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In this paper, a generalized fuzzy barycentric Lagrange interpolation method is proposed to solve two-dimensional fuzzy fractional Volterra integral equations. Firstly, we use the generalized Gronwall inequality and iterative methods to demonstrate the existence and uniqueness of solutions to the original equation. Secondly, combining the generalized fuzzy interpolation method and the fuzzy Gauss-Jacobi quadrature formula to discretize the original equation into corresponding algebraic equations in fuzzy environment. Then, the convergence of the proposed method is analyzed, and an error estimate is given based on the uniform continuity modulus. Finally, some numerical experiments show that the proposed method has high numerical accuracy for both smooth and non-smooth solutions.

In this paper, a model of deformations of Stieltjes strings system located along a geometric star graph with a nonlinear condition at the node is studied. This kind of condition arises due to the presence of a limiter for the movement of strings in the node under the influence of an external load. In the present paper, the necessary and sufficient conditions for the extremum of the energy functional are established; existence and uniqueness theorems for the solution are proved; the critical loads at which the strings come into contact with the limiter are analyzed; the dependence of the solution on the limiter length is studied.

In this work, we study the coupled system of fractional integro-differential equations, which includes the fractional derivatives of the Riemann–Liouville type and the fractional q -integral of the Riemann–Liouville type. We focus on the utilization of two significant fixed-point theorems, namely the Schauder fixed theorem and the Banach contraction principle. These mathematical tools play a crucial role in investigating the existence and uniqueness of a solution for a coupled system of fractional q -integro-differential equations. Our analysis specifically incorporates the fractional derivative and integral of the Riemann–Liouville type. To illustrate the implications of our findings, we present two examples that demonstrate the practical applications of our results. These examples serve as tangible scenarios where the aforementioned theorems can effectively address real-world problems and elucidate the underlying mathematical principles. By leveraging the power of the Schauder fixed theorem and the Banach contraction principle, our work contributes to a deeper understanding of the solutions to coupled systems of fractional q -integro-differential equations. Furthermore, it highlights the potential practical significance of these mathematical tools in various fields where such equations arise, offering a valuable framework for addressing complex problems.

This paper establishes a rigorous theoretical framework for analyzing the existence and uniqueness of solutions to Cohen–Grossberg bidirectional associative memory neural networks (CGBAMNNs) incorporating four distinct types of time-varying delays: leakage, neutral, distributed, and transmission delays. This study makes three key contributions to the field: First, it overcomes the fundamental challenge posed by the system’s inherent inability to be expressed in vector–matrix form, which previously limited the application of standard analytical techniques. Second, the work develops a novel and generalizable methodology that not only proves sufficient conditions for solution existence and uniqueness but also, for the first time in the literature, provides an explicit representation of the unique solution. Third, the proposed framework demonstrates remarkable extensibility, requiring only minor modifications to be applicable to a wide range of delayed system models. Theoretical findings are conclusively validated through numerical simulations, confirming both the robustness of the proposed approach and its practical relevance for complex neural network analysis.

This paper is concerning with the study of stability involving a thermoelastic system with internal nonlinear localized damping. The main novelty of the paper is to introduce to the study of thermoelastic system the general Wentzell boundary conditions associated to the internal heat equation. This boundary condition takes into account that there is a boundary source of heat which depends on the heat flow along the boundary, the heat flux across the boundary, and the temperature at the boundary. The tools are the use of multipliers with the construction of appropriate perturbed energy functionals.

This paper considers a linear system of partial differential equations (PDEs) to describe the stress-strain state of a two-phase body under static load, such as water-saturated soil. It investigates the basic properties of a new general differential operator Lame. The equations differ from the classical Lame equations by including first derivatives, which account for the influence of pore water on soil mineral particles. The properties of the generalized Lamé operator are investigated for the application of variational methods to solve the problem. It also describes alternative of the Betti and Clapeyron formulas using strain energy results. The calculus of variations of the Galerkin method is used to solve the minimum functional problem. Properties of bilinear forms are established and a theorem on the existence and uniqueness of the solution of the two-phase equilibrium problem is proved. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The adequacy of the mathematical model of a water-saturated foundation for a natural experiment is shown. The use of Korn's inequality implies limitations on elastic properties (homogeneity, anisotropy) and the geometry of the region (requiring regularity and smooth boundaries). The study illustrates that the methodology of mechanics of a deformable solid can be adapted with appropriate modifications to a two-phase body in a stabilized state. The finite element method is adapted for a kinematic model that considers excess residual pore pressures. A new stiffness matrix is obtained, which is the sum of two matrices: one for the soil skeleton and one for pore water. The finite element method is tested on the Flamand problem. The adequacy of the mathematical model of a water-saturated foundation for a full-scale experiment is shown. The problem of the action of distributed load on a water-saturated heterogeneous foundation was solved using the finite element method and the results were compared with experimental data. The effect of mesh partitioning on the accuracy of the numerical solution is also studied in the finite element method. The maximum discrepancy was no more than 26%.

In this study, we investigate the existence and uniqueness of a solution for a first-kind linear Volterra functional integral equation with proportional delay and weakly singular kernel, which has been stated as a research problem by H. Brunner (2017, pp. 99–100). We solve the problem numerically by the Tau-collocation method using Müntz-Jacobi polynomials along with the Gauss-Jacobi quadrature rule. Moreover, we prove a convergence theorem for the proposed method in L 2 -norm. We use several examples to corroborate the theoretical results and numerical efficiency of the method.

The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.

In our study we approached the mixed problem in the context of linear theory of thermoelasticity for Cosserat bodies. After we define the finite energy solution for this mixed problem, some results with respect to the existence and uniqueness are proven of this kind of solution. To this aim, we generalized some analogous results established by Dafermos, for a mixed problem from linear theory of the classical elasticity. In another important result of our study, we introduce some specific conditions which give the possibility that the evolution of the finite energy solution to be controlled.

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled systems are presented to validate the theoretical results, and simulations of the triply coupled case illustrate the influence of different fractional orders on the system dynamics.