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Abstract The paper drying process has the highest level of energy consumption in the pulp and paper production process. Analysis and optimization of the energy system during the paper drying process is critical for improving the energy efficiency of the entire paper mill. In the existing model for the paper drying process, the solution requires accurate boundary conditions such as the air temperature and humidity of the pocket area and the cylinder surface temperature, which are very difficult to obtain in the papermaking process. This can result in significant deviations between the model solution and the actual production process. This paper focuses on the single-tier dryer cylinder-based paper drying process that has been widely used with high-speed papermaking machines in recent years. A mathematical model is proposed based on real-time data. The verification via industrial production demonstrates that the proposed model is reliable for the paper drying process. Based on the simulation results, two optimization operations have been proposed. The energy consumption decreases from 1.51 t steam/t paper to 1.44 t steam/t paper, 4.6% of the steam and 1.26 × 106 RMB can be saved for a medium-scale paper mill with the annual production capacity of 105 t paper. Highlights • Mathematical model for the single-tier cylinder based paper drying process is proposed. • Both the steam-condensate system and the paper drying process are considered. • The simulation results are verified by the industrial data from a papermaking enterprise. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

This paper introduces a topology optimization approach based on the virtual element method (VEM), incorporating uncertainties. The objective of this optimization process is to design an optimal material layout for problems governed by linear elasticity equations to minimize the volume while satisfying probabilistic compliance constraints. The VEM is used to solve the boundary value problem in reliability-based topology optimization (RBTO). In the comparison between the VEM and the standard finite element method (FEM), a key difference emerges in the absence of explicitly defined shape functions tied to discrete degrees of freedom in VEM. Unlike FEM, VEM directly constructs the discrete bilinear form and load linear form without the need for computing shape function derivatives within the elements. This flexibility accommodates meshes with intricate geometries and arbitrarily shaped elements. The paper discusses the computational efficiency of VEM RBTO and explores the geometric impact of tessellations on converged topologies, demonstrating reduced susceptibility to checkerboard patterns compared to conventional quadrilateral elements. Additionally, the single-loop approach is examined, showcasing comparable accuracy to the first-order/second-order reliability methods (FORM/SORM) of RBTO using VEM. Numerical results for several problems that demonstrate the feasibility of the proposed method are presented. [ABSTRACT FROM AUTHOR]

The bioconvective flows are truly connected to real-life and engineering development. So, the models of biomicrosystems and biocells are considered for the technical analysis in this paper. Our intention for the current analysis was to theoretically examine electrical conduction flow into mass and heat transfer by an extensive gyrotactic microorganism into an inclined magnetic field toward a vertical stretching sheet with nonlinear solar radiation with different solar thermal appliances. The effect on velocity slip and Joule heating was again studied in detail. This classical problem on Navier Stokes equations to the current imitation was decreased to ordinary differential equations by applying the comparison method. The numerical solutions were changed by boundary value problem (BVP) to clarify the subject into finite difference numerical scheme by applying MATLAB. The important results show that the density of motile microorganisms reduces to the bioconvection Lewis number and Peclet number, although reverse performance was noticed for the bioconvection Rayleigh number. Further, solar radiation fosters heat transport. In this paper, complete analysis is provided for potential functions of solar thermoelectric cells, solar ponds, solar thermal power fabrication, etc. [ABSTRACT FROM AUTHOR]

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]

In this paper, we have studied the behaviour of two symmetric mode-III collinear cracks in a functionally graded material (FGM). The fundamental goal of this paper is to provide insight on the interaction of two cracks in FGMs with the strain gradient effect. To assess the influence of gradient elasticity, we have considered two key parameters ℓ and ℓ ′ , which describe the size scale effect caused by the underlying microstructure and are related to volumetric and surface strain energy, respectively. The crack boundary value problem have been solved by the approach involving Fourier transforms and the innovative hyper-singular integro-differential equation method, where the integral equation contains the two terms in integrals for the both cracks. A system of equations has been constructed by employing the Chebyshev polynomial expansion and then by choosing the suitable collocation points the system of equation have been solved. Our investigation involves the determination of stress intensity factors at both crack tips. These factors are vital for understanding the material's fracture behavior and structural integrity. Furthermore, we explore the variations in the displacement profile when the distance between the cracks is reduced to close proximity. This particular scenario is of significant interest as it provides insights into how the interaction between the cracks impacts the overall structural response. [ABSTRACT FROM AUTHOR]

Ordinary and partial differential equations with operators containing the p -Laplace operator, the so called ϕ -Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered. [ABSTRACT FROM AUTHOR]

The study of nonlinear problems related to heat transfer in a substance is of great practical important. Earlier, this paper's authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on experimental observations of the dynamics of the temperature field in the object. In this paper, the problem of simultaneous identification of temperature-dependent volumetric heat capacity and thermal conductivity of the substance under study from the heat flux at the boundary of the domain is investigated. The consideration is based on the first (Dirichlet) boundary value problem for a one-dimensional unsteady heat equation. The coefficient inverse problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of fast automatic differentiation. The uniqueness of the solution of the inverse problem is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

Asymmetry plays a significant role in the transmission dynamics in novel discrete fractional calculus. Few studies have mathematically modeled such asymmetry properties, and none have developed discrete models that incorporate different symmetry developmental stages. This paper introduces a Taylor monomial falling function and presents some properties of this function in a delta fractional model with Green's function kernel. In the deterministic case, Green's function will be non-negative, and this shows that the function has an upper bound for its maximum point. More precisely, in this paper, based on the properties of the Taylor monomial falling function, we investigate Lyapunov-type inequalities for a delta fractional boundary value problem of Riemann–Liouville type. [ABSTRACT FROM AUTHOR]

We study global attractors A = A f of semiflows generated by semilinear partial parabolic differential equations of the form u t = u xx + f (x , u , u x) , 0 < x < 1 , satisfying Neumann boundary conditions. The equilibria v ∈ E ⊂ A of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of A into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors A as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation σ = σ f derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries x = 0 , 1 , respectively, completely determines the Sturm global attractor A . Equivalently, we use a planar curve, the meander M = M f , associated to the the ODE boundary value problem by shooting. In the previous paper (Fiedler and Rocha in J Dyn Differ Equ, 2020. https://doi.org/10.1007/s10884-020-09836-5), we set up to determine the boundary neighbors of any specific unstable equilibrium O , based exclusively on the information on the corresponding signed hemisphere complex. In addition, a certain minimax property of the boundary neighbors was established. In the signed hemisphere decomposition of the cell boundary of O , this property identifies the equilibria which are closest to, or most distant from, O at the boundaries x = 0 , 1 , in each hemisphere. The main objective of the present paper is to derive this minimax property directly from the Sturm permutation σ , or equivalently from the Sturm meander M , based on the Sturm nodal properties of the solutions of the ODE boundary value problem. This minimax result simplifies the task of identifying the equilibria on the cell boundary of each unstable equilibrium, directly from the Sturm meander M . We emphasize the local aspect of this result by an example for which the identification of the equilibria is obtained from the knowledge of only a segment of the Sturm meander M . [ABSTRACT FROM AUTHOR]

• A Hp-Adaptive Pseudospectral Hybrid Optimal Algorithm is proposed. • The optimal solution of the OSIP and OPEG problem can be robustly solved by this work. • Cases with different thrust configurations and initial conditions are reported. In this paper, a Cartesian model of the orbit pursuit-evasion game under J2 perturbations is constructed, and a step-by-step solution method applicable to this problem, namely the Hp-Adaptive Pseudospectral Hybrid Optimal Algorithm, is proposed by analyzing the principle of the pseudospectral method and the costate mapping principle. We first use the hp-adaptive Radau pseudospectral method to solve the one-sided interception problem to obtain a tentative guess of the costate and then get the saddle-point solution of the game problem by solving the two-point boundary value problem through the Newton method. The numerical results show that the hybrid method proposed in this work can efficiently find the optimal trajectory of interception and the optimal control laws of the game problem and have high accuracy. In addition, the effects of J2 perturbations and the initial orbital altitude of the evader on the game results and the algorithm itself are discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and  B.  The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required. [ABSTRACT FROM AUTHOR]

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H - (ϑ , φ) -contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems. [ABSTRACT FROM AUTHOR]

This paper describes a teaching experiment in a Numerical Methods course for Master of Science students. The experiment uses scientific papers to develop modelling studies in the context of wine fermentation microbial interactions. The course involves theoretical and laboratory classes that focus on implementing numerical methods using Matlab for Initial Value Problems and Boundary Value Problems. The students are asked to formalise the mathematical model and build their own experiments using the information provided in the papers. Additionally, a parameter estimation experiment is organised, which involves generating synthetic data and computing noisy data to estimate the natural death rate of sensitive yeast. The results show that data noise significantly affects the parameter estimate and that scaling the data can help reduce the impact of measurement errors. The presented results can be used to investigate other possible assignments, such as how the evaluation of the Jacobian affects the estimation performance and compare different optimisation algorithms. [ABSTRACT FROM AUTHOR]

In this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2 - sequential and 4 - sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2 - sequential and 4 - sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2 - sequential and 4 - sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2 - sequential fractional differential equations. At the end of the paper we show that 6 - sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems. [ABSTRACT FROM AUTHOR]

The overall reliability of a power transformer depends to a great extent on the sound operation of the bushings thereof. Oil impregnated paper (OIP) insulated bushings have been in use for a long time now. In many situations, it becomes necessary to avoid OIP insulation in bushings. In the recent past, a new technological breakthrough has been achieved whereby the OIP is replaced by epoxy resin impregnated crepe paper (RIP) insulation. This new system has several advantages over OIP and has now become the insulation of choice. However, its long time thermal and electrical performance need to be carefully assessed. This paper reports the results of a study of temperature distribution in the body of insulation, based on the ac conductivity of RIP insulation. A method of computing the maximum thermal voltage of this system is also given. [ABSTRACT FROM AUTHOR]

Abstract: The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) with multi-reflection solid boundary conditions is used to study anisotropic permeabilities of a carbon paper gas diffusion layer (GDL) in a fuel cell. The carbon paper is reconstructed using the stochastic method, in which various porosities and microstructures are achieved to simulate different samples. The simulated permeability and tortuosity show anisotropic characteristics of the reconstructed carbon papers with in-plane permeability higher than through-plane, and in-plane tortuosity lower than through-plane. The calculated permeabilities are in good agreement with existing measurements. The relationship between the permeability and the porosity is fitted with empirical relations and some fitting constants are determined. Furthermore, the obtained relationship of tortuosity and porosity is used in a fractal model for permeabilities. The results indicate that the fractal model and the Kozeny–Carman equation provide similar predictions on the through-plane permeability of the carbon paper GDL. [Copyright &y& Elsevier]

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained. [ABSTRACT FROM AUTHOR]

A parabolic–elliptic Keller–Segel model for multipopulations in a two‐dimensional unit ball will be considered in this paper. For the initial boundary value problem with mutually attractive populations, the global existence of bounded solutions has been proved through the logarithmic Hardy–Littlewood–Sobolev inequality for system when the initial masses satisfy the subcritical condition. Moreover, based on the moments technique, there exists a blow‐up solution when the initial masses satisfy the supercritical case. In a word, a critical mass phenomenon has been established in this multispecies chemotaxis system. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

The extended multiscale finite element method (EMsFEM) shows great efficiency and accuracy for analyzing the mechanical behavior of heterogeneous materials, especially for non-periodic multiscale materials. The conventional EMsFEM requires solving boundary value problems repeatedly on each coarse-scale element to construct the numerical base functions related to the material parameters of fine-scale element, which constitutes the main part of computational resources. This paper presents a physics-informed neural network (PINN) enhanced EMsFEM to further improve the efficiency of multiscale mechanical analysis. Since the boundary value problems are based on the same solution domain and boundary conditions, a PINN is elaborately designed to solve them described by mechanical equations. The input parameters of PINN contain the material parameters of the fine-scale elements inside the coarse-scale element; therefore, the PINN can quickly map the heterogeneous material properties to the displacements inside the coarse-scale element and greatly improve the construction efficiency of the numerical base functions. To enhance the computational accuracy, the domain decomposition technique is applied to characterize the heterogeneity of the elements, and an unbiased construction method is developed to obtain the numerical base functions that simultaneously ensure the computational consistency and normalization condition. In addition, to further improve the computational efficiency, the construction process of numerical base functions is simplified according to the approximately ergodic property of the network for randomly physical fields. Several representative numerical examples are presented to demonstrate the high efficiency and accuracy of the proposed PINN-enhanced EMsFEM. The method is of high universality since the PINN does not need to be retrained as the geometry of the entire domain and loading of the problem change, the network structure is only related to the length ratio of the coarse- and fine-scale elements. [ABSTRACT FROM AUTHOR]

The paper is concerned with the study of a chemical reaction between diffusing and solid constituents. The reaction is localized at a propagating reaction front and is accompanied by a transformation strain, which generates stresses affecting the reaction rate. The effect of mechanical stresses on the velocity of the reaction front is taken into account basing on the concept of a chemical affinity tensor. The initial boundary value problem of the propagation of the chemical reaction front is discussed by the example of an axisymmetric problem for a reaction in a cylinder in the case of linear elastic solid constituents. The stage of initial accumulation of the diffusing constituent and the stage of the propagation of the reaction front after its separation from the outer boundary of a body are distinguished. The time preceding the start of the reaction is determined. Conditions at the propagating chemical reaction front for the diffusion problem are discussed. A comparative analysis of the solutions obtained in the quasi-stationary and nonstationary formulations of the diffusion problem is carried out, taking into account and without taking into account the influence of the initially accumulated diffusing constituent on the reaction front velocity, as well as for the case of the absence of diffusion in the untransformed material. [ABSTRACT FROM AUTHOR]

In this paper, the coupled linear theory of thermoviscoelasticity for porous materials is considered in which the coupled phenomenon of the following four mechanical principles is proposed: the deformation of the skeleton of a porous solid, the volume fraction concept of the pore network, Darcy's law for the flow of a fluid through a porous medium, and Fourier's law of thermal conduction. The governing systems of equations of motion and steady vibrations are proposed. The fundamental solution of the system of steady vibration equations is presented explicitly by means of elementary functions, and its basic properties are established. By virtue of Green's identity the uniqueness theorems for the classical solutions of the basic internal and external boundary value problems (BVPs) of steady vibrations are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the above-mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we develop a nonlinear mixed finite element method for flexoelectric semiconductors and analyze the mechanically tuned redistributions of free carriers and electric currents through flexoelectric polarization in typical structures. We first present a macroscopic theory for flexoelectric semiconductors by combining flexoelectricity and nonlinear drift-diffusion theory. To use C0 continuous elements, we derive an incremental constrained weak form by introducing Langrage multipliers, in which the kinematic constraints between the displacement and its gradient are guaranteed. Based on the weak form, we established a mixed C0 continuous nine-node quadrilateral finite element as well as an iterative process for solving nonlinear boundary-value problems. The accuracy and convergence of the proposed element are validated by comparing linear finite element method results against analytical solutions for the bending of a beam. Finally, the nonlinear element method is applied to more complex problems, such as a circular ring, a plate with a hole, and an isosceles trapezoid. Results indicate that mechanical loads and doping levels have distinct influences on electric properties. [ABSTRACT FROM AUTHOR]

In this paper we consider second order iterative boundary value problem with Sturm Liouville type boundary conditions and establish the existence of countably many positive symmetric solutions by using Krasnoselskii's fixed point theorem for operator on a cone. [ABSTRACT FROM AUTHOR]

Recently, the physics-informed neural network (PINN) has attracted much attention in solving partial differential equations (PDEs). The success is due to the strong generalization ability of the neural network (NN), which is supported by the universal approximation theorem, and its mesh-free implementation. In this paper, we propose a multi-head NN enriched PINN (MultiPINN) for solving differential equations. The trial function is built based on the radial basis function (RBF)-interpolation, which makes NN training parameters partially interpre. The loss function is constructed by embedding the physics information of differential equations and boundary conditions. Then the parameters in MultiPINN are trained using the ADAM optimizer. A significant feature of MultiPINN is that it combines the traditional RBF interpolation method with machine learning (ML) techniques. The ML technique is employed to learn the basis feature enrichment that provides global information. The multi-head mechanism is used so that each node has multiple bases, which can improve the accuracy of the MultiPINN solution. Two ordinary differential equations and three partial differential equations, i.e. the convection equation, the Burgers equation, and the Poisson equation, are used in the numerical experiments. The experimental outcomes demonstrate that MultiPINN produces solutions consistent with both analytical solutions and solutions obtained through traditional numerical methods. Additionally, MultiPINN shows robustness and adaptability over the other NN-based methods in the implementations. [ABSTRACT FROM AUTHOR]

In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decomposition formula for the fluid-solid interaction scattering problem. By the decomposition of the displacement, the corresponding problem is transformed into a coupled boundary value problem. For the two dimensional case, the coupled boundary value problem is about two scalar Helmholtz equations, and the other one is a scalar Helmholtz equation and a vector Helmholtz equation for the three dimensional. Then the coupled boundary value problem is solved by the method of fundamental solution. At last we give some examples to show the effectiveness of the present method in two and three dimensional case. A benchmark example is available for checking the feasibility and accuracy of the presented method. In the numerical examples, a part of exact solution for the boundary value problem is including for checking the accuracy. [ABSTRACT FROM AUTHOR]

This paper presents the following AB-Caputo fractional boundary value problem 0ABCDαu(ς) = 픖(ς,u(ς)),ς ∈ [0, 1] with integral-type boundary conditions u(0) = 0 = u″(0),γu(1) = λ∫01g 1(ϰ)u(ϰ)dϰ, of order 2 < α ≤ 3. Schauder and Krasnoselskii’s fixed point theorems are used to find existence results. Uniqueness is obtained via the Banach contraction principle. To investigate the stability of a given problem, Hyers–Ulam stability is discussed. An example is provided to validate our results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [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]

In this paper, we obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for an elliptic system. More precisely, we combine the potential theory with the sub-super solution method and use the properties of the well-known Kato class to justify our existence results. The novelty of our study is that we consider either the bounded or the exterior domain; Also, the nonlinearities may be singular near the boundary. Some examples are presented to validate our main results. [ABSTRACT FROM AUTHOR]

In this paper, the time-independent Klein-Gordon equation in R 3 is treated with a decomposition of the operator Δ − γ 2 I by the Clifford algebra C l (V 3 , 3) . Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated. [ABSTRACT FROM AUTHOR]

Let |$({\mathcal{M}},d,\mu)$| be the metric measure space with a Dirichlet form |$\mathscr{E}$|⁠. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on |${\mathcal{M}}$| to its Poisson integral P t u on |${\mathcal{M}}\times\mathbb{R}_+$|⁠. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math. 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971]. [ABSTRACT FROM AUTHOR]

The stability of a current-carrying beam in an external magnetic field is studied in this paper. The beam rests on periodic supports and is modeled as an Euler-Bernoulli elastic beam. Based on the linearized stability equation and propagator transfer matrix approach for a multi-span finite-length beam, the stability equation and solutions of the several boundary value problems are obtained. While the Floquet-Bloch theory is widely used in the dynamic problem of phononic and photonic structure conditions, its application to the solution of infinite length beam static stability problem is novel. The stability values of the periodic infinite beam model are in very good agreement with one of the finite-length multi-span beams. [ABSTRACT FROM AUTHOR]

This paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature. [ABSTRACT FROM AUTHOR]

This paper deals with the following chemotaxis system: u t = ∇ · (γ (v) ∇ u - u ξ (v) ∇ v) + μ u (1 - u) , x ∈ Ω , t > 0 , v t = Δ v - u v , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary. Here, the positive function γ ∈ C 2 ([ 0 , + ∞)) satisfies γ ′ (s) < 0 and γ ′ ′ (s) ≥ 0 for all s ≥ 0 , also ξ (s) = - (1 - α) γ ′ (s) with α ∈ (0 , 1) . For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly in time bounded. This result is obtained for small initial data without any restriction on μ. The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that (γ ′ (s)) 2 γ ′ ′ (s) ≤ n 2 (n + 1) 3 and some conditions on initial data and μ. We should mention that in the special cases γ (s) = (1 + s) - k (k > 0) and γ (s) = e - χ s (χ > 0) , the result in Li and Lu (2023) is obtained under conditions on k and χ. But, our result is without any restriction on k and χ. [ABSTRACT FROM AUTHOR]

With the dramatic increase in the number of resident space objects, the risk of collision between space objects and debris is growing. To protect spacecraft from such collisions, it is crucial to determine the generation patterns of space debris and analyze the evolution of debris clouds. In this paper, a continuous distribution formula is derived for each parameter of the breakup debris based on the China Aerodynamics Research and Development Center spacecraft breakup model (CARDC-SBM). Using CARDC-SBM, we theoretically establish a normal distribution for the separation velocity of the breakup debris in relation to the breakup mass center. Based on this foundation, the mean and standard deviation of the debris separation velocity distribution are empirically modeled using polynomial and exponential functions. Consequently, an empirical formula with a broad range of applicability is derived, which significantly reduces the computation time of debris cloud simulations. The initial velocity distribution given by this empirical formula is employed as the initial condition for the evolution of debris clouds from the perspective of a boundary value problem. Simulations using CARDC-SBM and subsequent analysis of the debris cloud evolution resulting from a breakup event reveal that the distribution of each parameter is determined solely by the relative collision velocity and the spacecraft's nominal density in the impact breakup scenario. Our results show that the empirical formula for the debris cloud distribution provides good agreement with Monte Carlo simulations. • A continuous distribution formula is derived for each parameter of the impact breakup debris based on CARDC-SBM. • We point out and validate that normal distribution can characterize the velocity of debris produced by CARDC-SBM. • Empirical formulas are developed to compute the debris separation velocity distribution with low computational burden. • The proposed method can provide initial condition for space debris clouds evolution. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator. [ABSTRACT FROM AUTHOR]