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The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of "information" across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with the k -path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization. [ABSTRACT FROM AUTHOR]

The study of the existence and uniqueness of solutions to 2D systems utilizing the generalized proportional fractional derivative operator is the focus of this work. We also derive a finite difference scheme in order to numerically approximate such an operator, and we prove that the method we propose is convergent. Several tests are performed at the end to illustrate the robustness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the existence result of at least two positive solutions is obtained for a nonlinear Riemann-Liouville fractional differential equation subject to nonlocal boundary conditions, where fractional derivatives and Riemann-Stieltjes integrals are involved. The nonlinearity possesses singularities on both its time and space variables. The discussion is based on the fixed point index theory on cones. [ABSTRACT FROM AUTHOR]

This study analyzes a solution that requires efficient and comprehensive processing of images of a large number of vehicles and their related parts, such as batteries, plastic fastening components, and brake discs, during the design investigation of electric vehicle accessories. The problem involves the extraction of the outer contours of different components, which is important to build a comprehensive image processing system that can handle different vehicle accessories. In this study, a comprehensive image processing system is proposed, which introduces an improved GrabCut and computer vision methods. It can complete the positioning of vehicle batteries, the fastening of automobile components, and the identification of brake discs, which improves the efficiency of inspection and design work. The improved GrabCut uses adaptive median filtering on the electric car accessory to reduce noise from the surface in variable degrees. The image is then sharpened using the Laplacian operator, followed by a contrast-limited histogram equalization (CLAHE) algorithm to boost the image brightness. We have compared our proposed work against existing techniques, i.e., the GrabCut algorithm, region growing algorithm, and K-means algorithm. The comparison clearly shows that our proposed work achieves a much better peak signal-to-noise ratio value as compared to the existing techniques. [ABSTRACT FROM AUTHOR]

Elliptic grid generation equations based on the Laplacian operator have the well-known property of clustering the mesh near convex boundaries and declustering it near concave boundaries. In prior work, a new differential operator was derived and presented to address this issue. This new operator retains the strong smoothing properties of the Laplacian without the latter's adverse curvature effects. However, the new operator exhibits slower convergence properties than the Laplacian, which can lead to increased turnaround times and in some cases preclude the achievement of convergence to machine accuracy. In the work presented here, a Newton linearization of the new operator is presented, with the objective of achieving more robust convergence properties. Sample solutions are presented by evaluating a number of solvers and preconditioners and assessing the convergence properties of the solution process. The efficiency of each solution method is demonstrated with applications to two-dimensional airfoil meshes. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of at least two nonnegative nontrivial solutions to a p z –Laplacian system involving critical nonlinearity in the context of Sobolev spaces with variable exponents on complete manifolds. We have established our main results by exploring both Nehari's method and doing a refined analysis on the associated fiber map and some variational techniques. [ABSTRACT FROM AUTHOR]

Using the mountain pass theorem, we obtain the existence of a nontrivial and nonnegative weak solution of a quasi-linear Schrödinger system driven by the ω ⋅ -Laplacian operator in Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

Mobile robots are often in a situation where they need to find a bump-free path or navigation in their environment from any starting to a specific target point. Within this study, improving the navigation problem of a mobile robot iteratively by using a numerical method based on the potential field method is one of the main aims. This potential field will lean on the use of Laplace's equation to restrain the formation of a potential function across regions within the mobile robot configuration area. The present paper proposed a Quarter-Sweep Modified Accelerated Overrelaxation (QSMAOR) approach to improve the pathfinding of mobile robots in a given environment. The experiment shows that, by using a finite difference method, it is capable of producing an optimal path and creating a smooth path between the starting and target point. The results of the simulation also show that this numerical approach works more rapidly and provides a smoother/clearer direction than the previous study. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized Caputo boundary value problem of fractional differential equation with composite p -Laplacian operator. Boundary value conditions of this problem are of three-point integral type. First, we obtain Green's function in relation to the proposed fractional boundary value problem and then for establishing the existence and uniqueness results, we use topological degree theory and Banach contraction principle. Further, we consider a stability analysis of Ulam-Hyers and Ulam-Hyers-Rassias type. To examine the validity of theoretical results, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and multiplicity of solutions for a discrete Dirichlet boundary value problem involving the p , q -Laplacian. By using the critical point theory, we obtain the existence of infinitely many solutions under some suitable assumptions on the nonlinear term. Also, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green's function, and then by defining two lower and upper control functions and using the wellknown Schauder's fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically. [ABSTRACT FROM AUTHOR]

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland's variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of positive solutions to a class of multipoint boundary value problems for nonlinear multiterm fractional differential equations with p -Laplacian operator. Using the Banach contraction mapping principle, the existence, the uniqueness, the positivity, and the continuous dependency on m -point boundary conditions of the solutions to the given problem are investigated. Also, two examples are presented to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following Kirchhoff type problem involving the fractional p x -Laplacian operator. a − b ∫ Ω × Ω u x − u y p x , y / p x , y x − y N + s p x , y d x d y L u = λ u q x − 2 u + f x , u x ∈ Ω u = 0 x ∈ ∂ Ω , , where Ω is a bounded domain in ℝ N with Lipschitz boundary, a ≥ b > 0 are constants, p x , y is a function defined on Ω ¯ × Ω ¯ , s ∈ 0 , 1 , and q x > 1 , L u is the fractional p x -Laplacian operator, N > s p x , y , for any x , y ∈ Ω ¯ × Ω ¯ , p x ∗ = p x , x N / N − s p x , x , λ is a given positive parameter, and f is a continuous function. By using Ekeland's variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition. [ABSTRACT FROM AUTHOR]

The present research paper is related to the analytical studies of p -Laplacian heat equations with respect to logarithmic nonlinearity in the source terms, where by using an efficient technique and according to some sufficient conditions, we get the global existence and decay estimates of solutions. [ABSTRACT FROM AUTHOR]

In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system having p -Laplacian fractional-order differentiation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L K p . Specifically, we get multiple solutions to the following fractional p -Laplacian equations with the help of fibering maps and Nehari manifold. − Δ p s u x = λ u q + u r , u > 0 in Ω , u = 0 , in ℝ N \ Ω. . Our results extend the previous results in some respects. [ABSTRACT FROM AUTHOR]

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h , r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the "true" geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem. [ABSTRACT FROM AUTHOR]

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of infinitely many solutions of a class of perturbed nonlinear fractional p -Laplacian differential systems using one control parameter combined with the variational method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the application of three fixed point theorem by discussing the existence of three positive solutions for a class of Caputo fractional difference equation boundary value problem. We establish the condition of the existence of three positive solutions for this problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence and uniqueness results for a boundary value problem consisting by a nonlinear fractional q -difference equation subject to a new type of boundary condition, combining the fractional Hadamard and quantum integrals. Our analysis is based on Banach's fixed point theorem, a fixed point theorem for nonlinear contractions, Krasnosel'ski i ˘ 's fixed point theorem, and Leray-Schauder nonlinear alternative. Examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ -Laplacian operator D 0 + α ϕ x , D 0 + β u x + f x , u x = 0 , x ∈ 0 , 1 , D 0 + σ u 0 = D 0 + β u 0 = 0 , u 1 = ∫ 0 1 g t u t d t + ∑ n = 1 n = + ∞ α n u η n , where ϕ : 0 , 1 × R → R is a continuous function and D 0 + p is the Riemann-Liouville derivative for p ∈ α , β , σ . By using some properties of fixed point index, we obtain the existence results and give an example at last. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions for the nonlinear four-point singular BVP for there-order with p -Laplacian operator on time scales will be studied. By using the fixed-point theory, the existence of positive solutions for nonlinear singular boundary value problem with p -Laplacian operator on time scales is obtained. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of solutions for the Dirichlet boundary value problems of p -Laplacian difference equations containing both advance and retardation depending on a parameter λ. Under some suitable assumptions, infinitely many solutions are obtained when λ lies in a given open interval. The approach is based on the critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation ∂ ρ t , x / ∂ t = div x ρ t , x ∇ x G ′ ρ + V q x − 2 ∇ x G ′ ρ + V . The potential V is not necessarily smooth but belongs to a Sobolev space W 1 , ∞ Ω . Given the initial datum ρ 0 as a probability density on Ω , we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W 1 , q. Ω ↪↪ L q. Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space. [ABSTRACT FROM AUTHOR]

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p -Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of multiple positive solutions or at least one positive solution for fractional three-point boundary value problem with p-Laplacian operator. Our approach relies on the fixed point theorem on cones. The results obtained in this paper essentially improve and generalize some well-known results. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of positive solutions to a class of multipoint boundary value problems for singular fractional differential equations with the p -Laplacian operator. Here, the nonlinear source term f permits singularity with respect to its time variable t. Some fixed-point theorems such as the Leray-Schauder nonlinear alternative, the Schauder fixed-point theorem, and the Banach contraction mapping principle and the properties of the Gauss hypergeometric function are used to prove our main results. And by employing the upper and lower solutions technique, we derive a new approach to obtain the maximal and minimal solutions to the given problem. Finally, we present some examples to demonstrate our existence and uniqueness results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of integral boundary value problems of fractional differential equations with a p -Laplacian operator. Existence of solutions is obtained by using the fixed point theorem, and an example is given to show the applicability of our main result. [ABSTRACT FROM AUTHOR]

In this work, we are pleased to investigate multiple positive solutions for a system of Caputo fractional p -Laplacian boundary value problems, and we also provide an example for illustrating our main results. [ABSTRACT FROM AUTHOR]

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) D C 0 + α y x + f x , y x = 0 , 0 < x < 1 , y 0 = y ′ 1 = y ″ 0 = 0 is established, where 2 < α ≤ 3 , C D 0 + α is the Caputo fractional derivative, and f : 0 , 1 × 0 , ∞ ⟶ 0 , ∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity. [ABSTRACT FROM AUTHOR]

The chief topic of this paper is to investigate the fractional differential system on an infinite interval. By introducing an appropriate compactness criterion in a special function space and applying the Schauder fixed-point theorem and the Banach contraction mapping principle, we established the results for the existence and uniqueness of positive solutions. An example is then given to show the utilization of the main results. [ABSTRACT FROM AUTHOR]

In this paper, by using subsuper solutions method, we study the existence of weak positive solutions for a new class of p , q Laplacian nonlinear elliptic system in bounded domains, when a x , b x , α x , and β x are sign-changing functions that maybe negative near the boundary, without assuming sign conditions on f 0 , g 0 , h 0 , and γ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of solutions for a class of integral boundary value problems of nonlinear fractional differential equations with p -Laplacian operator. We obtain some existence and uniqueness results concerned with our problem by using Schaefer's fixed-point theorem and Banach contraction mapping principle. Finally, we present some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

This study reveals new fractional behavior of Minkowski inequality and several other related generalizations in the frame of the newly proposed fractional operators. For this, an efficient technique called generalized proportional fractional integral operator with respect to another function Φ is introduced. This strategy usually arises as a description of the exponential functions in their kernels in terms of another function Φ. The prime purpose of this study is to provide a new fractional technique, which need not use small parameters for finding the approximate solution of fractional coupled systems and eliminate linearization and unrealistic factors. Numerical results represent that the proposed technique is efficient, reliable, and easy to use for a large variety of physical systems. This study shows that a more general proportional fractional operator is very accurate and effective for analysis of the nonlinear behavior of boundary value problems. This study also states that our findings are more convenient and efficient than other available results. [ABSTRACT FROM AUTHOR]

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error. [ABSTRACT FROM AUTHOR]

Let ℒ = − Δ + V be a Schrödinger operator on ℝ d , d ≥ 3 , where Δ is the Laplacian operator on ℝ d , and the nonnegative potential V belongs to the reverse Hölder class RH s with s ≥ d / 2 . For given 0 < α < d , the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐ α = ℒ − α / 2 . Suppose that b is a locally integrable function on ℝ d and the commutator generated by b and ℐ α is defined by b. ℐ α f x = b x ⋅ ℐ α f x − ℐ α b f x . In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RH s with s ≥ d / 2 . Then, we will establish the boundedness properties of the fractional integrals ℐ α on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b , ℐ α in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than A p , q , BMO ℝ d , and L p , κ μ , ν corresponding to the classical case (that is V ≡ 0). [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for the discrete mixed boundary value problems involving p , q -Laplacian operator. By using critical points theory, we obtain the existence of at least two positive solutions for the boundary value problem under appropriate assumptions on the nonlinearity. [ABSTRACT FROM AUTHOR]

In this work, we investigate the following system of fractional q-difference equations with four-point boundary problems: D q α u t + f t , v t = 0,0 < t < 1 ; D q β v t + g t , u t = 0,0 < t < 1 ; u 0 = 0 , u 1 = γ 1 u η 1 ; and v 0 = 0 , v 1 = γ 2 u η 2 , where D q α and D q β are the fractional Riemann–Liouville q-derivative of order α and β, respectively, 0 < q < 1 , 1 < β ≤ α ≤ 2 , 0 < η 1 , η 2 < 1 , 0 < γ 1 η 1 α − 1 < 1 , and 0 < γ 2 η 2 β − 1 < 1. By virtue of monotone iterative approach, the iterative positive solutions are obtained. An example to illustrate our result is given. [ABSTRACT FROM AUTHOR]

We investigate positive solution for high-order fractional differential equations with resonant boundary value conditions. By using the spectral theory and the fixed point index theory, both the nonexistence and existence results of the resonant boundary value problems are given. [ABSTRACT FROM AUTHOR]

In this paper, we apply the fixed-point theorems of γ concave and − γ convex operators to establish the existence of positive solutions for fractional differential systems with multipoint boundary conditions. Two examples are given to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractional p & q Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solution for differential systems involving a ϕ − Laplacian operator which incorporates as a special case the well-known p − Laplacian operator. In this purpose, we use a variational method which relies on Szulkin's critical point theory. We obtain the existence of solution when the corresponding Euler–Lagrange functional is coercive. [ABSTRACT FROM AUTHOR]

In this paper, the fuzzy fractional evolution equations of order q (FFEE) with fuzzy Caputo fractional derivative are introduced. We study the existence and uniqueness of mild solutions for FFEE under some conditions. Also, we generalize the definition of the fuzzy fractional integral and derivative order q. The fuzzy Laplace transform is presented and proved. The solvability of the problem (FFEE) and the properties of the fuzzy solution operator and its generator are investigated and developed. [ABSTRACT FROM AUTHOR]