This issue presents a collection of 15 carefully selected papers authored by both international and national researchers. Each paper has undergone rigorous peer review and introduces novel findings in the fields of analysis and applied mathematics, with particular emphasis on their application to the construction and investigation of solutions to well-posed and ill-posed boundary value problems for partial differential equations.
The contributing authors represent a diverse range of countries, including Turkey, Kazakhstan, Sweden, the Russian Federation, Azerbaijan, Uzbekistan, Turkmenistan, Iraq and Cyprus. We are especially pleased to note that many of these articles are co-authored by researchers from different universities across the globe, reflecting the collaborative spirit and international scope of contemporary mathematical research.
In this work, numerical algorithms of higher-order accuracy are constructed and studied for a pseudoparabolic equation that describes the filtration process in fractured-porous media. The increase in the order of accuracy is achieved in various ways. First, only the spatial variables are approximated, as in the method of lines. Then, to solve the resulting system of linear ordinary differential equations, the finite difference method and the finite element method are applied. The application of these methods makes it possible to achieve a higher order of approximation for the difference schemes. Schemes of fourth-order accuracy in the spatial variables and second-order in time are presented, as well as schemes of fourth-order accuracy in all variables. Based on the stability theory of three-level difference schemes, stability conditions for the proposed algorithms are obtained. Using a special technique for solving the difference schemes, a priori estimates are derived, and based on them, theorems on convergence and accuracy are proven in the class of smooth solutions to the differential problem. An implementation algorithm is proposed for the difference scheme constructed using the finite element method. Test examples for one-dimensional and two-dimensional equations are also provided, demonstrating the higher-order accuracy of the proposed schemes.
Pareekshith G. Bhat, Ali J. Chamkha, Nityanand P. Pai
et al.
The current study strives to theoretically examine the heat and mass transfer properties on the magnetohydrodynamic upper convected Maxwell fluid flow through a squeezing channel of parallel plates. Due to its vast applications, such as lubrication systems and bearing, the flow of upper convected Maxwell fluid through the channel comprising a moving impermeable top plate and a stationary porous bottom that is responsible for injection and suction effects in addition to squeezing motion is analysed in the study. The fundamental equations governing the conservation laws of fluid mechanics are transfigured into a non-linear system of ordinary differential equations adopting similarity transformations along with the boundary conditions. The so-obtained non-linear ordinary differential equations are then approached by the homotopy perturbation method to achieve an approximate analytic solution. Various graphs concerning the velocity, temporal, and concentration profiles are plotted against distinct pertinent parameters that pose a physical impact on the model. It is observed that the temporal distribution field elevates with a rise in the Eckert number and a decrease in the radiation parameter. Further, it is noticed that the concentration profile upsurges with a hike in the radiation parameter and depletion in the Eckert number. Moreover, the numerical values corresponding to the coefficient of skin friction, and rates of heat and mass transfer are tabulated for distinct pertinent parameters involved in the study.
This research lies within the domain of model theory, which investigates the properties of, broadly speaking, incomplete theories. The article introduces novel methods for classifying classes of structures whose associated theories are Jonssonian, forming a distinct subclass within the broader category of inductive theories. This subclass is characterized by satisfying the standard model-theoretic properties of joint embedding and amalgamation. The focus is placed specifically on the second kind of hybrids, those involving theories with different signatures. As a representative case of such hybrids among Jonsson theories, we examine the classical examples of the theory of unars and the theory of undirected graphs. The study proposes and formalizes several new notions, including the perfect Robinson hybrid, the center of a Robinson hybrid, the Kaiser class of a theory, and the concept of triple factorization. Within the framework of these definitions, we establish new results, among them a theorem confirming the existence of a unique countably categorical theory of S-acts, which is syntactically equivalent to the Robinson hybrid formed by the aforementioned classes.
This paper investigates the issues of existence and uniqueness of a solution to a combined boundary value problem with local and nonlocal conditions for a specific class of mixed elliptic-hyperbolic-type equations with singular coefficients. A distinctive feature of the considered problem is that on one part of the boundary characteristic, the values of the desired function are specified, while on the other part, nonlocal conditions are imposed. These conditions establish pointwise connections between the values of the sought function on different parts of the boundary characteristics using the Riemann-Liouville fractional differentiation operator. At the same time, a portion of hyperbolic domain’s boundary remains free from boundary conditions. The proof of the solution’s uniqueness is based on the application of an analogue of A.V. Bitsadze’s extremum principle for mixed-type equations with singular coefficients. The existence of the solution is reduced to the analysis of a Tricomi singular integral equations’ system with a shift, containing a non-Fredholm operator with isolated first-order singularity in kernel. By applying the Carleman-Vekua regularization method, these equations are reduced to a Wiener-Hopf integral equation, for which it is proved that the index is equal to zero. This, in turn, reduces the problem to a Fredholm integral equation of the second kind, the uniqueness of whose solution ensures the well-posedness of the given problem.
In this paper, we consider the application of the method of fictitious domains to a viscoelastic incompressible medium based on the Kelvin-Voigt model. Application of the method of fictitious domains allows solving the original problem in regions with complex geometric configuration. This makes it easier to automate the construction of a consistent difference mesh, and to solve the problem in areas of standard shape. Estimates for the proximity of the auxiliary problem’s solution are obtained. The auxiliary problem is constructed by the method of fictitious domains. These estimates refer to the solution of the original problem. The original problem describes a viscoelastic incompressible medium. Convergence follows from the estimates of the proximity of the solutions of the original and auxiliary problems. Further, on the basis of the method of fictitious domains, two-sided estimates on a small parameter for the difference between the solution of the original problem and the solution of the auxiliary problem constructed by the method of fictitious domains are obtained. Moreover, the solution to the auxiliary problem is expanded as a series in powers of the small parameter. This is possible because that solution is represented as a functional series that converges absolutely in the original domain.
The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics tostatistical one.
Abstract This manuscript investigates the analytical and semi-analytical solutions of nonlinear phi-four (PF) equation by applying the sech–tanh expansion method, modified Ψ ′ Ψ -expansion method and Adomian decomposition method. This equation is considered as a particular case of the well-known Klein–Fock–Gordon (KFG) equation. The KFG equation is derived by Oskar Klein and Walter Gordon and relates to Schrodinger equation. Many quantum effects can be studied based on the PF model’s solutions, such as wave-particle duality to describe reality in the form of waves is at the heart of quantum mechanics. The considered model is also used to explain de Broglie waves’ character, the spineless relativistic composite particles, relativistic electrons, etc., which are also the main icons for a good understanding of the phenomenon of quantum physics. Through two recent analytical schemes, handling this model gives many novel computational solutions that are tested through the semi-analytical scheme to investigate their accuracy. Demonstrating the obtained analytical and matching between analytical and semi-analytical through some distinct sketches shows the considered model’s novel physical properties.
Abstract The article studies the exact traveling wave solutions to the Schrodinger-Poisson system which has applications in gravity’s role of quantum state and approximate the coupling between quantum mechanics with gravitation. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtain using two norms of integration. For this sake modified extended direct algebraic (MEDA) and ( G ′ / G ) -expansion techniques are used. The 3D plots and their corresponding contour graphs are also depicted. The constraints conditions for the exact of solutions are also emerged during the derivation of solution.
Abstract In the context of nonclassical continuum mechanics, the nonlocal strain gradient theory is employed to develop a nonclassical size dependent model to investigate the dynamic behavior of a CNTs reinforced composite beam resting on two parameters elastic foundations under a moving load. The governing dynamic equations of motion are derived based on third-order shear deformation theory using Hamilton’s principle. An analytical solution methodology is developed using Navier’s procedure is developed to derive the analytical solution for the equations of motion. The developed methodology is checked and compared. Parametric studies are conducted to demonstrate the applicability of the developed procedure to investigate the dynamic behavior of CNTs beams under moving load. Effects of the elastic foundation parameters, volume fraction, CNTs configuration, the nonclassical parameters, and the moving load velocity parameter on the dynamic behavior of CNTs beams under moving load are investigated and analyzed. The obtained results are supportive for design and manufacturing of composite CNTs beams.
Serious Games (SGs) are gaining an ever increasing interest for education and traning. Exploiting the latest simulation and visualization technologies, SGs are able to contextualize the player’s experience in challenging, realistic environments, supporting situated cognition. However, we still miss methods and tools for effectively and deeply infusing pedagogy and instruction inside digital games. After presenting an overview of the state of the art of the SG taxonomies, the paper introduces the pedagogical theories and models most relevant to SGs and their implications on SG design. We also present a schema for a proper integration of games in education, supporting different goals in different steps of a formal education process. By analyzing a set of well-established SGs and formats, the paper presents the main mechanics and models that are being used in SG designs, with a particular focus on assessment, feedback and learning analytics. An overview of tools and models for SG design is also presented. Finally, based on the performed analysis, indications for future research in the field are provided.
In this paper, new objects of research are identified, both from the standpoint of model theory and from the standpoint of universal algebra. Particularly, the Jonsson spectra of the Jonsson varieties and the Jonsson quasivarieties are considered. Basic concepts of 3 types of convexity are given: locally convex theory, ϕ(x)-convex theory, J-ϕ(x)-convex theory. Also, the inner and outer worlds of the model of the class of theories are considered. The main result is connected with the question of W. Forrest, which is related to the existential closedness of an algebraically closed variety. This article gives a sufficient condition for a positive answer to this question.
Abstract Micromechanical tests are reliable tools to study the failure mechanisms in composites reinforced with continuous fibers. This paper presents an overview of various analytical models developed to study the pullout (push-back) behavior of a fiber embedded in a matrix block to characterize the fiber/matrix interfacial adhesion. Two approaches can be distinguished: one based on a maximum stress criterion (shear lag) and the other based on fracture mechanics. This article gives an overview of the analytical models reported in the literature to measure the shear strength and critical fracture energy at the interface, the parameters influencing these properties, the geometry of the model, embedded length of the fiber, fiber diameter and loading conditions (opening width between the knife-edges for example), including components (fiber, matrix, interface), manufacturing route and the resulting defects.