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In this paper, the self-propelled movement on gyrotactic swimming microorganisms into this generalized slip flow by nanoliquid over the stretching cylinder with strong magnetic field is discussed. Constant wall temperature was pretended as well as the Nield conditions of boundary. The intuitive non-Newtonian particulate suspension was included into applying Casson fluid by the base liquid and nanoparticles. This formation on the bio-mathematical model gives the boundary value problem by the nonlinear partial differential equations. Primly, modeled numerical system was converted to nondimensional against this help on acceptable scaling variables and the bvp4c technique was used to acquire the mathematical outcomes on the governing system. This graphical description by significant parameters and their physical performance was widely studied. The Prandtl number has the maximum contribution (112.595%) along the selected physical parameters, whereas the Brownian motion has the least (0.00165%) heat transfer rate. Anyhow, Casson fluid was established for much helpful suspension of this method on fabrication and coatings, etc. Therefore, this magnetic field performs like the resistive force of that fluid motion, and higher energy was enlarged into the structure exhibiting strong thermal radiation. [ABSTRACT FROM AUTHOR]

The solution of a nonlinear boundary value problem in ordinary differential equations of the form $ L(x(t))+f(x(t)) = 0 $ can be obtained by solving the associated system of nonlinear equations. In this paper, a three-step iterative method with ninth order convergence is proposed to solve the system of nonlinear equations. The scheme is kept free from second and higher order Frechet derivatives to make it computationally efficient. The nonlinear boundary value problem is discretized to produce a system of nonlinear equations which is then solved using the proposed method. Later, the ninth order method is generalized to obtain a m-step scheme with $ 3m $ order of convergence. Standard problems like Bratu [3] one-dimensional problem and Frank-Kamenetzki [8] problem are solved using the new method. The performance of the new method is compared with an existing method [18] to establish its computational superiority. [ABSTRACT FROM AUTHOR]

We study half-spheres with small radii λ sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point p of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of p. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the "domain-setting". [ABSTRACT FROM AUTHOR]

The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of h = 1 / 1 0 and h = 1 / 5. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods. [ABSTRACT FROM AUTHOR]

This article presents a detailed analysis of a numerical model for studying the failure of masonry structures that have been retrofitted with timber solutions. The model accurately reproduces the failure mechanisms of masonry and evaluates the effectiveness of timber-based retrofitting. The study validates the model through comparisons with experimental and numerical results and assesses its efficacy through nonlinear static analysis. The article also discusses the use of a timber frame system as a retrofitting strategy for masonry walls, highlighting its significant enhancement of the wall's load-bearing capacity against lateral forces. However, the post-peak behavior of the retrofitted cases reveals potential weaknesses in the timber frame system, which can lead to sudden drops in resistance. The article concludes by emphasizing the importance of preventing simultaneous failure of multiple bracing elements in the retrofitting system and the need for further research to improve its design and mitigate brittle behavior. [Extracted from the article]

The radial propagation of the steady-state wave motion is studied for a pre-stressed cylindrical shell composed of the isotropic compressible neo-Hookean material. The radial static deformation of the cylindrical shell and radial wave motion of the pre-stressed cylindrical shell are investigated. The mathematical model is established as a class of boundary value problems of the second-order nonlinear ordinary differential equation (ODE). Based on the theory of the large elastic deformation superposed by the small harmonic deformation, a radial dynamic displacement is added to the pre-stressed cylindrical shell. The existence of the steady-state wave motion is conducted. Numerical solutions are obtained by using the multiple shooting method. In addition, the influences of the static load, material and structural parameters on the steady-state wave motion are discussed. Numerical results indicate that as the static load increases, the frequency of the steady-state wave motion increases and the amplitude decreases in the compressible hyperelastic cylindrical shell. The material parameter plays a significant role in the frequency and amplitude of the steady-state wave motion. [ABSTRACT FROM AUTHOR]

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]

In this study, we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov (EFK) problem. The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme. Following temporal discretization, the generalized finite difference method (GFDM) with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node. These supplementary nodes are distributed along the boundary to match the number of boundary nodes. By incorporating supplementary nodes, the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation. To demonstrate the efficacy of our approach, we present three numerical examples showcasing its performance in solving this nonlinear problem. [ABSTRACT FROM AUTHOR]

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We investigate a class of semipositone nonlinear elliptic boundary value problems in unbounded domains D of R n , n ≥ 3 . Under some mild-assumptions, we obtain the existence of a unique positive continuous solution and provide information about its global behavior. [ABSTRACT FROM AUTHOR]

In this article, we propose novel coupled nonlinear singular boundary value problems arising in epitaxial growth theory. The coupled equations are nonlinear, non‐self‐adjoint, and singular and have no exact solutions. We derive some qualitative properties of the coupled solutions, which depend on the size of parameters that occur in the coupled system. To prove the existence of the coupled solutions, we apply the monotone iterative technique on equivalent coupled integral equations in the presence of upper and lower solutions. We also compute the bounds for the parameters, which indicates the existence of coupled solutions for small values of the parameters, and nonexistence for large positive values of these parameters. To illustrate the theory developed, we consider test cases and compute the sequence of upper and lower solutions. To verify the bounds on the parameters, we have used a type of Adomian decomposition method. [ABSTRACT FROM AUTHOR]

We formulate a Neumann variational problem, which characterizes the capillary phenomena, and disLaplacian‐like operator defined bycuss the existence of nontrivial weak solutions. We get the results using sufficient hypotheses on the reaction term. [ABSTRACT FROM AUTHOR]

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ ∈ [ 0 , 1 ] . The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented. [ABSTRACT FROM AUTHOR]

This research focuses on finding multiple solutions (MSs) to nonlinear fractional boundary value problems (BVPs) through a new development, namely the predictor Laplace fractional power series method. This method predicts the missing initial values by applying boundary or force conditions. This research provides a set of theorems necessary for deriving the recurrence relations to find the series terms. Several examples demonstrate the efficacy, convergence, and accuracy of the algorithm. Under Caputo's definition of the fractional derivative with symmetric order, the obtained results are visualized numerically and graphically. The behavior of the generated solutions indicates that altering the fractional derivative parameters within their domain symmetrically changes these solutions, ultimately aligning them with the standard derivative. The results are compared with the homotopy analysis method and are presented in various figures and tables. [ABSTRACT FROM AUTHOR]

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions. [ABSTRACT FROM AUTHOR]

The study aims to explore wavelet applications for analyzing nonlinear boundary value problems. Although several wavelet methods are reviewed in the literature, a comparative study of their strengths and limitations has found only a few attempts. This study bridges the gap between two wavelet-based numerical methods, namely, higher order Daubechies waveletbased Galerkin method and Haar wavelet collocation method, by conducting a comparative study. Nonlinear boundary value problems arising in mathematical physics are solved using both schemes, followed by the computation of optimal error estimates. Furthermore, the advantages offered by the Haar wavelet collocation method over the wavelet-Galerkin method and the rate of convergence are also discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the stationary Black-Scholes model arising in finance with transaction costs. This model becomes interesting when the time does not play a role such as, for instance, in perpetual options. The equation describing this model is a nonlinear second-order boundary value problem and there is no analytic solutions in closed form for such a nonlinear equation. After discretization via the centered finite difference formula we have to solve a nonlinear algebraic system which would be a serious problem when we use a small discretization mesh. We solve this nonlinear system by the residual-based Broyden's method, which is an efficient quasi-Newton method and is convenient to implement by a desk computer. We give a convergence analysis of the Broyden's method by assuming a lower and upper bound of the converged solution of the Black-Scholes model. Numerical results are given to show that the convergence rate of the method is robust with respect to the discretization mesh and the problem parameters. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of nontrivial solutions for an anisotropic discrete nonlinear problem with variable exponent. The analysis makes use of variational methods and critical point theory. [ABSTRACT FROM AUTHOR]

In this paper the authors consider the existence of positive solutions to a two point boundary value problem for nonlinear second-order impulsive systems. They use a vector version of Krasnosel'skii's fixed point theorem in cones in their proofs. Examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton’s method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form. [ABSTRACT FROM AUTHOR]

In this article, we study the uniqueness and stability of nonnegative solutions for a class of semilinear elliptic problems in a ball, when the nonlinearity has more than one zero, negative at the origin and concave. [ABSTRACT FROM AUTHOR]

A boundary value problem for a nonlinear second-order functional-differential equation on the interval with integral boundary conditions is considered. Sufficient conditions for the existence of at least one positive solution of the problem under consideration are found by using special topological tools. A nontrivial example in which the sufficient conditions for the unique solvability of the problem are satisfied is given. [ABSTRACT FROM AUTHOR]

In this paper, we study a free‐boundary problem modeling the growth of spherically symmetric tumors with angiogenesis and a necrotic core, where the Robin boundary condition is imposed for the nutrient concentration. The existence of a global solution is established by first reducing the free‐boundary problem into an equivalent initial boundary value problem for a nonlinear strongly singular parabolic equation on a fixed domain, then proving that an approximation problem admits a unique solution by the Schauder fixed point theorem combined with the Lp$$ {L}&#x0005E;p $$ estimates for parabolic equations, and finally taking the limit. Compared with the Dirichlet boundary value condition problem, the Robin condition causes some new difficulties in making rigorous analysis of the model, particularly on the uniqueness of solutions to the approximation problem. [ABSTRACT FROM AUTHOR]

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is to present the effectiveness and robustness of a numerical algorithm based on the block Newton method for the nonlinear kinematic hardening rules adopted in modeling ductile materials. Design/methodology/approach: Elastoplastic problems can be defined as a coupled problem of the equilibrium equation for the overall structure and the yield equations for the stress state at every material point. When applying the Newton method to the coupled residual equations, the displacement field and the internal variables, which represent the plastic deformation, are updated simultaneously. Findings: The presented numerical scheme leads to an explicit form of the hardening behavior, which includes the evolution of the equivalent plastic strain and the back stress, with the internal variables. The features of the present approach allow the displacement field and the hardening behavior to be updated straightforwardly. Thus, the scheme does not have any local iterative calculations and enables us to simultaneously decrease the residuals in the coupled boundary value problems. Originality/value: A pseudo-stress for the local residual and an algebraically derived consistent tangent are applied to elastic-plastic boundary value problems with nonlinear kinematic hardening. The numerical procedure incorporating the block Newton method ensures a quadratic rate of asymptotic convergence of a computationally efficient solution scheme. The proposed algorithm provides an efficient and robust computation in the elastoplastic analysis of ductile materials. Numerical examples under elaborate loading conditions demonstrate the effectiveness and robustness of the numerical scheme implemented in the finite element analysis. [ABSTRACT FROM AUTHOR]

This study investigates the boundary layer flow problem arising in the viscous fluid flow of a slightly rarefied gas-free stream over a moving plate embedded in a porous medium. Previous works were mainly focused on numerically, and some properties of the solutions were discussed from numerical results. But, numerical solutions are only sometimes guaranteed the solution's existence. In this work, we have adopted the famous topological shooting argument to prove the existence result of the solution. We have shown that the solution may be convex or concave depending on the parameter values. Also, for these reasons, there arise some difficulties. However, we have rectified all the challenges and proved that unique solutions exist for all the particular governing parameter values. Nevertheless, we found an exact solution for some specific parameter values. Furthermore, the Haar wavelet collocation method is used to address the numerical results of the nonlinear boundary value problem. We first validate the obtained results with the existing numerical outcomes. The impacts of permeability and slip parameters on the velocity profiles and shear stress are elucidated through tables and figures, and probable arguments are explained in detail. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mechanical system of ordinary differential equations (ODEs) with multi-point boundary conditions is considered by a novel type of reproducing kernel Hilbert space method (RKHSM). To begin, we define the unknown variables in terms of the reproducing kernel function. The roots of the Shifted Chebyshev polynomials (SCPs) are then utilized to collocate the resulting system. Finally, Newton's iterative method is employed to find the unknown expansion coefficients. The solutions of this system of equations, which arise from the flow of an electrically conducting nanofluid over an impermeable stretching cylinder, are numerically analyzed, and convergence analysis is discussed to demonstrate the reliability of the presented method (PM). Tables and figures are provided to further discuss the solutions and assess the effectiveness of the method in comparison to other techniques in the literature. [ABSTRACT FROM AUTHOR]

The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter ζ {{\zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem. [ABSTRACT FROM AUTHOR]

In this study, an efficient meshfree numerical method is introduced for solving the nonlinear boundary value problems. The method of fundamental solutions (MFS) is one of the most popular among meshfree methods. While traditionally limited to linear and homogeneous problems, this study extends the applicability of the MFS to include nonhomogeneous and nonlinear equations. To achieve this, an extended MFS is combined with a fixed-point iteration scheme. This developed framework is benchmarked to address two different flow problems. The first involves fluid flow through porous media in a channel governed by the nonlinear Brinkman-Forchheimer equation. The second problem pertains to electrohydrodynamic (EHD) flow in a circular conduit. The obtained solutions are compared with the finite element method and the solutions available in the existing literature. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we shall investigate the large-time behavior of the solution to an outflow problem of the one-dimensional bipolar quantum Navier-Stokes-Poisson system in the half space. Under some suitable assumptions on the boundary data and the space-asymptotic states, we successfully construct a nonlinear wave which is the superposition of the stationary solution and the 2-rarefaction wave. Then, by means of the $ L^2 $-energy method, we prove that this nonlinear wave is asymptotically stable provided that the initial perturbation and the strength of the stationary solution are small enough, while the strength of the 2-rarefaction wave can be arbitrarily large. [ABSTRACT FROM AUTHOR]

Riga plate is known to create a crossing electric and magnetic field to generate a wall-parallel Lorentz force. The significance of Casson nanofluid flow past a Riga plate is observed in the sphere of engineering, such as polymer extrusion, food manufacturing, plastic films, oil reserves and geothermal manufacturing. Researchers are interested in this model because of its potential use in biological rheological models. As Casson nanofluid flows are of great interest, this study aims to investigate the three-dimensional magnetohydrodynamics (MHD) flow with heat and mass transport of Casson nanofluid over a flat Riga plate. As a novelty, this study also includes the effectiveness of wall velocity slip, activation energy, nonlinear radiation, and temperature and space-dependent heat source/sink. Suitable similarity transformations have been employed to generate the dimensionless ordinary differential equations (ODEs) from the partial differential equations (PDEs) regulating the fluid flow problem. The transformed nonlinear boundary value problem is then solved numerically using the in-built routine "bvp4c" in MATLAB. The visual demonstrations are provided for the effects of various significant physical factors on the flow, heat and mass distributions. On the other hand, wall shear stress and rates of heat and mass transport at the surface are measured and displayed numerically in tabular form. The findings indicate that the fluid velocity in both directions slows as the velocity slip parameter increases. However, the velocity profile is escalated with the boost of modified Hartmann number. An increase in heat source parameters leads to decrease the heat transmission rate at the wall. The higher values of the radiation parameter result in a better wall heat transmission rate. Further, the rate of mass transport drops when the activation energy parameter is hiked. [ABSTRACT FROM AUTHOR]

Radial basis function neural networks (RBFNN) have been increasingly employed to solve boundary value problems (BVPs). In the current study, we propose such a technique for nonlinear (apparently for the first time) elliptic BVPs of orders two and four in 2-D and 3-D. The method is also extended, in a natural way, to solving 2-D and 3-D inverse BVPs. The RBFNN is trained via the least squares minimization of a nonlinear functional using the MATLAB® routine lsqnonlin. In this way, as well as the solution, appropriate values of the RBF approximation parameters are automatically delivered. The efficacy of the proposed RBFNN is demonstrated through several numerical experiments. [ABSTRACT FROM AUTHOR]

A high-order numerical scheme based on collocation of a quintic B-spline over finite element is proposed for the numerical solution of a class of nonlinear singular boundary value problems (SBVPs) arising in various physical models in engineering and applied sciences. Five illustrative examples are presented to illustrate the applicability and accuracy of the method. In order to justify the advantage of the proposed numerical scheme, the computed results are compared with the results obtained by two other fourth-order numerical methods, namely the finite difference method (Chawla et al. in BIT 28(1):88–97, 1988) and B-spline collocation method (Goh et al. in Comput Math Appl 64:115–120, 2012). [ABSTRACT FROM AUTHOR]

The aim of this paper is to use the second derivative of Chebyshev polynomials (SDCHPs) as basis functions for solving linear and nonlinear boundary value problems (BVPs). Then, the operational matrix for the derivative was established by using SDCHPs. The established matrix via mixing between two spectral methods, collocation and Galerkin, has been applied to solve BVPs. Consequently, an error analysis is investigated to ensure the convergence of the technique used. Finally, we solved some problems involving real-life applications and compared their solutions with exact and other solutions from different methods to verify the accuracy and efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fourth-order compact discretization method for solving a secondorder boundary value problem governed by the nonlinear Fredholm integro-differential equations. Using an efficient approximate polynomial, a compact numerical integration method is first designed. Then by applying the derived numerical integration formulas, the original problem is converted into a nonlinear system of algebraic equations. It is shown that the proposed method is easy to implement and has the third order of accuracy in the infinity norm. Some numerical examples are presented to demonstrate its approximation accuracy and computational efficiency, as well as to compare the derived results with those obtained in the literature. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and uniqueness of solutions for a two-point boundary value problem of Caputo fractional differential equation of variable order. The results are obtained by means of Banach’s and Krasnoselskii’s fixed point theorems. In addition, the obtained results are illustrated with the aid of a numerical example. [ABSTRACT FROM AUTHOR]

This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations. [ABSTRACT FROM AUTHOR]

This paper considers a heat conduction process for isotropic layered media with internal thermal heating. As a result of the heterogeneity of environments, significant temperature gradients arise as a result of the thermal load. In order to establish the temperature regimes for the effective operation of electronic devices, linear and non-linear mathematical models for determining the temperature field have been constructed, which would make it possible to further analyze the temperature regimes in these heat-active environments. The coefficient of thermal conductivity for the above structures was represented as a whole using asymmetric unit functions. As a result, the conditions of ideal thermal contact were automatically fulfilled on the surfaces of the conjugation of the layers. This leads to solving one heat conduction equation with discontinuous and singular coefficients and boundary conditions at the boundary surfaces of the medium. For linearization of nonlinear boundary value problems, linearizing functions were introduced. Analytical solutions to both linear and nonlinear boundary value problems were derived in a closed form. For heat-sensitive environments, as an example, the linear dependence of the coefficient of thermal conductivity of structural materials on temperature, which is often observed when solving many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in these environments were obtained. Based on this, a numerical experiment was performed, and it was geometrically represented depending on the spatial coordinates. This proves that the constructed linear and nonlinear mathematical models testify to their adequacy to the real physical process. They make it possible to analyze heat-active media regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements but also the entire structure. [ABSTRACT FROM AUTHOR]

The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial–boundary value problems of mathematical physics with nonlinearities of unbounded growth. Based on sufficient conditions for the stability of A.A. Samarskii's two- and three-level difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e., when the difference solution and its first time derivative are nonnegative at all nodes of the grid domain. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein–Gordon equations with nonlinear right-hand sides. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a ψ -Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. [ABSTRACT FROM AUTHOR]

The Hadamard fractional derivative and integral are important parts of fractional calculus which have been widely used in engineering, biology, neural networks, control theory, and so on. In addition, the periodic boundary conditions are an important class of symmetric two-point boundary conditions for differential equations and have wide applications. Therefore, this article considers a class of nonlinear Hadamard fractional coupling (p 1 , p 2) -Laplacian systems with periodic boundary value conditions. Based on nonlinear analysis methods and the contraction mapping principle, we obtain some new and easily verifiable sufficient criteria for the existence and uniqueness of solutions to this system. Moreover, we further discuss the generalized Ulam–Hyers (GUH) stability of this problem by using some inequality techniques. Finally, three examples and simulations explain the correctness and availability of our main results. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear fractional-order boundary value problem   H D a , g α 1 , β , μ x (t) + f (t , x (t) , H D a , g α 2 , β , μ x (t)) = 0 , for t ∈ [ a , b ] , α 1 ∈ (1 , 2 ] , α 2 ∈ (0 , 1 ] , β ∈ [ 0 , 1 ] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and   H D a , g α , β , μ represents the tempered-Hilfer fractional derivative of order α > 0 with parameter μ ∈ R + and type β ∈ [ 0 , 1 ] . The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. [ABSTRACT FROM AUTHOR]

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied. [ABSTRACT FROM AUTHOR]

Under water compressibility, resonant triads can occur within the family of acoustic-gravity waves. This work investigates steady-state triad resonance between a surface gravity wave and two hydroacoustic waves. The water-wave equations are solved as a nonlinear boundary value problem using the homotopy analysis method (HAM). Within the HAM framework, a potential singularity resulting from exact triad resonance is avoided by appropriately choosing the auxiliary linear operator. The resonant hydroacoustic wave component, along with the two primary waves (one hydroacoustic wave and one gravity wave), is considered to determine an initial guess for the velocity potential. Additionally, by selecting an optimal "convergence-control parameter," the steady-state resonance between a surface gravity wave and two hydroacoustic waves is successfully obtained. It is found that steady-state resonant acoustic-gravity waves are ubiquitous under certain circumstances. The two primary wave components and the resonant hydroacoustic wave component take up most of the energy in the steady-state resonant acoustic-gravity wave system. The amplitude of the resonant hydroacoustic wave component is mainly determined by the primary hydroacoustic wave component, and the amplitudes of both hydroacoustic waves are approximately equal in all cases considered. In addition, when the overall amplitude of the wave system increases, both dimensionless angular frequencies decrease, indicating that the nonlinearity of the entire wave system becomes stronger with an increase in the wave system amplitude. The amplitude of the primary hydroacoustic wave has a relatively large effect on the system's nonlinearity. This work will enrich and deepen our understanding of acoustic-gravity waves. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of transonic shocks in a cylindrical nozzle. More precisely, given the compatible outer normal momentum at the exit of a cylindrical nozzle, there exists a transonic shock solution such that the incoming supersonic flow is compressed to some subsonic flow. The shock position is uniquely determined by the exit normal momentum when we fix one point of the shock. This problem can be reduced to solve a nonlinear free boundary value problem for a hyperbolic–elliptic coupled system and the key ingredient is to solve a second‐order quasilinear elliptic equation with a nonlocal term by the method of separation of variables. [ABSTRACT FROM AUTHOR]

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ (t , x) and φ (t , x) . Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f 1 , f 2 , w 1 , w 2 , α , ξ , θ 0 , α 0 , φ 0 , and ξ 0 , we prove the well-posedness of a solution θ , α , φ , ξ , which is θ (t , x) , α (t , x) ∈ W p 1 , 2 (Q) × W p 1 , 2 (Σ) , φ (t , x) , ξ (t , x) ∈ W ν 1 , 2 (Q) × W p 1 , 2 (Σ) , ν = min { q , μ } . According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future. [ABSTRACT FROM AUTHOR]

In this paper, we investigate stability and optimization of an axially loaded nonlocal beam that is simply supported at one end and elastically restrained against rotation on the other. Nonlocal continuum mechanics is a theoretical framework that extends classical continuum mechanics to account for the influence of small length scales, especially in nanostructures. We analyze elastically buckling nanobeam based on Eringen's nonlocal elasticity theory. The Euler method of adjacent equilibrium configuration is used to derive the nonlinear governing equations. The critical axial force and postbuckling shape are obtained for the beam with the unit cross-sectional area. The shape of the nonlocal beam stable against buckling that has minimal volume is determined by using Pontryagin's maximum principle. The optimality conditions are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. The first integral (Hamiltonian) is used to monitor the accuracy of the integration. The numerical analysis includes the influence of the characteristic parameter of the small length scale on the critical load, the postbuckling shape and the optimal shape of the analyzed beams. It is shown that there are the saving materials for optimally designed nanobeam in all numerical examples. [ABSTRACT FROM AUTHOR]