In this paper, we consider the following fractional Schrödinger equation ε 2 s ( − ∆ ) s u + V ( x ) u = P ( x ) f ( u ) + Q ( x ) | u | 2 s ∗ − 2 u in R N , where ε>0 is a parameter, s∈(0 ,1), 2 s ∗ = 2 N N − 2 s , N>2 s, ( − ∆ ) s is the fractional Lapalacian and f is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration-compactness principle, we prove the existence of a positive solution for the above equations.

We show that the Reynolds averaged equations for compressible fluids (neglecting third order correlations) are well-posed in H s when the pressure vanishes in dimensions d=2 and 3. In order to do this, we show that the system is Friedrichs-symmetrizable. This model belongs to the class of non-conservative hyperbolic systems. Hence the usual symmetrisation method for conservation laws can not be used here.

In this paper, we study a class of sequential fractional differential inequalities involving Caputo fractional derivatives with different orders. The nonexistence of nontrivial global solutions is investigated in a suitable space via the test function technique and some properties of fractional integrals. Our results are supported by numerical examples.

To be a quantitative and testable tectonic model, plate tectonics requires spherical geometry and spherical kinematics in terms of finite rotations conveniently parametrized by their angle and axis and described by unit quaternions. In treatises on ’Plate Tectonics’ infinitesimal, instantaneous, and finite rotations, absolute and relative rotations are said to be applied to model the motion of tectonic plates. Even though these terms are strictly defined in mathematics, they are often casually used in geosciences. Here their definitions are recalled and clarified as well as the terms rotation, orientation, and location on the sphere. For instance, infinitesimal rotations refer to a mathematical limit, when the angle of rotation converges to zero. Their rules do not apply to finite rotations, no matter how small their finite angles of rotation are. Mathematical approaches applying appropriate and feasible assumptions to model spherical motion of tectonic plates over geological times of hundreds of millions of years are derived including (i) sequences of incremental finite rotations, (ii) sequences of accumulating successive concatenations of finite rotations, (iii) continuous rotations in terms of fully transient quaternions. The incremental and the accumulating approach provide complementary views. While the relative Euler pole appears to migrate in the latter, it appears fixed in the former. Path, mean and instantaneous velocity of the migrating Euler pole are derived as well as the angular and trajectoral velocity of the rotational motion about it. The approaches are illustrated by a geological example with actual data and a numerical yet geologically inspired example with artificial data. The former revisits the three plates scenario with stationary axes of two “absolute” rotations implying transient “relative” rotations about a migrating Euler pole and employs a proper plate circuit argument to determine them numerically without resuming to approximations. The latter applies an involved interplay of incremental and accumulating modeling inducing split-join cycles to approximate sinusoidal trajectories as reported to record plates’ motion during the Gondwana breakup.

In this paper, we investigate the controllability under positivity constraints from the birth of a size-structured population system with delayed birth process. Firstly, we establish the well-posedness and positivity property of the model. Secondly, and more importantly, we derive a sufficient condition for boundary approximate controllability under state and control positivity constraints. The method relies on the feedback theory of infinite-dimensional positive systems.

The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non-local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take advantage of the representation of the resulvent operator and recover the heat kernel by applying the inverse Laplace transform.

Uncertain fractional differential equation (UFDE) is an useful tool for studying complex systems in uncertain environments. In this paper, we study the extreme value theorems of the solution to Caputo-Hadamard UFDEs and applications. A numerical algorithm for solving the numerical solution of a nonlinear Caputo-Hadamard UFDE is presented, the feasibility of the numerical algorithm is validated by numerical experiments. The extreme value theorems are applied to the financial markets, and the pricing formulas of the American option based on the new uncertain stock model are given. Considering the properties of the American option pricing, the algorithms for computing the expected value of the extreme values based on the Simpson’s rule are designed. Finally, the price fluctuation of the American option is illustrated by numerical experiments.

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials ( P α ). As a consequence of energy argument, we derive L ∞ ( 0 , T ; H 0 1 ( Ω ) ) bound as well as L 2 ( 0 , T ; H 2 ( Ω ) ) bound on the solution of the problem ( P α ) by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem is established. Further, we study semi discrete formulation of the problem ( P α ) by discretizing the space domain using a conforming FEM and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator in such a way that it reduces the complexities arising from the Kichhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem ( P α ) with a convergence rate of O ( h + k 2 − α ) , where α (0 1) is the fractional derivative exponent, h and k are the discretization parameters in the space and time directions respectively. This convergence rate is improved to second order in the time direction by proposing a novel linearized L2-1 σ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.

We investigate fourth order equations with Dirichlet type boundary conditions with perturbation unbounded from above making the problem non-potential. We apply variational method to some auxiliary problem and conclude about the existence and uniqueness to the original one. Multiple solutions are also considered. We conclude our note with the result pertaining to the continuous dependence on parameters.

In this study, a generalized nonlinear local fractional Lighthill-Whitham-Richards (LFLWR) model has been developed. The local fractional variational iteration method (LFVIM) solves and analyzes the proposed model. Numerous works have been described in past to address linear LWR and linear LFLWR models. This research highlighted on generalized nonlinear LFLWR model and LFVIM is employed to derive non-differentiable solutions of the suggested model. The existence and uniqueness of the resolution of LFLWR model have also been established. Furthermore, several exemplary instances are discussed to demonstrate the success of implementing LFVIM to the proposed model. The numerical simulations for each of the cases have also been shown. Additionally, the obtained solutions of the suggested model have been compared with the solutions of the classical LWR model with non-differentiable conditions in few examples. The study demonstrates that the employed iterative scheme is quite efficient and can be utilized for obtaining the non-differentiable solution to proposed generalized nonlinear LFLWR model of traffic flow.

Let µ be a finite Borel measure on [0 ,1). In this paper, we consider the generalized integral type Hilbert operator I µ α + 1 ( f ) ( z ) = ∫ 0 1 f ( t ) ( 1 − tz ) α + 1 d µ ( t ) ( α > − 1 ) . The operator I µ 1 has been extensively studied recently. The aim of this paper is to study the boundedness(resp. compactness) of I µ α + 1 acting from the normal weight Bloch space into another of the same kind. As consequences of our study, we get completely results for the boundedness of I µ α + 1 acting between Bloch type spaces, logarithmic Bloch spaces among others.

This survey introduces 101 new publications on applications of Clifford’s geometric algebras (GA) newly published during 2022 (until mid-January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image- and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.

This paper aims to investigate the time fractional Keller-Segel system with a small parameter. After the fractional order traveling wave transformation, the heteroclinic orbit to the degenerate time fractional Keller-Segel system is demonstrated through the method of constructing a suitable invariant region. Moreover, the persistence of traveling waves in the system with a small parameter can be further illustrated. The results are mainly reliance on the application of geometric singular perturbation theory and Fredholm theorem, which are fundamental theoretical frameworks for dealing with problems of complexity and high dimensionality. Eventually, the asymptotic behavior is depicted by the asymptotic theory to illustrate the rate of decay for traveling waves.

The goal of this paper is to introduce the notion of polyconvolution for Fourier-cosine, Laplace integral operators, and its applications. The structure of this polyconvolution operator and associated integral transforms is investigated in detail. The Watson-type theorem is given, to establish necessary and sufficient conditions for this operator to be unitary on L 2 ( R ) , and to get its inverse represented in the conjugate symmetric form. The correlation between the existence of polyconvolution with some weighted spaces is shown, and Young’s type theorem, as well as the norm-inequalities in weighted space, are also obtained. As applications of the Fourier cosine–Laplace polyconvolution, the solvability in closed-form of some classes for integral equations of Toeplitz plus Hankel type and integro-differential equations of Barbashin type is also considered. Several examples are provided for illustrating the obtained results to ensure their validity and applicability.

In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained by Frota-Goldstein [18] and Limaco-Clark-Frota-Medeiros [26] to allow weak internal dampings and removing the dimensional restriction 1≤ n≤4. The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques.

The purpose of this paper is to construct generating functions in terms of hypergeometric function and logarithm function for finite and infinite sums involving higher powers of inverse binomial coefficients. These generating functions provide a novel way of examining higher powers of inverse binomial coefficients from the perspective of these sums, assessing how several of these sums and these coefficients related to each other. A unique relation between the Euler-Frobenius polynomial and B-spline associated with exponential Euler spline is reported. Moreover, with the aid of derivative operator and functional equations for generating functions, many new computational formulas involving the special finite sums of higher powers of (inverse) binomial coefficients, the Bernoulli polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the harmonic numbers, and finite sums are derived. Moreover, A few recurrence relations containing these particular finite sums are given. Using these recurence relations, we give a solution of the problem which was given by Charalambides7, Exercise 30, p. 273. We give calculations algorithms for these finite sums. Applying these algorithms and Wolfram Mathematica 12.0, we give some plots and many values of these polynomials and finite sums.

We study the ground state solutions for the following p\&q-Laplacain equation \[ \left\{ \begin{array}{ll} -\Delta_pu-\Delta_qu+V(x) (|u|^{p-2}u+|u|^{q-2}u)=\lambda K(x)f(u)+|u|^{q^*-2}u,~x\in\R^N, \\ u\in W^{1,p}(\R^N)\cap W^{1,q}(\R^N), \end{array} \right. \] where $\lambda>0$ is a parameter large enough, $\Delta_ru = \text{div}(|\nabla u|^{r-2}\nabla u)$ with $r\in\{p,q\}$ denotes the $r$ Laplacian operator, $1

By protein structure prediction (PSP) we refer to the prediction of the 3-dimensional (3D) folding of a protein, known as tertiary structure, starting from its amino acid sequence, known as primary structure. The state-of-the-art in PSP is currently achieved by complex deep learning pipelines that require several input features extracted from amino acid sequences. It has been demonstrated that features that grasp the relative orientation of amino acids in space positively impacts the prediction accuracy of the 3D coordinates of atoms in the protein backbone. In this paper, we demonstrate the relevance of Geometric Algebra (GA) in instantiating orientational features for PSP problems. We do so by proposing two novel GA-based metrics which contain information on relative orientations of amino acid residues. We then employ these metrics as an additional input features to a Graph Transformer (GT) architecture to aid the prediction of the 3D coordinates of a protein, and compare them to classical angle-based metrics. We show how our GA features yield comparable results to angle maps in terms of accuracy of the predicted coordinates. This is despite being constructed from less initial information about the protein backbone. The features are also fewer and more informative, and can be (i) closely associated to protein secondary structures and (ii) more readily predicted compared to angle maps. We hence deduce that GA can be employed as a tool to simplify the modeling of protein structures and pack orientational information in a more natural and meaningful way.