It is found that there are two piecewise functions satisfy the conditions in the impulsive fractional partial differential system (IFrPDS), which deduce that the three different integral solutions of the IFrPDS given in the cited papers are inappropriate. Next, by applying two limit properties of the IFrPDS and the properties of piecewise function, the new formula of solution of the IFrPDS is discovered that is the integral equation with an arbitrary continuously differentiable function of t on [0 ,c] to reveal the non-uniqueness of the IFrPDE’s solution. Finally, an example is provided to expound the computation of the solution of the IFrPDS.
In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with time-dependent damping for any σ≥1. Motivated strongly by the classification of damping terms in the paper34, the first main goal of the present work is to make some generalizations from σ=1 to σ>1 and simultaneously to investigate decay estimates for solutions to the corresponding linear equations in the so-called effective damping cases. For the next main goals, we are going not only to prove the global well-posedness property of small data solutions but also to indicate blow-up results for solutions to the semi-linear problem. In this concern, the novelty which should be recognized is that the application of a modified test function combined with a judicious choice of test functions gives blow-up phenomena and upper bound estimates for lifespan in both the subcritical case and the critical case, where σ is assumed to be any fractional number. Finally, lower bound estimates for lifespan in some spatial dimensions are also established to find out their sharp results.
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.
where - Δ G is a sub-Laplacian on Carnot group G, μ ∈ [ 0 , μ G ) , d is the Δ G -natural gauge, ψ is the weight function defined as ψ : = | ∇ G d | . By analytic technics and variational methods, the extremals of the corresponding best Sobolev constant are found, the existence of positive solution to the system is established. Moreover, by the Moser iteration method, some asymptotic properties of its nontrivial solution at the singular point are verified.
In the present work, we focus on the longitudinal model of microtubules (MTs) proposed by Satar i c ̵́ et al. [Phy. Rev. E 48, 89 (1993)], that consider cell MTs to have ferroelectric properties, i.e., a displacive ferro-distortive system of dimers and usually referred to as u-model of MTs. It has been shown that during the hydrolysis of GTP into GDP, the energy released is transferred along the MTs trough kink-like solitons. Substantially, we propose to theoretically investigate the dynamic of MTs by intrinsically taking into account the effect of the oriented molecules of polarized cytoplasmic water and enzymes surrounding the MT. In this regards, we introduce a cubic nonlinear term in the electric potential characterizing the polyelectrolyte features of MTs and show that in addition to the kink and antikink solitons, asymmetrical bright and dark solitons, and discrete modes can also propagate along the MTs. Theses results are supported by numerical analysis. The investigation shows us that the nonlinear dynamics of MTs is strongly impacted by the intrinsic electric field, the polyelectrolyte and the viscosity effects. Moreover, new solitons and discrete solitary modes may help to find new phenomena occurring in the microtubulin systems.
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.
In this work, bases on the reproducing kernel theory and collocation method, we study the space Riesz fractional Navier-Stokes equations, and propose the numerical method to solve it. Firstly the new base space can be constructed by the spline and reproducing kernel space. The ε-approximate solution in binary spline space in the form of finite terms can be derived. Through using the collocation method, the approximate problem is solved. In addition, we provide analysis of the stability and convergence. In final, two numerical examples are provided to show the effectiveness of our method.
We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an L p -norm between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing, the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.
We aim to study Mittag-Leffler type functions of two variables D 1 ( x , y ) , . . . , D 5 ( x , y ) by analogy with the Appell hypergeometric functions of two variables,. Moreover, we targeted functions E 1 ( x , y ) , . . . , E 10 ( x , y ) as limiting cases of the functions D 1 ( x , y ) , . . . , D 5 ( x , y ) and studied certain properties, as well. Following Horn’s method, we determine all possible cases of the convergence region of the function D 1 ( x , y ) . Further, for a generalized hypergeometric function D 1 ( x , y ) (Mittag-Leffler type function) integral representations of the Euler type are proved. One-dimensional and two-dimensional Laplace transforms of the function are also defined. We have constructed a system of partial differential equations which is linked with the function D 1 ( x , y ) .
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  and Limaco-Clark-Frota-Medeiros  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.
In this paper, a coupling transmission epidemic model with mutualistic two-strain of virus in body and vitro of host is proposed, in which humoral immune response only works for strain 1 due to immunity evasion of mutation. For the within-host subsystem, the global stability of all feasible equilibria with and without environmental influence are discussed. For the between-host subsystem, the basic reproduction number R 0 is obtained. When R 0 < 1 , the disease-free equilibrium is local stable, while the endemic equilibrium is local stable and the disease is uniformly persistent if R 0 > 1 . Meanwhile, backward bifurcation would occur when there exists immune response within host. Finally, numerical examples are provided to illustrate obtained conclusions, by which we find that the mutualism between two strains during co-infection leads to a more persistent disease than single strain, even the basic reproduction number is small than 1 in each single strain.
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.