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In this paper, the linear theory of Moore-Gibson-Thompson (MGT) thermoviscoelasticity for materials with voids is examined and the basic boundary value problems (BVPs) of steady vibrations are investigated. The governing equations of motion and steady vibrations are formulated. The fundamental solution to the system of steady vibration equations is constructed explicitly using four elementary functions, and its key properties are analyzed. Then, Green's first identity is established and the uniqueness theorems for classical solutions of the associated basic BVPs are proved. The surface and volume potentials are defined, and their essential properties are established. Singular integral operators are introduced, and their symbolic determinants and indices are calculated. Finally, existence theorems for classical solutions of the basic internal and external BVPs are established using the potential method.

For amenable discrete groupoids$\\mathcal {G}$and row-finite directed graphs E , let$(\\mathcal {G},E)$be a self-similar groupoid and let$C^*(\\mathcal {G}, E)$be the associated$C^*$-algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that$C^*(\\mathcal {G})$embeds in$C^*(\\mathcal {G}, E)$. Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.

Uncertain partial differential equations are widely used in practice, such as demography, traffic flows and so on. This paper proves an existence and uniqueness theorem for a class of uncertain partial differential equations. Then the properties of α-path are given based on linear growth, Lipschitz and regular conditions. Since uncertain partial differential equations are difficult to get analytical solutions, this paper presents a formula which combines an uncertain partial differential equation with a class of classical partial differential equations. Based on the formula, an algorithm for calculating the inverse uncertainty distribution of solution of an uncertain partial differential equation is also deduced. Finally, expected value, extreme value, first hitting time, time and spatial integrals of the solution of uncertain partial differential equation are also discussed.

In this paper, the Pfaff equations with continuous coefficients are considered. Analogs of Peano’s existence theorem and Kamke’s theorem on the uniqueness of the solution to the Cauchy problem are established, and a method for approximate solution of the Cauchy problem for the Pfaff equation is proposed.

This paper considers a free boundary problem for a system of quasi-linear parabolic equations in one dimension. Nonlinear problems with a free boundary are studied using a method based on constructing a priori estimates. For the solutions to the problem, a priori estimates of Shauder type are established. On the basis of a priori estimates, the existence and uniqueness theorems are proved.

In this study, we examine the uniqueness conditions for solutions of fractal differential equations using the Krasnoselskii-Krein uniqueness theorem. The analysis establishes sufficient criteria that guarantee the existence of unique solutions. Additionally, we employ the successive midpoint method to numerically solve chaotic systems governed by both fractal and global derivatives. To evaluate the effectiveness of the proposed approach, graphical simulations are presented for various derivative orders. These results illustrate the method’s accuracy, stability, and reliability in capturing the intricate dynamics of the considered systems.

Uncertain differential equations with a time delay, called uncertain-delay differential equations, have been successfully applied in feedback control systems. In fact, many systems have multiple delays, which can be described by uncertain differential equations with multiple delays. This paper defines uncertain differential equations with multiple delays, which are called uncertain multiple-delay differential equations (UMDDEs). Based on the linear growth condition and the Lipschitz condition, the existence and uniqueness theorem of the solutions to the UMDDEs is proven. In order to judge the stability of the solutions to the UMDDEs, the concept of the stability in measure for UMDDEs is presented. Moreover, two theorems sufficient for use as tools to identify the stability in measure for UMDDEs are proved, and some examples are also discussed in this paper.

In this paper, neutral delay differential equations, which contain constant and proportional terms, termed mixed neutral delay differential equations, are solved numerically. Moreover, an efficient numerical approach is introduced (a combination of the method of steps and the Haar wavelet collocation method) to solve mixed neutral delay differential equations. Furthermore, we prove the existence and uniqueness theorem using successive approximation methods. Three numerical examples are presented to demonstrate the implementation of the proposed method. Furthermore, the precision and accuracy of the Haar wavelet collocation method are validated theoretically by proving that the error tends to zero as the resolution level increases, and numerically, by calculating the rate of convergence. The findings contribute to a broader application of the wavelet-based method to a more complex type of differential equation. This study offers a framework for the extension of the combination of both methods to be applied to potential real-world applications in control theory, biological models, and computational sciences.

The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received tremendous attention. This article aims to study the theory for stochastic functional differential equations (SFDEs) driven by the G-Lévy process. It derives the existence-uniqueness theorem for solutions to SFDEs driven by the G-Lévy process. Moreover, it shows the error estimation between the exact solution x(t) and Picard approximate solutions xn(t),n≥1. Ultimately, the exponential estimate has been derived.

This paper is devoted to the study of a degenerating parabolic heat equation with nonlinearities of a general type in the presence of a source (sink) in the case of two spatial variables. The problem of initiating a heat wave propagating over a cold (zero) background with a finite velocity and a boundary condition specified on a moving manifold—a closed line—is considered. For this problem, a new existence and uniqueness theorem is proved, a numerical algorithm for constructing a solution based on the boundary element method, collocation method, and difference time approximation is proposed; a special change of variables of the hodograph-type transformation is used. New exact solutions to this equation in the case of power nonlinearities are found. A numerical algorithm is implemented, and a numerical experiment is carried out. A comparison of the constructed numerical solutions with exact ones (found both in this paper and earlier) showed good agreement. The numerical convergence in the time step and number of collocation points is proved.