This paper explores novel extensions of the classical Steklov eigenvalue problem within a nonlocal fractional framework. While our approach is inspired by the foundational methods presented in [11], the primary novelty of this work lies in the unprecedented combination of a fractional operator with a Kirchhoff-type function to determine a complete sequence of eigenvalues. By integrating these elements, we establish a nonlocal eigenvalue problem that not only captures the complex nonlocal dynamics of Kirchhoff equations but also recovers the classical Steklov problem through a rigorous limiting process.
In this article, we present a new two-step cubic spline method of O(2,4) for estimating the numerical solution of a 1D nonlinear parabolic partial differential equation with appropriate conditions at the initial time level and Dirichlet conditions along the boundary. Furthermore, we present a new single-segment, group-explicit iteration procedure to solve a nonlinear system of difference equations at each subsequent time level. The convergence of this new iterative scheme is analysed. We have applied the proposed numerical procedures to solve benchmark problems, namely the Burgers-Huxley, Burgers-Fischer, and singular parabolic equations. The effectiveness of the proposed iteration schemes is validated by numerical results, which are compared with the corresponding two-step Newton-SoR and Newton-AGE iteration procedures.
This work develops a class of osculating surfaces in three-dimensional Euclidean space using a modified orthogonal frame defined along curves with non-zero curvature. We begin by deriving the connections between the classical Frenet frame and the modified orthogonal frame adapted to curvature. Within this setting, the fundamental geometric quantities of the resulting osculating surfaces, including their singular behavior, Gaussian curvature, mean curvature, and fundamental forms, are explicitly determined. Moreover, criteria are established for when the parameter curves of the associated osculating surfaces represent geodesics, asymptotic curves, or curvature lines. Finally, to demonstrate the applicability of the theoretical results, a representative example of an osculating surface constructed using both the Frenet frame and the modified orthogonal frame is provided.
This paper presents a new two-dimensional chaotic system with three progressively designed variants aimed at achieving a balance between analytical simplicity and complex nonlinearity. The proposed system DY2DCM includes exponential terms, trigonometric functions-based nonlinearity, and rational coupling, while successive changes introduce control parameters and modulated trigonometric functions to enrich the dynamical behavior of DY2DCM variants. A thorough analysis of the proposed system is carried out using different tools like time-series analysis, phase portraits, bifurcation diagrams, Lyapunov exponents, and the 0-1 test. The results of these analysis confirm the presence of robust chaotic dynamics in DY2DCM. This work presents an substitution–permutation network based new image encryption to demonstrate the practicability of DY2DCM. Further, the proposed system is utilized to generate key-streams for providing required confusion and diffusion in proposed scheme. This work also presents the result of extensive experiments carried on scheme, including information entropy, histogram analysis, correlation coefficients, and avalanche property, etc. The results support the robustness of proposed scheme. This work highlights that our scheme is resistant against common cryptanalytic attacks like chosen-plaintext/ciphertext attacks.
Hybrid structures provide a unifying framework for the study of fuzzy and soft sets. In this paper, we introduce the concepts and terminology of hybrid covered bi-ideals. Within the framework of semigroups, we investigate several fundamental properties and present illustrative examples. Furthermore, we show that the class of hybrid covered bi-ideals is closed under intersection, though not necessarily under union. Finally, the concept of covered bi-ideals is generalized to the broader setting of generalized bi-ideals.
The ‘divide and conquer’ paradigm proves to be one of the most frequently used techniques for dealing with the complexities of graph-related problems. Therefore, it is of great importance to measure the tendency of a vertex to be critical and its susceptibility in a graph. The criticality of a vertex is often analysed in terms of its strength. Removing a highly critical vertex from a graph modelling a network may introduce vulnerability into the system represented by the graph. Minimizing the vulnerability of such a network without affecting its fundamental structure, thereby improving the stability of the graph, is the primary objective of the article. To achieve this, certain properties of Euler graphs are analysed in terms of vertex strength, and a method is presented for determining all possible constructions of Euler graphs corresponding to different integer partitions. The parts of a partition represent the vertex strengths, and their sum corresponds to the total vertex strength of the graph. Various connectivity indices are employed to validate the proposed constructions. Furthermore, their interrelationships and potential real-life applications are also discussed. It is evident from the constructions that they may play a vital role in developing network deception technology to protect digital assets, as each partition of the network generates a distinct network.
A class of singularly perturbed convection–diffusion boundary value problems for second-order ordinary differential equations without interior turning points, involving a small perturbation parameter multiplying the highest-order derivative and subject to Dirichlet boundary conditions, is solved using a Shishkin mesh based on spline-in-compression. The proposed method effectively captures the boundary layer behavior and ensures improved numerical accuracy. To demonstrate the efficiency and robustness of the scheme, two numerical examples are presented and the results are compared with those obtained by using variable-mesh and uniform mesh approaches. The comparative analysis confirms the superior performance of the proposed method, particularly for problems exhibiting sharp boundary layers.
In this paper, we study sequential product submanifolds with respect to a semi-symmetric metric connection. We derive expressions for the covariant derivatives, Riemannian curvature, Ricci curvature, and scalar curvature of these sequential warped product submanifolds with respect to the semi-symmetric metric connection. Furthermore, we establish a relationship between these curvatures and their counterparts computed using the Levi-Civita connection. Additionally, we introduce a generalized curvature inequality for sequential warped product submanifolds endowed with the semi-symmetric metric connection in complex space forms. Moreover, this work lays a robust groundwork for future investigations and advancements in this field of study.
In this note we provide explicit expressions for the number of edges of L(G)^* and C(G)^* for finite abelian p-groups of rank at most three and for elementary abelian p-groups.
In this paper, the Bronze Leonardo–Lucas matrix, denoted by Γ =(B́Ć_𝔫𝔨)_𝔫,𝔨∈ℕ_0 is defined by B́Ć_𝔫𝔨 = {[ 3B́Ć_𝔨4B́Ć_𝔫+B́Ć_𝔫-1+3𝔫-10 1 ≤𝔨≤𝔫,; 0, 𝔨 > 𝔫, ]. and {B́Ć_𝔨} corresponds to terms of the Bronze Leonardo–Lucas sequence. The sequence {B́Ć_𝔨} is defined by B́Ć_𝔨 = 3B́Ć_𝔨-1 + B́Ć_𝔨-2-3 for 𝔨≤ 2, B́Ć_0=3,B́Ć_1=4. Using the Bronze Leonardo–Lucas matrix as a starting point, we define new matrix-domain sequence spaces related to the Bronze Leonardo–Lucas numbers. We further establish key aspects explored include the properties and inclusion relationships of these spaces, the construction of a Schauder basis, and the identification of their α -, β -, and γ -duals. Additionally, we examine certain specific operator ideals between these spaces. Lastly, the geometric characteristics of these sequence spaces are investigated.
This study investigates the reflection of plane waves in a dual-phase-lag hygro-thermoelastic medium incorporating multi-temperature theory. The framework integrates heat conduction and moisture diffusion equations to account for wave propagation, while multi-temperature theory distinguishes between conductive and thermodynamic temperatures, encompassing one-temperature, classical two-temperature, and hyperbolic two-temperature models. Governing equations for the coupled fields of displacement, temperature and moisture are derived and for a half-space medium. Plane wave solution shows the existence of three coupled longitudinal waves and one transverse shear wave, with expressions for phase velocity and reflection coefficients are obtained for the incident wave. Numerical results demonstrate the frequency dependence of reflection coefficients and phase velocities highlighting the influence of multi-temperature parameters and phase lags on wave propagation and reflection behavior.
In this paper, we have introduced new Cesaro-type sequence spaces formed by combining a generalized difference operator Δ ^r with a Cesaro summability operator ϕ of order α >-1 in the setting of 2-normed spaces. Using a bounded sequence of positive real numbers p=(p_n), we define the spaces ℓ _∞ (Δ ^r,ϕ ,p,‖· ,·‖ ) and ℓ (Δ ^r,ϕ ,p,‖· ,·‖ ). Some special cases for lower-order difference operators are also discussed to show the general nature of our construction. We study the algebraic and topological properties of these spaces in detail. In particular, we examine basic properties such as linearity, paranormed structure, completeness, Banach-type behavior, and inclusion relations between the newly defined spaces and known sequence spaces. The inclusion results help to explain the relationships among these spaces. Overall, this work extends several existing results in summability theory and supports further studies in sequence spaces defined on 2-normed and paranormed spaces.
This study investigates a class of nonlinear ϕ -Hilfer fractional generalized double-phase problems governed by the p-Laplace operator under Dirichlet boundary conditions. More precisely, we establish the existence of nontrivial solutions in the presence of both logarithmic nonlinearities and singular perturbation terms. To do this, we combine the min–max method with variational techniques. We rigorously demonstrate the existence of nontrivial weak solutions to the proposed class of problems. The theoretical results obtained are novel and provide a significant generalization of several existing contributions in the literature. Beyond their theoretical interest, such fractional double-phase mo dels have several applications in various fields of applied mathematics and physics. They can be used to describe heterogeneous materials with nonstandard growth properties, anomalous diffusion processes in viscoelastic media, phase transition phenomena, and nonlinear heat conduction in materials with memory effects. Moreover, the presence of the ϕ -Hilfer fractional operator allows a more accurate modeling of systems exhibiting both local and nonlocal interactions, bridging both the classical and fractional dynamics and the analytical framework.
Let 𝒮_T^* denote the class of normalized analytic functions f such that zf^'(z)╱ f(z)≺ e^z+z^2/2 for | z| <1 . The structural formula, inclusion relations, coefficient estimates and various radii constants for functions in the class 𝒮_T^* are obtained.
Let G be a locally compact hypergroup provided with a left Haar measure μ . Let K be a compact subhypergroup of G such that (G, K) is a Gelfand pair, and H be a locally compact group. In this paper, considering a continuous action of H on the dual space of G we define linear operators of wavelet type acting on square-integrable K-biinvariant functions on G and we study their ranges.
The first and second Gourava indices of a hypergraph are defined in terms of the sum and product of the degrees of the vertices contained in each hyperedge. These indices act as structural descriptors that capture the degree distribution and interaction patterns within hypergraphs, thereby extending the concept of Gourava indices originally proposed for simple graphs. In this paper, we establish bounds on the first and second Gourava indices for general, k-uniform, and bipartite hypergraphs. Furthermore, bounds are derived in terms of the size, order, and extremum degrees of the hypergraph. In addition, various hypergraph operations are examined, and the extremal values of the Gourava indices are determined in relation to other well-known degree-based topological indices. The results obtained may be useful for descriptor-based modelling, network analysis, and structural studies of industrial and infrastructure systems.
The Pythagorean theorem x^2+y^2=z^2 is usually stated over the integers Z , a subring of the reals R , and it holds true for infinitely many solutions. We explore the theorem over subrings of other number spaces, such as H (quaternions) and O (octonions). We present several results mainly for L (Lipschitz quaternions) and provide a geometric interpretation of the theorem in the subring L . Some results for the corresponding subring G of O are also presented. Finally, we also present some results for the rings H/Z_p and O/Z_p .
This paper introduces and studies a generalized fractional Clairaut differential equation (GFCDE) within the framework of Wick calculus over spaces of generalized functions. As a foundational step, we extend the modified Mittag-Leffler function to act on generalized functions. Using this extension, we establish that the explicit solution of the Atangana–Baleanu Caputo (ABC) GFCDE is given by the Wick product of the initial condition and a fundamental solution, where the latter is formulated as a generalized process involving the modified Mittag-Leffler function. Finally, we demonstrate that the solution admits an integral representation, characterized by a unique positive Radon measure.
This paper presents the Tykhonov well-posedness of variational–hemivariational inequality with history-dependent operators. Some theorems are deduced to illustrate the continuous dependence of the solution with respect to the data. Next, concrete example of the model from contact mechanics is presented for which the tools presented in this paper can be applied.
The purpose of this study is to demonstrate several types of coordinated Copson integral inequalities by employing the Copson–Steklov operator T:=T_f^ϕ and the Hölder inequality. Some additional bilinear Copson integral inequalities will be derived as special cases of the results that we have obtained.
