secondary 335R02, 35B45 [ABSTRACT FROM AUTHOR]

We study harmonic functions associated to systems of stochastic differential equations of the form d X t i = A i 1 (X t -) d Z t 1 + ⋯ + A id (X t -) d Z t d , i ∈ { 1 , ⋯ , d } , where Z t j are independent one-dimensional symmetric stable processes of order α j ∈ (0 , 2) , j ∈ { 1 , ⋯ , d } . In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems. [ABSTRACT FROM AUTHOR]

In this article, we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branch-cut singularities for certain Fourier multipliers. As an application we present quantitative Runge approximation results for the operator L s (D) = ∑ j = 1 n (− ∂ x j 2) s + q with s ∈ [ 1 2 , 1) and q ∈ L ∞ acting on functions on R n. [ABSTRACT FROM AUTHOR]

As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with α-stable Lévy processes are presented to illustrate the maximum principles.

The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]