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Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model. [ABSTRACT FROM AUTHOR]

The committor constitutes the primary quantity of interest within chemical kinetics as it is understood to encode the ideal reaction coordinate for a rare reactive event. We show the generative utility of the committor in that it can be used explicitly to produce a reactive trajectory ensemble that exhibits numerically exact statistics as that of the original transition path ensemble. This is done by relating a time-dependent analog of the committor that solves a generalized bridge problem to the splitting probability that solves a boundary value problem under a bistable assumption. By invoking stochastic optimal control and spectral theory, we derive a general form for the optimal controller of a bridge process that connects two metastable states expressed in terms of the splitting probability. This formalism offers an alternative perspective into the role of the committor and its gradients in that they encode force fields that guarantee reactivity, generating trajectories that are statistically identical to the way that a system would react autonomously. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the method of solving partial differential equations (PDEs) using artificial neural network (ANN) structures, which have been actively applied in artificial intelligence models. The ANN model for solving PDEs offers the advantage of providing explicit and continuous solutions. However, the ANN model for solving PDEs cannot construct a conventionally solvable linear system with known matrix solvers; thus, computational speed could be a significant concern. We study the implementation of the multigrid method, developing a general concept for a coarse-grid correction method to be integrated into the ANN-PDE architecture, with the goal of enhancing computational efficiency. By developing a reduced form of the multigrid method for ANN, we demonstrate that it can be interpreted as an equivalent representation of the classic multigrid expansion. We validated the applicability of the proposed method through rigorous experiments, which included analyzing loss decay and the number of iterations along with improvements in terms of accuracy, speed, and complexity. We accomplished this by employing the gradient descent method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method to update the gradients while solving the given ANN systems of PDEs. [ABSTRACT FROM AUTHOR]

The transport phenomena of Casson nanofluid flow between two parallel disks subject to convective boundary conditions are analyzed in this paper. The mathematical model incorporates the impact of thermophoresis and Brownian motion since the Buongiorno's nanoliquid model is adopted to characterize the nanoliquid's transport features. The appropriate similarity transformations are applied to obtain the resulting nondimensional ordinary differential equations from the basic governing equations. The resulting ordinary differential equations and the associated boundary conditions are solved analytically by adopting the homotopy perturbation technique. Further, a statistical experiment is conducted to identify notable flow parameters which cause significant impact on the heat transfer rate. The characteristics of critical pertinent parameters on the flow field are graphically manifested. It is worth noting that the Casson nanofluid velocity escalates by augmenting the magnetic field parameter in the case of injection near the disks. Nanoparticle concentration is considerably diminished with an increment in thermophoresis parameter. In the cases of equal and unequal Biot numbers, the heat transfer rate is promoted with higher values of the Brownian motion parameter. Among the Casson fluid parameter, squeezing parameter and magnetic field parameter, the heat transfer rate discloses the highest positive sensitivity with the lowest value of the Casson fluid parameter. [ABSTRACT FROM AUTHOR]

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation. [ABSTRACT FROM AUTHOR]

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in Rn,n≥2$$ {\mathbb{R}}^n,n\ge 2 $$. The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in Rn,n=2,3$$ {\mathbb{R}}^n,n=2,3 $$. Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem. [ABSTRACT FROM AUTHOR]

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω−⊂ℝ3$$ {\Omega}^{-}\subset {\mathrm{\mathbb{R}}}^3 $$, has a compact boundary surface S=∂Ω−$$ S=\partial {\Omega}^{-} $$, which is divided into two disjoint parts, the Dirichlet part SD$$ {S}_D $$ and the Neumann part SN$$ {S}_N $$, where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L2$$ {L}_2 $$‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space [W2,loc1(Ω−)]3$$ {\left[{W}_{2, loc}^1\left({\Omega}^{-}\right)\right]}^3 $$ and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the Lp$$ {L}_p $$‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses Cα$$ {C}^{\alpha } $$‐Hölder continuity property in the closed domain Ω−‾$$ \overline{\Omega^{-}} $$ with α=12−ε$$ \alpha =\frac{1}{2}-\varepsilon $$, where ε>0$$ \varepsilon >0 $$ is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate an initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation. Under the different initial energy, we study the existence, uniqueness and asymptotic behavior of the global solution and the blow up phenomenon of the weak solution. Moreover, we obtain lower and upper bound estimates for the blow-up time and rate in different initial energy cases. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new numerical method to simulate acoustic scattering problems in two‐dimensional periodic structures with non‐periodic incident fields. Applying the Floquet‐Bloch transform to the scattering problem yields a family of quasi‐periodic boundary value problems dependent on the Floquet‐Bloch parameter. Consequently, the solution of the original scattering problem is written as the inverse Floquet‐Bloch transform of the solutions to these boundary value problems. The key step in our method is the numerical approximation of this integral transform by a quadrature rule with a nonuniform choice of quadrature points adapted to the regularity of the family of quasi‐periodic solutions. This achieved by a graded subdivision of the full interval for the Floquet‐Bloch parameter and applying a Gauss‐Legrendre quadrature rule on each subinterval. We prove that the numerical method converges exponentially with respect to both the number of subintervals and the number of Gaussian quadrature points. Some numerical experiments are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff‐type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first‐order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self‐adjoint boundary value problem on the tree for some second‐order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree. [ABSTRACT FROM AUTHOR]

We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary partial differential equation problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic energy to mechanical energy. [ABSTRACT FROM AUTHOR]

We study a superlinear elliptic boundary value problem involving the p$p$‐Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem. Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known. [ABSTRACT FROM AUTHOR]

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix. [ABSTRACT FROM AUTHOR]

Analytical solutions of two-dimensional statics problems of elasticity are presented in bipolar coordinates for homogeneous isotropic bodies bounded by the coordinate lines of a bipolar coordinate system. In particular, various boundary problems for an eccentric circular ring, a half-plane with circular holes, and others are considered. The equilibrium equations and Hooke's law are expressed using bipolar coordinates. This paper does not address the static equilibrium requirement of the external load at each circular boundary of the study area. This requirement, which significantly limits the range of problems that can be solved, typically appears in papers dealing with the aforementioned problems. In addition, the proposed method for obtaining an exact (analytical) solution is much simpler compared to the traditional approach. The exact solutions are derived using the method of separation of variables. By utilizing MATLAB software, the numerical results of some boundary value problems for an eccentric semi-ring are obtained, and the corresponding diagrams are presented. [ABSTRACT FROM AUTHOR]

We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other. [ABSTRACT FROM AUTHOR]

The global dynamics of a nonlocal predator–prey model with mutation are investigated in this paper. First, by building a new comparison principle and constructing monotone iterative sequences, we give the existence of the solution for the corresponding initial boundary value problem. Then the uniqueness and uniformly boundedness of the solution are established by using the fundamental solution and quasi-fundamental solution of the heat equation, Grönwall's inequality and auxiliary functions. Finally, under the truncation function method, we discuss the asymptotic behavior of the solution. It is worth mentioning that the asymptotic behavior here is different from the conclusion of the classical model and we use more refined analysis and estimation in the research process. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method for solving the first-passage time probability problem of nonlinear stochastic dynamic systems. The safe domain boundary is exactly imposed into the radial basis function neural network (RBF-NN) architecture such that the solution is an admissible function of the boundary-value problem. In this way, the neural network solution can automatically satisfy the safe domain boundaries and no longer requires adding the corresponding loss terms, thus efficiently handling structure failure problems defined by various safe domain boundaries. The effectiveness of the proposed method is demonstrated through three nonlinear stochastic examples defined by different safe domains, and the results are validated against the extensive Monte Carlo simulations (MCSs). [ABSTRACT FROM AUTHOR]

In this study it is approached a linear model for the mixture of two Cosserat bodies having pores. It is formulated the mixed problem with initial and boundary data in this context. The main goal is to show that the coefficients that realize the coupling of the elastic effect with the one due to voids can vary, without the mixture being essentially affected. In a more precise formulation, this means that a small variation of the coefficients in the constitutive equations of the two continua causes only a small variation of the solutions of the corresponding mixed problems, that is, the continuous dependence of the solutions in relation to these coefficients is ensured. The considered mixture model is consistent because all estimates, specific to continuous dependence, are made based on rigorous mathematical relationships. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model, which has a unique mechanism of heat transfer in the stretching/shrinking straight fin with an exponential profile. The thermal conductivity, internal heat generation, and heat transfer coefficient are considered temperature‐dependent. Heat is exposed to the surroundings by convection and radiation. The governing differential equation and boundary conditions are presented in a dimensionless form. In our study, we considered variable surface emissivity, that is, a constant, and the linear function of a temperature. The convective heat transfer parameter is considered a power‐low type. The novelty of this work is the application of temperature‐dependent surface emissivity, and the problem is solved by the Legendre wavelet collocation method. A comparative analysis of the present results in the context of previous findings is presented in the form of a table for validation and found exactly the same. The impacts of distinct variables are presented in the form of figures and discussed in detail. The present analysis is focused on real‐world applications and offers valuable insights for improving the design of fins. [ABSTRACT FROM AUTHOR]

The paper aims to study the numerical solution of the Zakharov-Rubenchik equations on unbounded domains, which model the propagation of two kinds of waves in plasma physics, by applying the artificial boundary method. Based on ideas of the operator splitting method and artificial boundary method of hyperbolic system and nonlinear Schrödinger equation, the artificial boundary conditions of the Zakharov-Rubenchik equations are designed on the introduced artificial boundaries to reduce the original problem defined on an unbounded domain into an initial boundary value problem on the bounded computational domain, which can be efficiently solved by finite difference method. A series of auxiliary variables is introduced to rigorously analyze the stability of the reduced initial boundary value problem through energy estimation. Numerical examples are presented to demonstrate the effectiveness of the proposed artificial boundary conditions, validate the theoretical analysis, and simulate the physical phenomena. [ABSTRACT FROM AUTHOR]

The elastic–plastic infinitesimal deformation of a solid is considered within the framework of the incremental flow theory using the constitutive relation in the general rate form. The appropriate initial boundary value problem is formulated for the displacement in the form of the evolutionary-variational problem (EVP), i.e., as the abstract Cauchy problem in the Hilbert space which coincides with a weak form of the equilibrium equation, known as the principle of possible displacements in mechanics. The general existence and uniqueness theorem for the EVP is discussed. The main sufficient condition has a simple algebraic form and does not coincide with the classical Drucker and similar thermodynamical postulates; therefore, it must be independently verified. Its independence is illustrated for the non-associated plastic model of linear isotropic-kinematic hardening with dilatation and internal friction. The classical and endochronic models are analyzed too. The initial EVP is reduced by a spatial finite element approximation to the Cauchy problem for an implicit system of essentially nonlinear ordinary differential equations which can be stiff. Therefore, for the numerical solution the implicit Euler scheme is proposed. All theoretical results are illustrated by means of original numerical experiments. [ABSTRACT FROM AUTHOR]

This article is to develop a method to solve the boundary value problems of the first-order measure differential equations in the space of bound-ed variation functions. Firstly, we obtain the solution and Green's function by applying the integration by parts. Secondly, the criterion for the existence of solution is given by using fixed point theorem and regularization theory. Finally, an example is provided to validate these conclusions. [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]

For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes. [ABSTRACT FROM AUTHOR]

This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness. [ABSTRACT FROM AUTHOR]

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms. [ABSTRACT FROM AUTHOR]

The numerical simulation of boundary value problems involving a high number of pile-type structures, such as pile foundation systems of high-rise buildings, represents a standard task in computational geotechnics. Due to their ability to simplify the underlying pile modeling process and decrease the simulation runtime embedded beam formulations (EBFs) have attracted widespread interest from the geotechnical community. State-of-the art EBFs are typically equipped with nonlinear interface constitutive models to capture the soil–structure interaction behavior with reasonable accuracy. However, reliable information concerning the calibration of related parameters is limited, which decreases the confidence in the results obtained with EBFs. The present work addresses this research aspect for the first time, along with a theoretical discussion of inherent simplifications in the numerical description of the soil–structure contact associated with EBFs. The significance of selecting adequate embedded interface constitutive model parameters to ensure EBF predictions with high credibility is demonstrated by means of parametric studies. This has motivated the development of an automatic calibration software, leveraged by particle swarm optimization, that provides guidance in selecting suitable values. The effectiveness of the calibration software is confirmed by back-calculation of different pile problems with default and calibrated parameter sets. Likewise, the results provide insight into crucial factors driving the calibration framework, with a view to increasing its potential for take-up in engineering practice. Potential lines of research in the context of the automatic calibration software are explored throughout this work and may serve as valuable reference in future research. Practical Applications: The design of geotechnical problems involving a high number of pile-type structures, such as pile foundation systems of high-rise buildings or wind turbines, is routinely assisted by finite-element analyses. The computational expense of these simulations is significantly influenced by the pile modeling technique. Due to their exceptional ability to simplify the underlying modeling process and reduce the runtime to an acceptable limit, embedded beam formulations have therefore attracted widespread interest for geotechnical design tasks. As with all soil–structure interaction problems, it is essential in embedded beam formulations to accurately describe the interface behavior between the soil and the structure. This work presents the first attempt to highlight the relevance of this concern and visually describes relevant limitations resulting from an inadequate selection of interface constitutive model parameters. This observation has motivated the development of an automatic calibration software to increase the confidence in the parameter selection. The high potential of this software to increase the credibility of numerical predictions obtained with embedded beam formulations is demonstrated based on pile problems with different soil stratification layers, pile geometries, and pile arrangements. In all cases considered, the results confirm that the numerical fidelity could be significantly increased by employing calibrated parameter sets. [ABSTRACT FROM AUTHOR]

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method. • A new Mainardi smoothing homotopy method is proposed. • Explicit expressions for the Mainardi kernel at specific derivative orders are derived. • An efficient approach for determining the smoothing parameter is provided. • Effectiveness of the proposed method is demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the multinode Shepard method is adopted for the first time to numerically solve a differential problem with a discontinuity in the boundary. Starting from previous studies on elliptic boundary value problems, here the Shepard method is employed to catch the singularity on the boundary. Enrichments of the functional space spanned by the multinode cardinal Shepard basis functions are proposed to overcome the difficulties encountered. The Motz's problem is considered as numerical benchmark to assess the method. Numerical results are presented to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Boundary value problems (BVPs) in ordinary differential equations (ODEs) with singularities arise in numerous mathematical models describing real-life phenomena in natural sciences and engineering. This motivates vivid research activities aiming to characterize the analytical properties of singular problems, to investigate convergence of the standard numerical methods when they are applied to simulate differential equation with singularities, and to provide software for their efficient numerical treatment. There are two well-known, high order numerical methods which we focus on in this paper, the finite difference schemes and the collocation methods. Those methods proved to be dependable and highly accurate in the context of regular differential equations, so the question arises how do they preform for singular problems. While, there is a strong evidence for the collocation schemes to be a robust method to solve singular systems in a stable and efficient way, finite difference schemes are still considered less suitable for this problem class. In this paper, we shall compare the performance of the code HOFiD_bvp based on the high order finite difference schemes and bvpsuite2.0 based on polynomial collocation, when the codes are applied to singular problems in ODEs. We are fully aware of the difficulties in a code comparison, so in this paper, we will try to only diagnose the potential improvements, we could address in the next update of the codes. [ABSTRACT FROM AUTHOR]

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green's function for the highest-order term 1 < α ≤ 2 , which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green's function kernel. The theory and properties of fractional Green's function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam's characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number. [ABSTRACT FROM AUTHOR]

Introduction. This work was initiated by the problems of cardiovascular diseases, which are one of the main causes of mortality of the population of our planet. More than ten years ago, in 2012, approximately 3.7 million people died of acute coronary syndrome worldwide. The fight against such pathologies is carried out with the help of so-called stents, the manufacture of which can be carried out by the method of magnetic pulse «expansion» from hollow metal cylinders. The limited production possibilities of magnetic pulse «expansion» were caused by the minimum cross-sectional size of the inductor-instrument, which can be practically manufactured. Other tools are required to perform this operation. Novelty. A system of magnetic-pulse expansion of thin-walled pipes of small diameter with an inductor that excites an azimuthal electromagnetic field in the case of direct current passing through the processing object and in the absence of its connection in an electric circuit with an inductor is proposed. Purpose. The main analytical dependencies for the characteristics of the electromagnetic processes taking place in the inductor systems for the expansion of cylindrical conductive pipes of small diameter when direct passage of current through the processed object and when it is not connected to an electric circuit with an inductor (insulated billet) is derived. Methods. The solution of the boundary value problem with given boundary conditions was carried out by applying Laplace transforms and integrating Maxwell’s equations. Results. Analytical expressions were obtained for the main characteristics of the processes: the intensities of the excited electromagnetic fields and currents in the system depending on the parameters of the studied systems. The analysis of possible technical schemes for solving the given problem indicated the choice of the optimal variant of an effective system of magnetic-pulse «stretching» of thin-walled cylindrical conductors of small diameter. Practical value. Based on the qualitative analysis of the obtained results, recommendations for the practical implementation of the proposed system were formulated. The obtained dependences allow us to give numerical estimates of the effectiveness of excitation of magnetic pressure forces on the object of processing and to choose directions for further improvement of the magnetic pulse technology for solving such problems. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of a steady-state solution $ [\rho_0 $, $ u_0\equiv 0 $, $ \Phi_0] $ of the three-dimensional Euler-Poisson-Korteweg system in a bounded domain with physical boundary conditions using calculus of variations. Based on the existence result, the asymptotic stability for the Euler-Poisson-Korteweg system is established near the given steady state. [ABSTRACT FROM AUTHOR]

The present paper investigates a new method for computationally solving the time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) with initial boundary conditions (IBCs). The method utilizes a set of generalized shifted Jacobi polynomials (GSJPs) that adhere to the specified initial and boundary conditions (IBCs). Our approach involves constructing operational matrices (OMs) for both ordinary derivatives (ODs) and fractional derivatives (FDs) of the GSJPs we employ. We subsequently employ the collocation spectral method using these OMs. This method successfully converts the TFCKdVEs into a set of algebraic equations, greatly simplifying the task. In order to assess the efficiency and precision of the proposed numerical technique, we utilized it to solve two distinct numerical instances. [ABSTRACT FROM AUTHOR]

Squeezing or squeeze flows have tremendous applications in applied fields, like engineering, biomedical sciences and rheological studies. This paper demonstrates the squeezing flow of kerosene-based nanoliquids between parallelly aligned plates, with a Riga-type fixed lower boundary. The effects of dispersing two types of copper-functionalized carbon nanotubes (CNTs), single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) are examined. Using appropriate transformations, a self-similar ordinary differential system is derived from the governing model of partial differential equations and substantial boundary conditions. Using the homotopy analysis method (HAM) and the b v p 4 c package, analytical and numerical estimates are obtained, respectively. For higher values of the squeezing parameter, dimensionless temperature increases, while velocity patterns are upside-down. Moreover, increments in dimensionless parameters representing Riga constituent width, modified Hartmann number, nanoparticle concentration and radiation parameter improve heat transfer rates and thermal boundary layer thickness, however, adversely affect velocity. Despite enhanced friction effects, numerical results show that the Nusselt number increases as nanoparticle loads and radiation parameters increase. This suggests that convection rates are improved over conduction rates. Excellent agreement is found between analytical and numerical evaluations. Apparently, it is noticed that MWCNTs perform better than SWCNTs. [ABSTRACT FROM AUTHOR]

In this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells. [ABSTRACT FROM AUTHOR]

The paper is concerned with the asymptotic expansion of solutions to the following 2 × 2 Dirac type system: L y = - iB - 1 y ′ + Q x y = λ y , B = b 1 0 0 b 2 , y = col y 1 , y 2 , 0.1 with a smooth matrix potential Q ∈ W 1 n 0 , 1 ⊗ C 2 × 2 and b1 < 0 < b2. If b2 = −b1 = 1, this equation is equivalent to one dimensional Dirac equation. We apply these formulas to get the asymptotic expansion of the characteristic determinant of the boundary value problem associated with the above equation subject to the general two-point boundary conditions. This expansion directly yields new completeness result for the system of root functions of such BVP with nonregular boundary conditions. [ABSTRACT FROM AUTHOR]

This paper established a functional design of the spectral Tau method (STM) upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to execute the Tau integration analytically. Moreover, explicit algebraic systems for solving the Lane–Emden and Riccati equations and the contamination of a system of three artificial lakes are introduced. The convergence and error analyses are explored in depth. Some enlightening boundary value problems (BVPs) for real-life and physical applications, as singularly perturbed equations, are provided to guarantee that this approach is authentic, reliable, and appropriate. Accurate results are obtained using only a few number of retained nodes. The different outcomes, patterns, and better results showed that the FDCHPs differ from Chebyshev polynomials of the second kind. [ABSTRACT FROM AUTHOR]

In the present paper, we generalized some of the operators defined in (p , q) -calculus with respect to another function. More precisely, the generalized (p , q) - ϕ -derivatives and (p , q) - ϕ -integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q / p) -shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p , q) - ϕ -difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle. [ABSTRACT FROM AUTHOR]

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs. [ABSTRACT FROM AUTHOR]

When studying the boundary value problems' solvability for some partial differential equations encountered in applied mathematics, we frequently need to create systems of partial differential equations and explicitly construct linearly independent solutions explicitly for these systems. Hypergeometric functions frequently serve as solutions that satisfy these systems. In this study, we develop self-similar solutions for a third-order multidimensional degenerate partial differential equation. These solutions are represented using a generalized confluent Kampé de Fériet hypergeometric function of the third order. [ABSTRACT FROM AUTHOR]

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional RC$$ \mathcal{RC} $$ electric circuit model as an example to show the practical applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3 − α) -order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α ∈ (1 , 2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method. [ABSTRACT FROM AUTHOR]

AbstractAn investigation on the propagation of bending waves along the edge of a piezoelectric thin plate resting on an elastic foundation (cf. Pasternak foundation) and under the action of a point-load is carried out in this paper. The displacement field of the plate is based on the classical plate theory. The Kelvin-Voigt type viscoelastic model is used to examine the viscous effect on the characteristics of the bending wave. The Moore-Gibson-Thompson (MGT) thermoelastic equation is solved for an external time variant harmonic heat source at the upper surface of the plate. The piezoelectric potential is obtained for an electrode-covered upper surface and an electrically shorted lower surface. The displacements of the propagating bending wave along the edge are examined under a time-harmonic point-load acting near the vicinity of the free edge of the plate. The well-known Green function technique and Fourier transform method are implemented to generate an analytical solution for the non-homogeneous boundary value problem. The examination of the derived explicit form of the dispersion relationship and the displacements of the bending edge wave under external loading are discussed and presented both analytically and graphically, and the results are summarized. [ABSTRACT FROM AUTHOR]

This research work studies a problem associated with an axisymmetric torsion of an orthotropic layer by a circular rigid disc at the midplane. The orthotropic layer is sandwiched by two identical orthotropic half-spaces with two interfaced cracks. The layer and half-spaces are dissimilar in nature. The mixed boundary value problem is reduced to a system of dual integral equations by Hankel transformation, which are converted to Fredholm integral equations of the second kind. The integral equations are solved numerically by the quadrature rule. The stress intensity factors for crack and disc have been derived and are presented graphically for different thicknesses of orthotropic layer. [ABSTRACT FROM AUTHOR]

This research introduces an accelerated training approach for Vanilla Physics-Informed Neural Networks (PINNs) that addresses three factors affecting the loss function: the initial weight state of the neural network, the ratio of domain to boundary points, and the loss weighting factor. The proposed method involves two phases. In the initial phase, a unique loss function is created using a subset of boundary conditions and partial differential equation terms. Furthermore, we introduce preprocessing procedures that aim to decrease the variance during initialization and choose domain points according to the initial weight state of various neural networks. The second phase resembles Vanilla-PINN training, but a portion of the random weights are substituted with weights from the first phase. This implies that the neural network's structure is designed to prioritize the boundary conditions, subsequently affecting the overall convergence. The study evaluates the method using three benchmarks: two-dimensional flow over a cylinder, an inverse problem of inlet velocity determination, and the Burger equation. Incorporating weights generated in the first training phase neutralizes imbalance effects. Notably, the proposed approach outperforms Vanilla-PINN in terms of speed, convergence likelihood and eliminates the need for hyperparameter tuning to balance the loss function. [ABSTRACT FROM AUTHOR]

The stretching of a composite specimen taking into account temperature influence is modelled in terms of tensile stress applied at the opposite boundaries of the elastic rectangular domain. Two other boundaries are supposed to be fixed, it allows to formulate the stated problem as a boundary value problem. The analytical solution is realized with the help of the integral transform method. According to it, the problem is reduced to the one‐dimensional non‐homogeneous ordinary differential equations with corresponding boundary conditions in the transform's domain. The apparatus of Green's matrix‐function was applied to solve it. It let to the system of two singular integral equations, which was solved with the help of orthogonal polynomials method. The final formulae for displacements and stress were used to analyze the stress state of the rectangular elastic specimen. Obtained numerical results were compared with some experimental data for the stretching rectangular ceramic specimen. [ABSTRACT FROM AUTHOR]