We theoretically examine the performance of a two-body wave energy converter (WEC) featuring a floating sphere and a submerged annular heave plate, connected by a power take-off (PTO) system. Utilising linear wave theory, we derive the system’s frequency-domain response to regular plane waves and analyse the impact of varying disk porosity on power generation. Our results suggest that annular disks can enhance power extraction efficiency in various cases compared with solid heave plates. Additionally, permeable plates can broaden operational conditions by reducing oscillation amplitudes and decreasing the mechanical strain on the PTO system without substantially compromising the power conversion efficiency. Overall, our findings provide valuable insights for optimising WEC designs to improve energy capture, emphasising the potential hydrodynamic advantages of using porous reaction bodies.
Fractional Hermite-Hadamard-type inequalities represent a significant area of study in convex analysis due to their extensive applications in mathematical and applied sciences. These inequalities provide powerful tools for estimating the integral mean of a convex function in terms of its values at the endpoints of a given interval. In this paper, we focus on the development and refinement of fractional Hermite-Hadamardtype inequalities for the class of twice differentiable m-convex functions. Utilizing advanced analytical techniques, such as Ho¨lder’s inequality and the power mean integral inequality, we derive new bounds that generalize existing results in the literature. These findings not only extend classical inequalities to a broader class of convex functions but also provide sharper and more versatile estimations. The presented results are expected to have significant implications in various mathematical domains, including fractional calculus, optimization, and mathematical modeling. This work contributes to the ongoing efforts to generalize and refine integral inequalities by incorporating fractional operators and broader convexity assumptions, offering a deeper understanding of the behavior of m-convex functions under fractional integration.
In this paper, we derive some new fixed point results in C∗-algebra valued fuzzy metric space with the help of subadditive altering distance function with respect to a t-norm. Our results generalize some existing fixed point results in the literature. A common fixed point result is also derived for a pair of mappings on complete C∗-algebra valued fuzzy metric space. The results are supported by suitable examples along with the graphical demonstration of the used conditions. As application, we establish the solvability of a second order boundary value problem. Moreover, the results are also applied in control theory to study the possibility of optimally controlling the solution of an ordinary differential equation in dynamic programming.
Md. Mamunur Roshid, Mosfiqur Rahman, Mahtab Uddin
et al.
The present work deals with the investigation of the time-fractional Klein–Gordon (K-G) model, which has great importance in theoretical physics with applications in various fields, including quantum mechanics and field theory. The main motivation of this work is to analyze modulation instability and soliton solution of the time-fractional K-G model. Comparative studies are investigated by β-fraction derivative and M-fractional derivative. For this purpose, we used unified and advanced exp−ϕξ-expansion approaches that are highly important tools to solve the fractional model and are used to create nonlinear wave pattern (both solitary and periodic wave) solutions for the time-fractional K-G model. For the special values of constraints, the periodic waves, lumps with cross-periodic waves, periodic rogue waves, singular soliton, bright bell shape, dark bell shape, kink and antikink shape, and periodic wave behaviors are some of the outcomes attained from the obtained analytic solutions. The acquired results will be useful in comprehending the time-fractional K-G model’s dynamical framework concerning associated physical events. By giving specific values to the fractional parameters, graphs are created to compare the fractional effects for the β-fraction derivative and M-fractional derivative. Additionally, the modulation instability spectrum is expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third-order dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to theoretical physics. This work provided the effect of the recently updated two fraction forms, and in the future, we will integrate the space–time M-fractional form of the governing model by using the extended form of the Kudryashov method. Maple 18 is utilized as the simulation tool.
This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.
This paper addresses the solution of a differential-difference type equation having an interior layer behaviour. A difference scheme is suggested to solve this equation using a non-standard finite difference method. Finite differences are derived from the first and second order derivatives. Using these approximations, the given equation is discretized. The discretized equation is solved using the algorithm for the tridiagonal system. The method is examined for convergence. Numerical examples are illustrated to validate the method. Maximum errors in the solution, in contrast to the other methods are organized to justify the method. The layer behaviour in the solution of the examples is depicted in graphs.
The work is devoted to the problem of setting new boundary and internal boundary value problems for hyperbolic equations. The consideration of these settings is given on the example of a wave equation. The research involves the d’Alembert method, the mean value theorem and the method of successive approximations. The paper formulates and studies a number of non-local problems summarizing the classical Goursat and Dardu tasks. Some of them are marginal, and the other part is internal-marginal, and in both cases both characteristic and uncharacteristic displacements are considered. It should also be noted that a number of problems discussed below arose as a special case in the construction of the theory of correct problems for the model loaded equation of string oscillation.
T.F.A. Alves, J.F. da Silva Neto, F.W.S. Lima
et al.
In this Letter we introduce some field-theoretic approach for computing the critical properties of systems undergoing continuous phase transitions governed by the κ-generalized statistics, namely κ-generalized statistical field theory. In particular, we show, by computations through analytic and simulation results, that the κ-generalized Ising-like systems are not capable of describing the nonconventional critical properties of real imperfect crystals, e.g. of manganites, as some alternative generalized theory is, namely nonextensive statistical field theory, as shown recently in literature. Although κ-Ising-like systems do not depend on κ, we show that a few distinct systems do. Thus the κ-generalized statistical field theory is not general, i.e. it fails to generalize Ising-like systems for describing the critical behavior of imperfect crystals, and must be discarded as one generalizing statistical mechanics. For the latter systems we present the physical interpretation of the theory by furnishing the general physical interpretation of the deformation κ-parameter.
Time-dependent and space-dependent source identification problems for partial differential and difference equations take an important place in applied sciences and engineering, and have been studied by several authors. Moreover, the delay appears in complicated systems with logical and computing devices, where certain time for information processing is needed. In the present paper, the time-dependent identification problem for delay hyperbolic equation is investigated. The theorems on the stability estimates for the solution of the time-dependent identification problem for the one dimensional delay hyperbolic differential equation are established. The proofs of these theorems are based on the Dalambert’s formula for the hyperbolic differential equation and integral inequality.
The article discusses the possibility of recognizing a group by the bottom layer, that is, by the set of its elements of prime orders. The paper gives examples of groups recognizable by the bottom layer, almost recognizable by the bottom layer, and unrecognizable by the bottom layer. Results are obtained for recognizing a group by the bottom layer in the class of infinite groups under some additional restrictions. The notion of recognizability of a group by the bottom layer was introduced by analogy with the recognizability of a group by its spectrum (the set of orders of its elements). It is proved that all finite simple nonAbelian groups are recognizable by spectrum and bottom layer simultaneously in the class of finite simple non-Abelian groups.
M.T. Jenaliyev, A.S. Kassymbekova, M.G. Yergaliyev
et al.
This paper considers an initial boundary value problem for a one-dimensional Boussinesq-type equation in a domain, that is, a trapezoid. Using the methods of the theory of monotone operators, we establish theorems on their unique weak solvability in Sobolev classes.
Derivatives in time of higher order (more than two) arise in various fields such as acoustics, medical ultrasound, viscoelasticity and thermoelasticity. The inverse problems for higher order derivatives in time equations connected with recovery of the coefficient are scarce and need additional consideration. In this
article the inverse problem of determination is considered, which depends on time, lowest term coefficient in differential equation in partial derivatives of fourth order in time with initial and boundary conditions from an additional integral observation. Under some conditions of regularity, consistency and orthogonality of data by using of the contraction principle the unique solvability of the solution of the coefficient identification problem on a sufficiently small time interval has been proved.
In our work we have defined an operator called subspace convex-cyclic operator. The property of this newly defined operator relates eigenvalues which have eigenvectors of modulus one with kernels of the operator. We have also illustrated the effect of the subspace convex-cyclic operator when we let it function in linear dynamics and joining it with functional analysis. The work is done on infinite dimensional spaces which may make linear operators have dense orbits. Its property of measure preserving puts together probability space with measurable dynamics and widens the subject to ergodic theory. We have also applied Birkhoff’s Ergodic Theorem to give a modified version of subspace convex-cyclic operator. To work on a separable infinite Hilbert space, it is important to have Gaussian invariant measure from which we use eigenvectors of modulus one to get what we need to have. One of the important results that we have got from this paper is the study of Central Limit Theorem. We have shown that providing Gaussian measure, Central Limit Theorem holds under the certain conditions that are given to the defined operator. In general our work is theoretically new and is combining three basic concepts dynamical system, operator theory and ergodic theory under the measure and statistics theory.
110 years have passed since the birth of the outstanding scientist academician Orymbek Akhmetbekovich Zhautykov. O.A. Zhautykov is a Soviet mathematician and mechanic, Doctor of Physical and Mathematical Sciences, professor, academician of the Academy of Sciences of the Kazakh SSR, Honored Worker of Science and Technology of Kazakhstan, laureate of the State Prize of the Kazakh SSR. Scientific research by O.A. Zhautykov are mainly associated with the theory of infinite systems of differential equations. He has published about 200 scientific, popular science, methodological works, textbooks and teaching aids, magazine and newspaper articles.
In this paper γµ -open sets and γµ -closed sets in a GTS ( X,µ ) have been studied, where γµ is an operation from µ to P( X ). In general, collection of γµ -open sets is smaller than the collection of µ -open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS ( X,µ ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.