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This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. [ABSTRACT FROM AUTHOR]

In the present paper, we give some remarks on the magnetic field operators ∇±iA. As its applications, we study the Schr¨odinger equation with a magnetic field -Δu+|A(x)|2u+iA(x) ·∇u = μu+|u|pu, x ∈ RN, where u is a complex-valued function and μ ∈ R. When N > 2, for 2 < p+2 < 2N N-2 or N = 2, for 2 < p+2 < +∞, the existence and nonexistence of minimizers of the corresponding minimization problem are given via constrained variational methods. As a by-product, the above equation admits a normalized solution. We point out that the condition divA(x) = 0 plays a crucial role in our study. [ABSTRACT FROM AUTHOR]

We consider the biharmonic equation Δ²u-(a + b R R5 |Δu|²dx) Δu + V (x)u = f(u), where V (x) and f(u) are continuous functions. By using a perturbation approach and the symmetric mountain pass theorem, the existence and multiplicity of solutions for this equation are obtained, and the power-type case f(u) = |u|p-2u is extended to p (2, 10), where it was assumed p (4, 10) in many papers. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the 3D generalized MHD system with nonlinear damping terms. We establish the global existence of strong solutions with the help of damping terms. Furthermore, we consider the balance between damping terms. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the Cauchy problem of inhomogeneous convolution equations of a fairly general nature. To solve the problems posed here, we apply the operator method proposed in some earlier papers by the author. The solutions of the problems under consideration are found using an effective method in the form of well-convergent vector-valued power series. The proposed method ensures the continuity of the obtained solutions with respect to the initial data and the inhomogeneous term of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic distribution of some U-statistics whose entries are functions of empirical moments computed from non-overlapping consecutive blocks of an underlying weakly dependent process. The length of these blocks converges to infinity, and thus we consider U-statistics of triangular arrays. We establish asymptotic normality of such U-statistics. The results can be used to construct tests for changes of higher order moments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze various classes of vector-valued (Dasymptotically) ρ-almost periodic type distributions in Rn. We also provide certain applications of our theoretical results to the (abstract) partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions, max{Sk ln (S/K), 0} and max{Sk ln (K/S), 0}, (k ∈ ℝ+ ∪ {0}). It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM are quite accurate to the closed form solutions. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.

In this paper, we consider the existence and multiplicity of nontrivial solutions to second order partial difference equation with Dirichlet boundary conditions by Morse theory. Given suitable conditions, we establish multiple results that the problem admits at least two nontrivial solutions. Moreover, we provide five examples to illustrate applications of our theorems. [ABSTRACT FROM AUTHOR]

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear Dirac equation - i ħ ∂ t ψ = i c ħ ∑ k = 1 3 α k ∂ k ψ - m c 2 β ψ + K (x) | ψ | p - 2 ψ , (N D E) where ψ : R × R 3 → C 4 , K ∈ L ∞ (R 3) , c denotes the speed of light, m > 0 , the mass of the electron, ħ is Planck's constant, ∂ k = ∂ ∂ x k , α 1 , α 2 , α 3 , β are 4 × 4 Pauli–Dirac matrices and p ∈ (2 , 8 3) . We present a new approach which is based on a perturbation argument and show the existence of L 2 -normalized solitary wave solutions of (NDE) using variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider stabilization of the transmission wave equation with variable coefficients, which has an acceleration term and interior delay. By using Riemannian geometry method and Lyapunov functional, we obtain the exponential decay for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of nontrivial solutions for the following Schrödinger–Poisson system - Δ u + u + λ ϕ u = | u | p - 2 u , - Δ ϕ = u 2 , lim | x | → + ∞ ϕ (x) = 0 , where λ > 0 and p ∈ (2 , 6) . In the case when 2 < p ≤ 3 , we obtain the existence of nontrivial radial solutions and the nonexistence of nontrivial solutions for the above system under weaker assumptions on the parameter λ than the ones in Ruiz (J Funct Anal 237:655–674, 2006). [ABSTRACT FROM AUTHOR]

In a blog post from July 2017, the US Food and Drug Administration (FDA) Commissioner Scott Gottlieb emphasized the importance of modeling and simulation (M&S) in drug development (Gottlieb [4]). Vendel et al. ([12]) start with a model for drug transport across the blood-brain barrier and incorporate two-dimensional drug transport and binding within brain tissue to better predict drug concentrations in the brain. [Extracted from the article]

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation. [ABSTRACT FROM AUTHOR]

Numerical methods have essential role to approximate the solutions of Partial Differential Equations (PDEs). Spectral method is one of the best numerical methods of exponential order with high convergence rate to solve PDEs. In recent decades the Chebyshev Spectral Collocation (CSC) method has been used to approximate solutions of linear PDEs. In this paper, by using linear algebra operators, we implement Kronecker Chebyshev Spectral Collocation (KCSC) method for n-order linear PDEs. By statistical tools, we obtain that the Run times of KCSC method has polynomial growth, but the Run times of CSC method has exponential growth. Moreover, error upper bounds of KCSC and CSC methods are compared. [ABSTRACT FROM AUTHOR]

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks (see [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201-232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201-232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I, 350, 2012, 861-865]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the boundedness of solutions for a class of fractional elliptic equations driven by the relativistic Schrôdinger operator (--Δ + I)s, with s ∈ (0, 1). The proof relies on a distributional Kato's inequality for (--Δ + I)s and on some properties of the Bessel kernel. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for an elliptic equation in a plane domain. The problem is ill-posed. Using the method of quasi-reversibility, an approximation to the exact solution is given. Using Carleman's inequatily, we derive a sharp error estimate. [ABSTRACT FROM AUTHOR]

With the rapid development of smart campus, this paper studies the attitude tracking control of flexible manipulator (FM) in colleges and universities under elastic vibration and external disturbances. First, different from the traditional modeling based on ordinary differential equations (ODEs), the partial differential equations (PDEs) dynamic model of a manipulator system is established based on the Hamilton principle (HP). Second, the boundary control condition of the end system of the manipulator is introduced to adjust the vibration of the manipulator. Furthermore, a Proportional-Derivative (PD) boundary control (PDBC) strategy is proposed by the Lyapunov function to suppress the vibration of the manipulator. Finally, a numerical comparison simulation based on MATLAB/SIMULINK further verifies the robustness and anti-disturbance performance of the control method proposed in this paper.

A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology). We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of "stochastic compactness method". [ABSTRACT FROM AUTHOR]

In this paper, by sub-supersolution methods, Karamata regular variation theory and perturbation method, we study the existence, uniqueness and asymptotic behavior of solutions near the boundary to quasilinear elliptic problem ⎧⎨ ⎩ div(|∇u|m-2∇u) = b(x) f (u)(1+|∇u|q(m-1)), x ∈ Ω, u > 0, x ∈ Ω, u|∂Ω =+∞, where Ω is a bounded domain with smooth boundary in RN (N ≤ 2), 1 < m ≤ 2, 0 < q ≤ m/(m-1). b ∊Cα (Ω)(α ∊ (0,1)) is positive in Ω, and may be vanishing on the boundary, and f ∊ C¹[0,+∞), f (0) = 0, is increase on (0,+∞) and normalized regularly varying at infinity with positive index p and p+(q-1)(m-1) > 0. [ABSTRACT FROM AUTHOR]

In this paper, an effcient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, effciency, and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we use the Fountain theorem under the Cerami condition to study the gauged nonlinear Schrödinger equation with a perturbation in R². Under some appropriate conditions, we obtain the existence of infinitely many high energy solutions for the equation. [ABSTRACT FROM AUTHOR]

Accurate reporting and forecasting of PM2.5 concentration are important for improving public health. In this paper, we propose a partial differential equation (PDE) model, specially, a linear diffusive equation, to describe the spatial-temporal characteristics of PM2.5 in order to make short-term prediction. We analyze the temporal and spatial patterns of a real dataset from China’s National Environmental Monitoring and validate the PDE-based model in terms of predicting the PM2.5 concentration of the next day by the former days’ history data. Our experiment results show that the PDE model is able to characterize and predict the process of PM2.5 transport. For example, for 300 continuous days of 2016, the average prediction accuracy of the PDE model over all city-regions is 93% or 83% based on different accuracy definitions. To our knowledge, this is the first attempt to use PDE-based model to study PM2.5 prediction in both temporal and spatial dimensions. [ABSTRACT FROM AUTHOR]

In the present paper an effective symmetric hybrid six-step method for the approximation of the solution of the Schrödinger equation and related problems is developed. More specifically, we will produced a method with the following properties: (1) is a symmetric hybrid six-step method, (2) is of twelfth algebraic order, (3) has three stages, (4) has eliminated phase-lag, (5) has eliminated the derivatives of the phase-lag up to order three. This numerical pair is obtained for the first time in the literature. We present a detailed analysis for the new obtained numerical scheme. More specifically we present: The theoretical and numerical achievements which are presented in this paper, show the efficiency of the new numerical pair compared with other known or recently developed pairs of the literature. [ABSTRACT FROM AUTHOR]

This paper derives a renormalization formula defined on the parameter space where mapping behavior is preserved, together with the equivalent potential function. In contrast to the universal function given by Feigenbaum, the behavior near the critical point is governed by the potential function. There are several interesting features which are shared by Feigenbaum's universal function, such as the representation of the critical point in terms of the unstable fixed point of the potential function, but the mapping differs from the scaling. The one-dimensional mapping is considered as an example, and the critical point and the scaling, which are major constants characterizing chaos from the potential function, are calculated precisely. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(8): 41–51, 1998 [ABSTRACT FROM AUTHOR]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking.

This paper deals with topology optimization based on the Heaviside projection method using a scalar function as design variables. The scalar function is then regularized by a PDE based filter. Several image-processing based filtering techniques have so far been proposed for regularization or restricting the minimum length scale. They are conventionally applied to the design sensitivities rather than the design variables themselves. However, it causes discrepancies between the filtered sensitivities and the actual sensitivities that may confuse the optimization process and disturb the convergence. In this paper, we propose a Heaviside projection based topology optimization method with a scalar function that is filtered by a Helmholtz type partial differential equation. Therefore, the optimality can be strictly discussed in terms of the KKT condition. In order to demonstrate the effectiveness of the proposed method, a minimum compliance problem is solved. [ABSTRACT FROM AUTHOR]

In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions. [ABSTRACT FROM AUTHOR]

In the present study, numerical simulations of two-dimensional steady-state incompressible Newtonian fluid flow in one-sided square and two-sided deep lid driven cavities under the aspect ratio K = 1, 4, 6 are reported. For the one-sided lid driven cavity, the upper wall is moved to the right with up to 5000 Reynolds numbers under a grid size of up to 501 × 501. This lends support to previous findings in the literature with Ghia et al.’s results. Three cases are used in this article for the two-sided deep lid driven square cavity specifically. In these cases, the top and lower walls are moved to the right, while the left and right walls remain fixed up to at high Reynolds numbers (5000) under the grid size of up to 201×201. All possible flow solutions are studied in the present article, and flow bifurcation diagrams are constructed as velocity profiles and streamline contours for the same Reynolds number using a finite volume SIMPLE technique. The work done in this paper includes flow properties such as the location of primary and secondary vortices, velocity components, and numerical values for benchmarking purposes, and it is in excellent agreement with previous findings in the literature. A PARAM Shavak, high-performance computing (HPC) computer, was used to execute the calculations. [ABSTRACT FROM AUTHOR]

Partial differential equations are crucial in scientific computing, and this paper will consider some of the problems of partial differential equation singularities. The Composite Mesh Method (CGM) is a new and improved numerical method for solving partial differential equations based on existing numerical methods for finite elements. The method has two meshes over the entire domain—a coarse and a fine set. The two sets of meshes generated by Mesh3 are separate in their respective regions and do not nest or interact. This method improves the accuracy of solving the numerical solution of partial differential equations. This paper discusses the CGM method based on the Finite Element Program Generator (FEPG) and uses it to simulate several singular problems. The numerical simulation results show that the proposed method can obtain more satisfactory simulation results for global problems and use a smaller number of computational generations than the general finite element method.

Abstract This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results.

Abstract In this paper, a mathematical proof of the existence of a global minimum of Self-Optim (Self-Optimizing Inverse Analysis Method) cost functional is presented based upon weak-solution theory of partial differential equations. The Self-Optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem. Furthermore, discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified. This paper provides a rigorous mathematical foundation for applications of the Self-Optim method to various inverse problems in mechanics. [ABSTRACT FROM AUTHOR]

The quasilinear parabolic equation with an absorption potential is considered: ..., where Ω is a bounded smooth domain in Rn, n ≥ 1, b is an absorption potential which is a continuous function such that b(t, x) > 0 in [0, T) x ≥ and b(t, x) ≡ 0 in {T} x ≥. In the paper, the conditions for b(t, x) that guarantee the uniform boundedness of an arbitrary weak solution of the mentioned equation in an arbitrary subdomain ... are considered. Under the above conditions the sharp upper estimate for all weak solutions u is obtained. The estimate holds for the solutions of the equation with arbitrary initial and boundary data, including blow-up data (provided that such a solution exists), namely, u = ∞ on {0} x ≥, u = ∞ on (0, T) x ∂≥. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K.

In this paper, we show that the analogue of Thurston's asymmetric metric on the Tcichmiiller space of flat Structmcs on the torus is weak Finsler and we give a geometric description of its unit circle at each point in the tangent space to Tcichmiiller space. We then introduce a family of weak Finslcr metrics which interpolate between Thurston's asymmetric metric and the Teichmuller metric of the torus (which coincides with the hyperbolic metric). We describe the unit tangent circles of the metrics in this family. [ABSTRACT FROM AUTHOR]

The aim of this paper is to use a variational approach in order to obtain the existence of non-trivial weak solutions of a quasilinear elliptic equation not in divergence form, in dimension N = 3. Moreover, we prove that our solution is C1,α(ω̅) and also locally C2,α(ω ̅) for a suitable α ∈ (0, 1) . [ABSTRACT FROM AUTHOR]