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This document is an editorial for a special issue of the journal Mathematics titled "Differential Equations of Mathematical Physics and Related Problems of Mechanics." The special issue covers a range of topics related to differential equations in mathematical physics and mechanics, including wave equations, spectral theory, scattering, and inverse problems. The editorial provides a summary of the published papers in the special issue, highlighting their contributions to the field. The document emphasizes the importance of the special issue in covering both applied and fundamental aspects of mathematics, physics, and their applications in various fields. The author expresses gratitude to the authors, reviewers, assistants, associate editors, and editors for their contributions to the special issue. The report does not provide specific details about the content of the papers or the nature of the special issue. [Extracted from the article]

This paper develops the mathematical apparatus of studying control problems for the stationary model of magnetic hydrodynamics of viscous heat-conducting fluid in the Boussinesq approximation. These problems are formulated as problems of conditional minimization of special cost functionals by weak solutions of the original boundary value problem. The model under consideration consists of the Navier–Stokes equations, the Maxwell equations without displacement currents, the generalized Ohm's law for a moving medium and the convection-diffusion equation for temperature. These relations are nonlinearly connected via the Lorentz force, buoyancy force in the Boussinesq approximation and convective heat transfer. Results concerning the existence and uniqueness of the solution of the original boundary value problem and of its generalized linear analog are presented. The global solvability of the control problem under study is proved and the optimality system is derived. Sufficient conditions on the data are established which ensure local uniqueness and stability of solutions of the control problems under study with respect to small perturbations of the cost functional to be minimized and one of the given functions. We stress that the unique stability estimates obtained in the paper have a clear mathematical structure and intrinsic beauty. [ABSTRACT FROM AUTHOR]

The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the coefficients of this q-Sturm–Liouville equation provide the bounds on the imaginary part of each eigenvalue. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fourth-order three-point boundary value problem. Despite the fact that the corresponding Green's function changes its sign on the square of its definition, we obtain the existence of at least one positive and decreasing solution under some suitable conditions. The results are based on the classical Krasosel'skii's fixed point theorem in cones. Then, we impose some sufficient conditions that allow us to deduce nonexistence results. In the end, some examples are given in order to illustrate our main results. [ABSTRACT FROM AUTHOR]

This research article addresses a nonclassical initial boundary value problem characterized by a non-local constraint within the framework of a pseudo-hyperbolic equation. Employing rigorous analytical techniques, the paper establishes the existence, uniqueness, and continuous dependence of a strong solution to the problem at hand. With respect to the associated linear problem, the uniqueness of its solution is ascertained through an energy inequality, which provides an a priori bound for the solution. Moreover, the solvability of this linear problem is verified by proving that the operator range engendered by the problem is indeed dense. Extending the analysis to the nonlinear problem, an iterative methodology is utilized. This approach is predicated on the insights gained from the linear problem and facilitates the demonstration of both the existence and uniqueness of a solution for the nonlinear problem under study. Consequently, the paper contributes a robust mathematical framework for solving both linear and nonlinear variants of complex initial boundary value problems with non-local constraints. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of the weak solvability to the initial boundary value problem, which describes the viscoelastic fluid motion with memory. The memory of the fluid is considered not at a constant position of the fluid particle (as in most papers on this topic), but along the trajectory of the fluid particle (which is more physical). This leads to the appearance of an unknown function z, which is the trajectory of fluid particles and is determined by the velocity v of a fluid particle. However, in this case, the velocity v belongs to L 2 (0 , T ; V 1) , which does not allow the use of the classical Cauchy Problem solution. Therefore, we use the theory of regular Lagrangian flows to correctly determine the trajectory of the particle. This paper establishes the existence of weak solutions to the considered problem. For this purpose, the topological approximation approach to the study of mathematical hydrodynamics problems, constructed by Prof. V. G. Zvyagin, is used. [ABSTRACT FROM AUTHOR]

The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for solving a class of higher order fractional differential equations with general boundary conditions. The convergence of the numerical method is proved and an error estimate is given. Finally, eight numerical examples, both linear and nonlinear, are presented to demonstrate the accuracy of our method. The proposed method introduces suitable base functions to calculate the approximate solutions and only requires us to deal with the linear or nonlinear systems. Thus, our method is convenient to implement. Furthermore, the numerical results show that the proposed method performs better compared to the existing ones. [ABSTRACT FROM AUTHOR]

One-dimensional fourth-order boundary value problems (BVPs) play a critical role in engineering applications, particularly in the analysis of beams. Current numerical investigations primarily concentrate on homogeneous boundary conditions. In addition to its high precision advantages, the Hermite finite element method (HFEM) is capable of directly computing both the function value and its derivatives. In this paper, both the cubic and quintic HFEM are employed to address two prevalent non-homogeneous fourth-order BVPs. Furthermore, a priori error estimations are established for both BVPs, demonstrating the optimal error convergence order in H 2 semi-norm and L 2 norm. Finally, a numerical simulation is presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach's fixed point theorem, while by applying Leray–Schauder's nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained. [ABSTRACT FROM AUTHOR]

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville's transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville's transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically. [ABSTRACT FROM AUTHOR]

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green's functions with an example. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities. [ABSTRACT FROM AUTHOR]

The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat's theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence and uniqueness result for a class of Dirichlet boundary value problems whose model is − Δ p u = β | ∇ u | q + c | u | p − 2 u + f in   Ω , u = 0 on   ∂ Ω , where Ω is an open bounded subset of R N , N ≥ 2 , 1 < p < N , Δ p u is the so-called p-Laplace operator, and p − 1 < q < p . We assume that β is a positive constant, c and f are measurable functions belonging to suitable Lorentz spaces. Our approach is based on Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω , ρ) -BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer's fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

Author Contributions Conceptualization, X.Y. and R.R.; methodology, X.Y. and R.R.; software, X.Y. and R.R.; validation, X.Y. and R.R.; formal analysis, X.Y. and R.R.; investigation, X.Y. and R.R.; resources, X.Y. and R.R.; data curation, X.Y. and R.R.; writing - original draft preparation, X.Y. and R.R.; writing - review and editing, X.Y. and R.R.; visualization, X.Y. and R.R.; supervision, X.Y. and R.R.; project administration, X.Y. and R.R.; funding acquisition, X.Y. and R.R. All authors have read and agreed to the published version of the manuscript. Well-posedness, dynamics, and control of nonlinear differential system with an initial-boundary value involve many mathematical, physical, and engineering problems. So, in [[1]], Xinggui Li, Xinsong Yang and Ruofeng Rao proposed the boundedness assumptions on the variables. Analytical Theory We may begin with a research paper by Xinggui Li and the Guest Editors Xinsong Yang and Ruofeng Rao [[1]]: "Impulsive Stabilization on Hyper-Chaotic Financial System under Neumann Boundary". [Extracted from the article]

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions. [ABSTRACT FROM AUTHOR]

To address the drawbacks of the slow convergence speed and lack of individual information exchange in the cuckoo search (CS) algorithm, this study proposes an improved cuckoo search algorithm based on a sharing mechanism (ICSABOSM). The enhanced algorithm reinforces information sharing among individuals through the utilization of a sharing mechanism. Additionally, new search strategies are introduced in both the global and local searches of the CS. The results from numerical experiments on four standard test functions indicate that the improved algorithm outperforms the original CS in terms of search capability and performance. Building upon the improved algorithm, this paper introduces a numerical solution approach for differential equations involving the coupling of function approximation and intelligent algorithms. By constructing an approximate function using Fourier series to satisfy the conditions of the given differential equation and boundary conditions with minimal error, the proposed method minimizes errors while satisfying the differential equation and boundary conditions. The problem of solving the differential equation is then transformed into an optimization problem with the coefficients of the approximate function as variables. Furthermore, the improved cuckoo search algorithm is employed to solve this optimization problem. The specific steps of applying the improved algorithm to solve differential equations are illustrated through examples. The research outcomes broaden the application scope of the cuckoo optimization algorithm and provide a new perspective for solving differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of monotone positive solutions for a class of nonlinear fractional differential equation with a disturbance parameter in the integral boundary conditions on the infinite interval. By using Guo–Krasnosel'skii fixed-point theorem and the analytic technique, we divide the range of parameter for the existence of at least two, one and no positive solutions for the problem. In the end, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported. [ABSTRACT FROM AUTHOR]

We suggest a new statement of the inverse spectral problem for Sturm–Liouville-type operators with constant delay. This inverse problem consists of recovering the coefficient (often referred to as potential) of the delayed term in the corresponding equation from the spectra of two boundary value problems with one common boundary condition. The previous studies, however, focus mostly on the case of zero initial function, i.e., they exploit the assumption that the potential vanishes on the corresponding subinterval. In the present paper, we waive that assumption in favor of a continuously matching initial function, which leads to the appearance of an additional term with a frozen argument in the equation. For the resulting new inverse problem, we pay special attention to the situation when one of the spectra is given only partially. Sufficient conditions and necessary conditions on the corresponding subspectrum for the unique determination of the potential are obtained, and a constructive procedure for solving the inverse problem is given. Moreover, we obtain the characterization of the spectra for the zero initial function and the Neumann common boundary condition, which is found to include an additional restriction as compared with the case of the Dirichlet common condition. [ABSTRACT FROM AUTHOR]

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B 1 a Green function is constructed in the cases c > 0 , c ∉ − N , where c is the coefficient in front of u in the boundary condition ∂ u ∂ n + c u = f . To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λ u + c u = f . Elliptic boundary value problems for Δ m u = 0 in B 1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δ u = 0 , Δ 2 u = 0 and Δ 3 u = 0 in B 1 as well as some additional results from the theory of spherical functions are proposed. [ABSTRACT FROM AUTHOR]

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method. [ABSTRACT FROM AUTHOR]

The purpose of this research is to examine the uniqueness and existence of the (ω , ρ) -BVP solution for a particular solution to a class of Hadamard fractional differential equations with impulsive boundary value requirements on Banach spaces. The notion of Banach contraction and Schaefer's theorem are used to prove the study's key findings. In addition, we offer the prerequisites for the set of solutions to the investigated boundary value with impulsive fractional differential issue to be convex. To enhance the comprehension and practical application of our findings, we offer two illustrative examples at the end of the paper to show how the results can be applied. [ABSTRACT FROM AUTHOR]

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary-value problems for differential equations of fractional order. In particular, we show the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues for these operators. [ABSTRACT FROM AUTHOR]

In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph. [ABSTRACT FROM AUTHOR]

This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order α and the fractional central formula for approximating the Caputo differentiator of order 2 α , where 0 < α ≤ 1 . The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach. [ABSTRACT FROM AUTHOR]

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven. [ABSTRACT FROM AUTHOR]

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed. [ABSTRACT FROM AUTHOR]

The paper aims to study a discrete boundary value problem of the Kirchhoff type based on the critical point theory and the strong maximum principle. Compared to the existing literature, the existence and multiplicity of positive solutions to the problem are considered according to the behavior of the nonlinear term f in some points between the zero and positive infinity, which is a new attempt. Under different assumptions of the nonlinear term f, we obtain the determined open intervals of the parameter λ , such that the problem has at least three positive solutions or at least two positive solutions in different intervals. In the end, two concrete examples are used to illustrate our main conclusions. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the Caputo fractional differential equation involving the ρ (τ) Laplacian operator and nonlocal multi-point of Riemann–Liouville's fractional integral. We also prove the uniqueness of the positive solutions for Boyd and Wong's nonlinear contraction via the Guo–Krasnoselskii fixed-point theorem in Banach spaces. Finally, we illustrate the theoretical results and show that by solving the nonlocal problems, it is possible to obtain accurate approximations of the solutions. An example is also provided to illustrate the applications of our theorem. [ABSTRACT FROM AUTHOR]

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is established. The application of the main theorem to four different semi-linear delay parabolic differential equations is presented. The first- and second-order accuracy difference schemes (FSADSs) for the solution of a one-dimensional semi-linear time-delay parabolic equation are considered. The new desired numerical results of this paper and their discussion are presented. [ABSTRACT FROM AUTHOR]

In this paper, the existence of infinitely many solutions for the partial discrete Kirchhoff-type problems involving p-Laplacian is proven by exploiting the critical point theory for the first time. Moreover, by using the strong maximum principle, we acquire some sufficient conditions for the presence of infinitely many positive solutions to the boundary value problems. Our major outcomes are explained with one example. [ABSTRACT FROM AUTHOR]

In this paper, by using fixed-point theorems, the existence and uniqueness of positive solutions to a class of four-point impulsive fractional differential equations with p-Laplacian operators are studied. In addition, three examples are given to justify the conclusion. The interest of this paper is to study impulsive fractional differential equations with p-Laplacian operators. [ABSTRACT FROM AUTHOR]

This paper is devoted to the mathematical analysis of a spatially homogeneous thermostatted kinetic theory framework with an unbounded activity domain. The framework consists of a partial integro-differential equation with quadratic nonlinearity where the domain of the activity variable is the whole real line. Specifically the mathematical analysis refers firstly to the existence and uniqueness of the solution for the related initial boundary value problem; Secondly the investigations are addressed to the existence of a class of self-similar solutions by employing the Fourier transform method. In particular the main result is obtained for a nonconstant interaction rate and a nonconstant force field. Conclusions and perspectives are discussed in the last section of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton's method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper. [ABSTRACT FROM AUTHOR]

The paper presents an exact solution to the internal boundary value problem of the field distribution in an electrostatic lens formed by two identical semi-infinite coaxially located round cylinders separated by a slit of finite width and located inside an infinite outer cylinder. The problem is reduced to a system of singular Wiener–Hopf integral equations, which is further solved by the Wiener–Hopf method using factorized Bessel functions. Solutions to the problem for each region inside the infinite outer cylinder are presented as exponentially converging series in terms of eigenfunctions and eigenvalues. Using the obtained formulas, a numerical calculation of the axial distribution of the potential of a two-electrode lens was made for various values of the radii of the outer and inner cylinders. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the role that a deep learning approach could play in computational mechanics. In this paper, a convolutional neural network technique based on modified loss function is proposed as a surrogate of the finite element method (FEM). Several surrogate-based physics-informed neural networks (PINNs) are developed to solve representative boundary value problems based on elliptic partial differential equations (PDEs). According to the authors' knowledge, the proposed method has been applied for the first time to solve boundary value problems with elliptic partial differential equations as the governing equations. The results of the proposed surrogate-based approach are in good agreement with those of the conventional FEM. It is found that modification of the loss function could improve the prediction accuracy of the neural network. It is demonstrated that to some extent, the deep learning approach could replace the conventional numerical method as a significant surrogate model. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce new forms of quasi-contractions in metric-like spaces and initiate more general conditions for the existence of invariant points for such operators. The proposed notions are then applied to study novel existence criteria for the existence of solutions to two-point boundary value problems in the domains of integer and fractional orders. To attract further research in this direction, important consequences are deduced and discussed to indicate the novelty and generality of our proposed concepts. [ABSTRACT FROM AUTHOR]

The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral summation identities (ISI). A detailed study of the matrix operator of the problem is performed and its Fredholm property in the weighted Hilbert space of infinite sequences is proven. The convergence of the truncation method constructed in the paper for the numerical solution of SLAE-II is justified and the results of computations are presented and discussed, specifically considering the determination of resonance modes. [ABSTRACT FROM AUTHOR]

In this article, we study a periodic boundary value problem related to valveless pumping. The valveless pumping is described by the unidirectional flow of liquid in a system. We establish some conditions for globally asymptotic stability and the existence of a positive periodic solution to the considered equation. Finally, a numerical example shows that the theoretical results in this paper are feasible. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of nonlocal boundary-value problem for two-dimensional pseudoparabolic equation which arise in many physical phenomena. A three-layer alternating direction implicit method is investigated for the solution of this problem. This method generalizes Peaceman–Rachford's ADI method for the 2D parabolic equation. The stability of the proposed method is proved in the special norm. We investigate algebraic eigenvalue problem with nonsymmetric matrices to prove this stability. Numerical results are presented. [ABSTRACT FROM AUTHOR]

After the introduction of Alam–Imdad's relation-theoretic contraction principle, the field of metric fixed point theory has attracted much attention. A number of fixed point theorems in the context of relational metric space employing various types of contractions has been appeared during the last seven years. In this manuscript, one proved a metrical fixed point theorem for ϕ -contraction involving (c)-comparison functions employing an amorphous relation. The result proved in this paper refines, modifies, unifies and sharpens several existing fixed point results. We also constructed an example in order to attest the credibility of our results. Finally, we applied our result to establish the existence and uniqueness of solution of certain periodic boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ 1 -Hilfer and ψ 2 -Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, the temperature shock phenomenon is considered. This phenomenon occurs during the operation of engineering structures on Earth and in outer space. A rectangular plate has been selected as a structural element exposed to temperature shock. It has a rigidly sealed edge and three free edges. A one-dimensional third initial boundary value problem of thermal conductivity was posed and solved to study the stress–strain state of the plate. Fourier's law was used to solve this problem, taking into account the inertial term, since the temperature shock is a fairly fast-dynamic phenomenon. It was believed that all the thermophysical properties of the plate are constant and do not depend on its temperature. As a result, the temperature field of the plate was obtained after the temperature shock. This temperature field generates temperature stresses inside the plate, which lead to temperature deformations. To determine these deformations, the initial boundary value problem of thermoelasticity was posed and solved in this work. The Sophie Germain equation was used while solving this problem. To describe the plate, the theory of flexible plates was used, taking into account the stresses in the middle surface of the plate. Next, the accuracy of analytical solutions for the points displacement of a homogeneous plate subjected to a temperature shock was investigated. The temperature field was constructed using a numerical simulation. Functions of the displacement vector components were obtained using approximate analytical solutions. The accuracy of approximate analytical solutions for the components of the plate points deformation vector was estimated. The obtained results allow us to describe the stress–strain state of the plate after the temperature shock. The results of this work can be used in the design of engineering structures for both terrestrial and space purposes in terms of stability calculations and the implementation of deformation constraints. [ABSTRACT FROM AUTHOR]

In this paper, we consider a zeroth-order perturbation q (x) of the buckling operator Δ 2 − κ Δ , which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator to the buckling operator, but the Dirichlet-to-Neumann map given in this study is more meaningful and general. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit ψ -Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle and Krasnoselskii's fixed point theorem. In addition, the Ulam–Hyers–Rassias stability result is proved using the nonlinear functional analysis technique. Finally, illustrative examples are given to validate our main results. [ABSTRACT FROM AUTHOR]