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We consider the initial boundary value problem for the compressible fluid-particle interaction system with cylindrical symmetry. We derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate when the shear viscosity μ = κ ρ β goes to zero without any smallness assumption on the initial and boundary data. [ABSTRACT FROM AUTHOR]

We develop methods for the solution of inhomogeneous Robin-type boundary value problems (BVPs) that arise for certain linear parabolic partial differential equations (PDEs) on a half-line, as well as a second-order generalization. We are able to obtain nonstandard solutions to equations arising in a range of areas, including mathematical finance, stochastic analysis, hyperbolic geometry, and mathematical physics. Our approach uses the odd and even Hilbert transforms. The solutions we obtain and the method itself seem to be new. [ABSTRACT FROM AUTHOR]

Unstable interfaces govern many processes in fluids, plasmas, materials, in nature and technology. In distinct physical environments, the interface dynamics exhibit similar characteristics and couple micro to macro scales. Our work establishes the rigorous theory examining the classical problem of the dynamics of an interface with mass and energy fluxes under destabilizing accelerations. We consider thermally conducting fluids in the low Mach regime with weak compressibility prevailing over thermal transport. We find the attributes of perturbation waves, solve the boundary value problem, and identify the flow field structure, the interface perturbations growth, and the interface velocity. The interface dynamics is stabilized primarily by the inertial mechanism and is unstable when the acceleration exceeds a threshold. The thermal heat flux provides extra stabilizations, seeds energy perturbations, creates the vortical field in the bulk, and rescales the interface velocity. Our results agree with experiments in plasmas and complex fluids and with contained turbulence experiments. We outline extensive benchmarks for experiments and simulations and chart future research directions. [ABSTRACT FROM AUTHOR]

This study constructs a robust higher-order fitted operator finite difference method for a two-parameter singularly perturbed boundary value problem. The derivatives in the governing ordinary differential equation are substituted by second-order central finite difference approximations, after which the fitting parameter is introduced and determined. The resulting system of linear equations may then be solved using the Thomas method. The stability, consistency, and convergence of the current method have been thoroughly validated. To enhance accuracy and achieve a higher-order numerical solution, a post-processing technique was employed to upgrade the method from second-order to fourth-order convergence. Finally, three test examples were used to confirm the method's appropriateness. The numerical results demonstrate that the proposed technique is stable, consistent, and produces a higher-order numerical solution than the existing ones in the literature. [ABSTRACT FROM AUTHOR]

In an attempt to simulate black hole echoes (generated by potential quantum-gravitational structure) in numerical relativity, we recently described how to implement a reflecting boundary outside of the horizon of a black hole in spherical symmetry. Here, we generalize this approach to spacetimes with no symmetries and implement it numerically using the generalized harmonic formulation. We cast the evolution equations and the numerical implementation into a Summation By Parts scheme, which seats our method closer to a class of provably numerically stable systems. We implement an embedded boundary numerical framework that allows for arbitrarily shaped domains on a rectangular grid and even boundaries that evolve and move across the grid. As a demonstration of this framework, we study the evolution of gravitational wave scattering off a boundary either inside, or just outside, the horizon of a black hole. This marks a big leap toward the goal of a generic framework to obtain gravitational waveforms for behaviors motivated by quantum gravity near the horizons of merging black holes. [ABSTRACT FROM AUTHOR]

Numerical methods are frequently developed to investigate concepts for approximately solving ordinary differential equations (ODEs). To achieve minimal error and higher accuracy in approximate solutions, researchers have focused on developing algorithms using various numerical techniques. This study proposes the application of Ball curves, specifically the Said–Ball curve, for estimating solutions to higher-order ODEs. To obtain the best control points of the Said–Ball curve, the least squares method is used. These control points are calculated by minimizing the residual error through the sum of the squares of the residual functions. To demonstrate the proposed method, several boundary value problems are presented, and their performance is compared with existing methods in terms of error accuracy. The numerical results indicate that the proposed method improves error accuracy compared to existing studies, including those employing Bézier curves and the steepest descent method. [ABSTRACT FROM AUTHOR]

Conventional approaches for the structural health monitoring of infrastructures often rely on physical sensors or targets attached to structural members, which require considerable preparation, maintenance, and operational effort, including continuous on-site adjustments. This paper presents an image-driven hybrid structural analysis technique that combines digital image processing (DIP) and regression analysis with a continuum point cloud method (CPCM) built on a particle-based strong formulation. Polynomial regressions capture the boundary shape change due to the structural loading and precisely identify the edge and corner coordinates of the deformed structure. The captured edge profiles are transformed into essential boundary conditions. This allows the construction of a strongly formulated boundary value problem (BVP), classified as the Dirichlet problem. Capturing boundary conditions from the digital image is novel, although a similar approach was applied to the point cloud data. It was shown that the CPCM is more efficient in this hybrid simulation framework than the weak-form-based numerical schemes. Unlike the finite element method (FEM), it can avoid aligning boundary nodes with regression points. A three-point bending test of a rubber beam was simulated to validate the developed technique. The simulation results were benchmarked against numerical results by ANSYS and various relevant numerical schemes. The technique can effectively solve the Dirichlet-type BVP, yielding accurate deformation, stress, and strain values across the entire problem domain when employing a linear strain model and increasing the number of CPCM nodes. In addition, comparative analysis with conventional displacement tracking techniques verifies the developed technique's robustness. The proposed technique effectively circumvents the inherent limitations of traditional monitoring methods resulting from the reliance on physical gauges or target markers so that a robust and non-contact solution for remote structural health monitoring in real-scale infrastructures can be provided, even in unfavorable experimental environments. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics { u t = Δ u − χ 11 ∇ ⋅ (u ∇ v) − χ 12 ∇ ⋅ (u ∇ z) + f 1 (u , w) , x ∈ Ω , t > 0 , 0 = Δ v − v + u + w , x ∈ Ω , t > 0 , w t = Δ w − χ 21 ∇ ⋅ (w ∇ v) − χ 22 ∇ ⋅ (w ∇ z) + f 2 (u , w) , x ∈ Ω , t > 0 , 0 = Δ z − z + u + w , x ∈ Ω , t > 0 , subject to homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n (n ≥ 1) , where χ i j > 0 (i , j = 1 , 2) , f 1 (u , w) = u (a 0 − a 1 u − a 2 w − a 3 ∫ Ω u d x − a 4 ∫ Ω w d x) , f 2 (u , w) = w (b 0 − b 1 u − b 2 w − b 3 ∫ Ω u d x − b 4 ∫ Ω w d x) with a i , b i > 0 (i = 0 , 1 , 2) , a j , b j ∈ R (j = 3 , 4). It is shown that if the parameters satisfy certain conditions, then the corresponding initial boundary value problem admits a unique global-in-time classical solution in any spatial dimension, which is uniformly bounded. Moreover, based on the construction of suitable energy functionals, the globally asymptotic stabilization of coexistence and semi-coexistence steady states is considered. Our results generalize and improve some previous results in the literature. [ABSTRACT FROM AUTHOR]

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the L2$$ {L}^2 $$ norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn$$ {\mathrm{\mathbb{R}}}^n $$ is investigated by utilizing the Mawhin's continuation theorem. In order to relax the assumptions of matrices in the boundary conditions, the Moore–Penrose pseudoinverse matrix is introduced to construct projectors. Furthermore, a ternary Carathéodory function that is nondecreasing in the last two variables and exhibits at most linear growth after integrating with respect to the first variable is used as a control function to constrain the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when k$$ k $$ is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. [ABSTRACT FROM AUTHOR]

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency. [ABSTRACT FROM AUTHOR]

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses. [ABSTRACT FROM AUTHOR]

A semi-analytic numerical method is described as an efficient meshless approach for the solution of anomalous non-linear thermal conduction problems in functionally graded materials in which the model results in fractional boundary value problems. The first key feature in this scheme is the derivation and discretization of the fractional derivative at every time step. The second key feature is the trigonometric basis functions (TBFs) as the basis functions were introduced by the need for approximate solutions on boundary conditions with more flexibility in choosing collocation points. Moreover, the approximate solution of the anomalous thermal conduction problems converges to the exact solution as γ is closed to 1 in the full closed time interval for three simulated numerical results. [ABSTRACT FROM AUTHOR]

In this paper, the linear model of Moore–Gibson–Thompson thermoporoelasticity is considered and the governing equations of motion and steady vibrations are given. The basic system of equations of steady vibrations with respect to the displacement vector, the changes of temperature and fluid pressure are proposed. Then the radiation conditions are established and Green’s first identity is obtained. Finally, on the basis of this identity, the uniqueness theorems for classical solutions of the boundary value problems of steady vibrations in the theory of MGT thermoporoelaticity are proved. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a fractional differential equation multipoint boundary value problem at resonance in ℝn$$ {\mathrm{\mathbb{R}}}^n $$ is investigated by utilizing the Mawhin's continuation theorem. In order to relax the assumptions of matrices in the boundary conditions, the Moore–Penrose pseudoinverse matrix is introduced to construct projectors. Furthermore, a ternary Carathéodory function that is nondecreasing in the last two variables and exhibits at most linear growth after integrating with respect to the first variable is used as a control function to constrain the nonlinear term. [ABSTRACT FROM AUTHOR]

In this research, we explore a finite horizon optimal stabilization and tracking control scheme for the dynamical model of a 6‐DOF Autonomous Underwater Vehicle (AUV). Dynamical equations of the AUV are represented in a Lie group (SE(3)$$ SE(3) $$) framework, encompassing both translational and rotational motions. Utilizing a left Lie group action on SE(3)$$ SE(3) $$, we define error function for velocities via a right transport map to effectively address optimal trajectory tracking. The optimal control objective is formulated as a trade‐off problem, aiming to minimize both errors and control effort simultaneously. Left action on SE(3)$$ SE(3) $$ yields the left trivialized Hamiltonian function from which the concomitant state and costate dynamical equations are derived using Pontryagin's Minimum Principle (PMP). Consequently, the resulting two‐point boundary value problem is solved to obtain optimal trajectories. We demonstrate the optimality of the resulting solution obtained from the derived control law. For ensuring boundedness in the presence of small disturbances, this study incorporates the effects of internal parametric uncertainties associated with added mass and inertia components, along with the influence of external disturbances induced by ocean currents. Through simulation validations, we confirm the alignment of our results with the theoretical developments, demonstrating that the proposed control law effectively mitigates both parametric uncertainties and ocean current disturbances. [ABSTRACT FROM AUTHOR]

Nanofluid is one of the modern heat transfer fluids that offer the potential to substantially enhance the heat transfer efficiency of conventional fluids. Extensive research has been undertaken to explore its fundamental thermophysical properties specifically viscosity and as well as thermal conductivity. This research emphasizes the significance of hybrid nanofluids and investigates the effect of Brownian motion and thermophoretic phenomena on the characteristics of the Agrawal flow that tends to a stagnation point adjacent to a moving porous disk. The model also accounts for the effects of Smoluchowski temperature and Maxwell velocity slip conditions. Through the utilization of similarity ansatz, the governing partial differential equations are simplified into a class of ordinary differential (similarity) equations. Subsequently, these simplified equations achieved numerical solutions by employing the bvp4c solver, which is specifically designed for fourth-ordered boundary value problems. The study delves into the remarkable impacts of the pertinent embedded parameters on key parameters such as mass transfer rate, heat transfer rate, and shear stress. These effects are brilliantly depicted through a combination of graphs and tables. Graphical analyses disclose the presence of dual solutions within a particular range of the stretching/shrinking parameter. Also, enhancing the solid volume fraction of nanoparticles leads to a notable rise in the shear stress and heat transfer for both solution branches, whereas the mass transfer rate experiences a reduction. [ABSTRACT FROM AUTHOR]

We consider an algorithm for finding solutions to boundary-value problems of the two-dimensional elasticity theory by using physics-informed neural networks. The suggested approach makes it possible to reduce the boundary-value problems of the mechanics of continuous media to the problems of optimization, whereas the application of physics-informed neural networks within the framework of the considered approach makes it possible to reduce the solution of a broad class of problems to the construction of an error function of the general form. For the case of the Neumann conditions with constant forces given on the contour of the rectangular area, we indicate the explicit form of a neural-network function and the solution in displacements as a whole. To verify the proposed methodology, we perform the numerical analysis of the stress-strain state for a one-dimensional dynamic problem of longitudinal vibrations of a rod. The analyzed methodology can be extended to the case of three-dimensional problems of the elasticity theory, including piecewise homogeneous and, in a more general case, inhomogeneous media. [ABSTRACT FROM AUTHOR]

This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the uniform regularity and zero dissipation limit of solutions to the initial boundary value problem of 3D incompressible magnetic Bénard equations in the half space, where the velocity field satisfies the no-slip boundary conditions, the magnetic field satisfies the perfect conducting boundary conditions, and the temperature satisfies either the zero Neumann or zero Dirichlet boundary condition. With the assumption that the magnetic field is transverse to the boundary, we establish the uniform regularity energy estimates of solutions as both viscosity and magnetic diffusion coefficients go to zero, which means there is no strong boundary layer under the no-slip boundary condition even the energy equation is included. Then the zero dissipation limit of solutions for this problem can be regarded as a direct consequence of these uniform regularity estimates by some compactness arguments. [ABSTRACT FROM AUTHOR]

ABSTRACT The paper addresses an initial boundary value problem for a partial differential–algebraic equation involving a parameter. An integral condition with respect to the time derivative of the unknown function is provided as an additional condition to determine this parameter. The Dzhumabaev parameterization method is employed to solve the problem. The domain is subdivided, and functional parameters are defined as the values of the solution along the internal lines of the subdomains. This reformulates the original problem into an equivalent initial boundary value problem for a system of hyperbolic equations with parameters and associated functional relations. The paper develops algorithms to solve the problem, demonstrating their applicability. Furthermore, conditions for the existence and uniqueness of a solution to the initial boundary value problem, involving the partial differential–algebraic equation with a parameter and discrete memory, are established. [ABSTRACT FROM AUTHOR]

The current study investigates the impact of gyrotactic microorganisms on mass and heat transport of Williamson hybrid nanoliquid, prepared with copper $\left({{\rm{Cu}}} \right)$Cu and aluminum oxide (Al2O3) as nanoparticles and ethylene glycol $\left({{\rm{EG}}} \right)$EG as base liquid, through a swirling cylinder with Christov–Cattaneo heat flux. The convective-type boundary conditions are also considered in this study. The appropriate similarity variables technique is employed to renovate the modeled partial differential equations (PDEs) into the ordinary differential equations (ODEs). The finite element method is implemented to solve the set of ordinary differential equations together with boundary conditions. The profiles of radial velocity, density of microorganisms, concentration of nanoparticles, temperature and tangential velocity of hybrid nanofluid for relevant parameters, such as thermal relaxation parameter $\left({0.5 \le {\rm{\gamma }} \le 2.0} \right)$0.5≤γ≤2.0, suction parameter $\left({0.1 \le {{\rm{V}}_0} \le 0.7} \right)$0.1≤V0≤0.7, bio-convection Lewis number $\left({1.0 \le {\rm{Sb}} \le 1.6} \right)$1.0≤Sb≤1.6, Prandtl number $\left({2.2 \le {\rm{Pr}} \le 8.2} \right),$2.2≤Pr≤8.2, volume fraction parameter of ${\rm{Cu}}$Cu nanoparticles $\left({0.01 \le {\phi _1} \le 0.04} \right)$0.01≤ϕ1≤0.04, chemical reaction parameter $\left({0.1 \le {\rm{Cr}} \le 0.7} \right)$0.1≤Cr≤0.7, magnetic field parameter $\left({0.1 \le {\rm{M}} \le 0.7} \right),$0.1≤M≤0.7, Peclet number $\left({0.1 \le {\rm{Pe}} \le 0.7} \right),$0.1≤Pe≤0.7, volume fraction parameter of Al2O3 nanoparticles $\left({0.01 \le {\phi _2} \le 0.04} \right)$0.01≤ϕ2≤0.04, Reynolds number $\left({0.5 \le {\rm{Re}} \le 2.0} \right),$0.5≤Re≤2.0, Biot number $\left({0.2 \le {\rm{Bi}} \le 0.8} \right)$0.2≤Bi≤0.8 and Weissenberg number $\left({0.5 \le {\rm{We}} \le 0.8} \right)$0.5≤We≤0.8, are shown in diagrams. Furthermore, the values of local density of microorganisms, rates of velocity, heat and mass transport rates for various parameters are also calculated numerically and revealed in tables. The important findings reveal that temperature profiles of hybrid nanofluid enlarge with rising values of We, and the density of motile microorganisms profiles impedes as the values of Sb improve. [ABSTRACT FROM AUTHOR]

We study a priori estimate, existence, and uniqueness of solutions with symmetric derivatives for a third-order boundary value problem. The main tool in the proof of our existence result is Leray-Schauder continuation principle. Two examples are included to illustrate the applicability of the results. [ABSTRACT FROM AUTHOR]

This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ-tempered fractional derivatives of Caputo type is considered. [ABSTRACT FROM AUTHOR]

The initial boundary value problem for the nonstationary heat equation is studied in a bounded domain with the specific overdetermination condition. This condition is nonlinear and can be interpreted as the energy functional. In present paper we construct the class of solutions to this problem. [ABSTRACT FROM AUTHOR]

The problem of explicit stiffness of members laminated with two hybrid materials was formulated and applied to dynamic analysis of the beam elements. The stiffness of these members was derived from their cross-sectional and material properties, expressed in terms of axial and flexural rigidities, mass per unit length, and mass moment of inertia. The focus is on the members with a rectangular cross-section. The formulated stiffness was applied to dynamic beam elements associated with free vibrations. The governing differential equation and boundary conditions for the lateral free vibrations of laminated beams, considering rotary inertia and shear deformation were derived. For solving this differential equation, numerical analyses were performed to obtain natural frequencies and mode shapes. To verify the natural frequencies obtained were compared with those calculated using the finite element method. As a result of numerical experiments, the effects of different beam parameters on the natural frequencies were discussed in detail. [ABSTRACT FROM AUTHOR]

Nanofluids find extensive applications in enhancing the thermodynamic efficiency of thermal systems across various domains of engineering and scientific disciplines. This study aims to explore the complex relationship between the varying thermal conductivity and viscosity impact in nanofluid dynamics. The main objective of this study is to examine the three-dimensional stagnation flow of Casson nanofluid across a stretching and spinning disk, influenced by a magnetic source. The Navier–Stokes model for flow systems includes Brownian diffusion and thermophoresis. By using scaling variables, the complex system of partial differential equations is simplified into a set of coupled high degree nonlinear ordinary differential equations with convective boundary conditions. The homotopy technique is applied for analytic solutions. The optimization analysis is conducted on heat transfer rate and surface drag force coefficient using response parameters. The influence of the different parameters for the flow problem has been discussed and is shown through graphs. The finding of our study is that the velocity profiles increase in both radial directions as well as in the azimuthal direction by varying the rotation parameter strength, while the temperature gradient profile dwindles. Finally, the precision of the presented model is reaffirmed by means of a graphical juxtaposition with published data under a specific limiting scenario. [ABSTRACT FROM AUTHOR]

This article examines the scattering of surface gravity waves by two types of structures: a bottom-mounted inverted Π-shaped structure or a floating Π-shaped structure, in the presence of an infinite trench. The physical phenomenon is formulated as a boundary value problem governed by the modified Helmholtz equation, which is transformed into a system of Fredholm-type integral equations through the eigenfunction expansion technique. To enhance numerical accuracy and convergence, basis functions (such as Chebyshev and ultraspherical Gegenbauer polynomials) multiplied by appropriate weights are incorporated into a multi-term Galerkin approximation. Thus, the model effectively captures the singular behavior of the horizontal fluid velocity near the sharp lower edges of the Π-shaped structure and the corners of the trench. Validation of the model is achieved by comparing its results with existing solutions from the literature on water wave problems in specific limiting scenarios. Numerical results for reflection and transmission coefficients, mean drift forces, and free surface elevation are presented graphically. These results provide a comprehensive understanding of the influence of various non-dimensional wave and structural parameters on these hydrodynamic characteristics. Notably, the floating Π-shaped structure achieves a more significant reduction in free surface elevation in the transmitted region compared to the bottom-mounted inverted Π-shaped structure. [ABSTRACT FROM AUTHOR]

A nonlinear large deformation cable equation system based on arc length is employed to determine the configuration of the cable under current and buoyancy loads. Different from a traditional initial method and a shooting method, we have solved this nonlinear problem as a boundary value problem with nonlinear and global boundary conditions directly. A finite difference scheme is proposed to solve the large deformation cable equation, and the Newton–Raphson iteration is used to search for numerical approximate solution. We demonstrate that this system degenerates into the catenary equation in the case of vanishing bending stiffness for the first time. The solution of the catenary equation serves as the initial guess for the three-section cable problem. This method overcomes the disadvantages of the initial value method and step method, avoiding the need to adjust the boundary location. The spatial shapes of the large deformation cable and the effects of the length and position of the buoyancy section are discussed. The impact of ocean currents is also analyzed using Morison's formula. [ABSTRACT FROM AUTHOR]

A Hertz radiator's Sommerfeld boundary value problem is considered for the case when its electric moment is directed horizontally relative to the plane interface between two media with different values of magnetic permeability. An integral representation of the exact expression for the Hertz potential, which generalizes the classical solution for non-magnetic media, both in cylindrical and spherical coordinate systems, is obtained. The corresponding expressions for the scattered wave fields are given in the form of Sommerfeld integrals. It is shown that the potential components can be represented as the sum of an infinite series in powers of the Green function. [ABSTRACT FROM AUTHOR]

The intent of this paper was to investigate the fixed-point results under relation-theoretic generalized weak contractivity condition employing a pair of auxiliary functions ϕ and ψ verifying appropriate properties. In proving our outcomes, we observed that the partial-ordered relation (even, transitive relation) adopted by earlier authors can be weakened to the extent of a locally ϝ -transitive binary relation. The findings proved herewith generalize, extend, improve, and unify a number of existing outcomes. To validate of our findings, we offer a number of illustrative examples. Our outcomes assist us to figure out the existence and uniqueness of solutions to a boundary value problem. [ABSTRACT FROM AUTHOR]

The aim of this article is to introduce and study a new class of fractional integro nonlocal boundary value problems involving the p-Laplacian operator and generalized fractional derivatives. The existence of solutions in Banach spaces is investigated with the aid of the properties of Kuratowski's noncompactness measure and Sadovskii's fixed-point theorem. Two illustrative examples are constructed to guarantee the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness results for a weak solution to a class of Dirichlet boundary value problems whose prototype is − Δ p u = β | ∇ u | q + f   i n   Ω ,   u = 0   o n   ∂ Ω , where Ω is a bounded open subset of R N , N ≥ 2 , 1 < p < N , Δ p u = div | ∇ u | p − 2 ∇ u , p − 1 < q < p , β is a positive constant and f is a measurable function satisfying suitable summability conditions depending on q and a smallness condition. [ABSTRACT FROM AUTHOR]

This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and D ∗ ∗ . Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications. [ABSTRACT FROM AUTHOR]

Due to its significance in numerous scientific and engineering domains, discrete fractional calculus (DFC) has received much attention recently. In particular, it seems that the exploration of the stability of DFC is crucial. A mathematical model of the discrete fractional equation describing the deflection of a vertical column along with two-point boundary conditions featuring the Riemann–Liouville operator is constructed to study several kinds of Ulam stability results in this research work. In addition, we developed Lyapunov-type inequality and its application to an eigenvalue problem for discrete fractional rotating string equations. Finally, the effectiveness of the theoretical findings is demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

In order to consider the effect of fabric anisotropy in the analysis of geotechnical boundary value problems, this study proposes a modified model based on a fabric-based modified Cam-clay model, which can account for the anisotropic response of soil. The major modification of the original model aims to simplify the equations for numerical implementation by replacing the SMP strength criterion with the Lade's strength criterion. This model comprehensively considers the inherent anisotropy, induced anisotropy, and three-dimensional strength characteristics of soil. The model is first numerically implemented using the elastic trial–plastic correction method, and then it is encapsulated into the FLAC3D 6.0 software, and tested through conventional triaxial, embankment loading, and tunnel excavation experiments. Numerical simulation results indicate that considering anisotropy and three-dimensional strength in geotechnical engineering analysis is necessary. By accounting for the interaction between microstructure and macroscopic anisotropy, the model can more accurately represent soil behavior, providing significant advantages for geotechnical analysis. [ABSTRACT FROM AUTHOR]

This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in the maximum norm. Numerical examples are presented to validate the theoretical estimates. Additionally, the standard hybrid finite difference scheme, a cubic spline scheme, and the proposed method are compared to demonstrate the effectiveness of the present approach. [ABSTRACT FROM AUTHOR]

This research presents a new form of iterative technique for Garcia-Falset mapping that outperforms previous iterative methods for contraction mappings. We illustrate this fact through comparison and present the findings graphically. The research also investigates convergence of the new iteration in uniformly convex Banach space and explores its stability. To further support our findings, we present its working on to a BV problem and to a delay DE. Finally, we propose a design of an implicit neural network that can be considered as an extension of a traditional feed forward network. [ABSTRACT FROM AUTHOR]

Predators impact prey populations directly through consumption and indirectly via trait-mediated effects like predator-induced emigration (PIE), where prey alter movement due to predation risk. While PIE can significantly influence prey dynamics, its combined effect with direct predation in fragmented habitats is underexplored. Habitat fragmentation reduces viable habitats and isolates populations, necessitating an understanding of these interactions for conservation. In this paper, we present a reaction–diffusion model to investigate prey persistence under both direct predation and PIE in fragmented landscapes. The model considers prey growing logistically within a bounded habitat patch surrounded by a hostile matrix. Prey move via unbiased random walks internally but exhibit biased movement at habitat boundaries influenced by predation risk. Predators are assumed constant, operating on a different timescale. We examine three predation functional responses—constant yield, Holling Type I, and Holling Type III—and three emigration patterns: density-independent, positive density-dependent, and negative density-dependent emigration. Using the method of sub- and supersolutions, we establish conditions for the existence and multiplicity of positive steady-state solutions. Numerical simulations in one-dimensional habitats further elucidate the structure of these solutions. Our findings demonstrate that the interplay between direct predation and PIE crucially affects prey persistence in fragmented habitats. Depending on the functional response and emigration pattern, PIE can either mitigate or amplify the impact of direct predation. This underscores the importance of incorporating both direct and indirect predation effects in ecological models to better predict species dynamics and inform conservation strategies in fragmented landscapes. [ABSTRACT FROM AUTHOR]

In this paper, we present Proinov-type fixed point theorems in the setting of bi-polar metric spaces and fuzzy bi-polar metric spaces. Fuzzy bi-polar metric spaces with symmetric property extend classical metric spaces to address dual structures and uncertainty, ensuring consistency and balance. We provide different concrete conditions on the real-valued functions Ω , Π : 0 , ∞ → R for the existence of fixed points via the (Ω , Π) -contraction in bi-polar metric spaces. Further, we define real-valued functions Ω , Π : (0 , 1 ] → R to obtain fixed point theorems in fuzzy bi-polar metric spaces. We apply Ω , Π fuzzy bi-polar version of a Banach fixed point theorem to show the existence of solutions. Furthermore, we provide some non-trivial examples to show the validity of our results. In the end, we find the existence and uniqueness of a solution of integral equations and boundary value problem used in chemical sciences by applying main results. [ABSTRACT FROM AUTHOR]

The main scope of this paper is to develop and analyse three-level linearized difference schemes for solving the classical Burgers-Fisher equation. For the Dirichlet boundary value problem, the first three-level linearized difference scheme is second-order accurate in both time and space. It is able to obey a discrete conservative law. By the discrete energy argument and induction, it is rigorously proved to be uniquely solvable and unconditionally convergent. Furthermore, with the purpose of improving the numerical accuracy in space, another three-level linearized compact difference scheme is then established together with some investigation on its discrete conservation law, unique solvability and unconditional convergence of order two in time and four in space. The coupled nonlinearity of Burgers' type and Fisher type is intensively treated via the order of reduction and Taylor expansion. In addition, extensions to the problem subject to the periodic boundary condition (PBC) are involved. To the best of our knowledge, this is a rare work to carefully design and rigorously analyse the high-order linearized difference schemes for solving the Burgers-Fisher equation. Numerical tests are provided to support the theoretical results and to verify the computational efficiency. [ABSTRACT FROM AUTHOR]

The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation with Neumann boundary condition. Using Barron's representation of the solution [IEEE Trans. Inform. Theory 39 (1993), pp. 930–945] with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Inspired by the work from Bach and Chizat [On the global convergence of gradient descent for over-parameterized models using optimal transport, 2018; ICM–International Congress of Mathematicians, EMS Press, Berlin, 2023], we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. Numerical experiments are given to show the potential of the method. [ABSTRACT FROM AUTHOR]

This paper investigates the degenerate non-local boundary value problem of logistic type - Δ u = λ u - b (x) u p - a (x) u ∫ Ω c (y) | u (y) | r d y in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N , N ≥ 1 , is a bounded domain with smooth boundary, λ ∈ R is a bifurcation parameter, p > 1 , r ≥ 1 , a, b ∈ C ν (Ω ¯) , ν ∈ (0 , 1 ] , vanish on some subsets of Ω with positive measure, and 0 < c ∈ L ∞ (Ω) . The presence of the non-local term prevents us from using the classical sub and supersolutions methods to characterize the existence of positive solutions and ascertain their point-wise behavior with respect to λ . Combining some ideas going back to Li et al. (Calc Var Partial Differ Equ 60:36, 2021) with the theory of large solutions of as reported by López-Gómez (Metasolutions of parabolic equations in population dynamics, CRCPress, Boca Raton, 2016), we can conduct a detailed study of the point-wise λ -limit of the set of positive solutions, revealing a behavior substantially different from the one exhibited by the positive solutions of the underlying local problem, because of the presence of the non-local term. [ABSTRACT FROM AUTHOR]

Clinical trials are already established for high-temperature treatment of localized cancer, i.e., rise of tissue temperature to more than 55 °C as an effective noninvasive method for the treatment of localized cancer. However, as the computational techniques and capacity have enhanced considerably personalized treatment planning has become a manured tool. In the present investigation, a novel treatment planning framework is being proposed for radio frequency (RF) ablation of cancer tissue based on the tomographic image-based actual model. In patient-specific modeling, different thermal parameters like temperature history during ablation, and thermal damage profile have been virtually determined based on Penne's bioheat transfer model with appropriate boundary conditions. This advancement promises to significantly enhance the capabilities of healthcare practitioners in tailoring personalized treatment strategies for their clinical cases. By leveraging simulation outcomes, clinicians can precisely determine the most effective parameters, such as ablation power and frequency. Unfortunately, the current landscape in India presents a scarcity of specialized medical experts in the field of ablation oncology. Moreover, those who practice in this niche often rely on empirical charts rather than data-driven approaches, highlighting a critical need for increased expertise and the integration of advanced simulation technology to optimize cancer tissue ablation procedures. This real-life patient-specific three-dimensional model-based heat transfer model in a data center-based approach will not only guide the medical practitioner but also a greater number of clinicians can use that. Last but not least, optimizing different operating parameters of RF in this patient-centric approach for accurate treatment has also been discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions of Hilfer-Katugampola fractional differential equations with boundary conditions. We first establish existence theory of solutions for the mentioned problem by the fixed point theory and the measure of noncompactness. Then we investigate the ε-approximate solution to our concerned problem via a generalized Gronwall inequality. Finally, as the application of abstract results, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

Model of theoretical description of a change in optical properties in near-surface layers due lightwave propagation along a parabolic graded-index film waveguide structure is proposed. Exact analytical solution of the value boundary problem formulated, which describe the stationary eigenmodes of the waveguide structure, are found. The modes correspond to a discrete spectrum of values of the effective refractive index determined by the optical and geometrical parameters of the waveguide system. The field amplitude enlarges and the penetration depth of the field decreases with an increase in the longitudinal wave numbed. The intensity of the light flux in thicker films in the center of the film is higher than near its surfaces. The equation is derived, the use of which makes it possible to establish the adequacy of the choice of a parabolic profile for real semiconductor crystals for a given film thickness and wavelength. Analytical expressions of the total power flow of the lightwave along the waveguide structure are obtained. The total power flow enlarges with an increase in the film thickness for modes of all orders. The minimum of dependence of the flow on the wave number is observed. The value of the power flow minimum decreases with increasing order of the mode and goes into saturation. Possibility to choose a thickness of the parabolic graded-index film is shown, at which the power of the transferred light flux will be concentrated in a certain region of the waveguide structure. [ABSTRACT FROM AUTHOR]

This paper considers the problem of symmetrical three-point bending of a prismatic beam with an edge crack. The solution is obtained by the mixed finite element method within the simplified Toupin–Mindlin strain gradient elasticity theory. A mixed variational formulation of the boundary value problem for displacements–strains–stresses and their gradients is applied, simplifying the choice of approximating functions. The concept of energy balance is adopted to calculate the energy release rate with a virtual increase in crack length. The increment of the potential energy of an elastic body is determined by accounting for the strain and stress gradient contribution. Numerical calculations were performed using a quasi-uniform triangular mesh of the cross-type. The mesh refinement was applied in the vicinity of the crack tip, at the concentrated support, and the point of application of the transverse force, and uniform mesh partitioning was utilized in the rest of the beam. The fine-mesh analysis was carried out on the successively condensed meshes in the stress concentration domain for different values of the length scale parameter. The crack opening displacements and the distribution of strains and Cauchy stresses for various values of the length scale parameter are presented. An increase in this parameter increases the stiffness of the crack, which leads to a decrease in the crack opening displacements and a smooth closure of its faces at the crack tip. In addition, accounting for the scale parameter reduces the calculated values of strains and stresses near the crack tip. Based on the energy balance criterion, local fracture parameters such as the release rate of elastic energy at the crack tip and the stress intensity factor are determined for different values of the mesh step. The numerical calculations indicate the convergence of the obtained approximations. The main feature of solutions, which includes the strain gradient contribution, is the decrease in the values of the calculated parameters associated with the fracture energy compared to the classical elasticity theory. [ABSTRACT FROM AUTHOR]

The present study deals with the problem of oblique wave scattering by a finite floating elastic plate over a thick porous bed. A potential flow-coupled thin-elastic plate model has been developed, and the wave flow model resembling the physical scenario is framed into a boundary value problem (BVP). A semi-analytical method has been employed to obtain the hydrodynamic coefficients following the numerical and physical illustrations by varying different geometrical parameters. Prior to wave interaction, dispersive roots are thoroughly analyzed, and a critical frequency is observed above which the group velocity is negative. Within this range, minimum reflection and maximum transmission occur due to the high porosity of the seabed. A discontinuous pattern in the reflection coefficient is observed within the blocking range, which mainly exists for high compression. The findings of this work may be highly valuable for Very Large Floating Structures in marine settings, particularly in situations where interactions are primarily influenced by the thick porous beds. [ABSTRACT FROM AUTHOR]

• The paper studies the heliocentric orbit transfer of a E-sail-based spacecraft. • A transfer between two close circular orbits is optimized. • The paper proposes a set of equations to solve the optimization problem. • The approach extends the recent literature to an E-sail scenario. The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system for deep space navigation that exploits the dynamic pressure of the solar wind to generate thrust using a web of long, conducting tethers. The heliocentric trajectory of an E-sail-based spacecraft in a classic transfer between two Keplerian orbits is usually analyzed, in an optimal framework, by minimizing the total flight time. This paper discusses an analytical, approximate procedure for solving the two-point boundary value problem associated with the trajectory optimization process in a heliocentric, two-dimensional scenario involving a first-generation E-sail. The procedure is applied in a typical transfer between two coplanar circular orbits whose (assigned) radii are sufficiently close to each other. Paralleling the approach proposed in the recent literature, the mathematical model presented in this paper discusses a set of analytical equations that give an accurate approximation of both the minimum flight time and the (unknowns) initial adjoint variables. The paper also analyzes a set of mission applications, which involve interplanetary transfers to some near-Earth asteroids whose orbits have a very small value of both eccentricity and inclination. [ABSTRACT FROM AUTHOR]