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In this article, the nonclassical transient heat propagation process in a cracked strip is investigated by Guyer–Krumhansl (G–K) model, which incorporates both the time lagging behavior and the spatially nonlocal effect. The impulsive thermal loading as well as cyclic loading exerted on the top bounding surface are examined to explore the non-Fourier thermal characteristics. By means of the Laplace transform and Fourier transform, the governing partial differential equations subjected to mixed boundary conditions are converted to a group of singular integral equations. With the aid of numerical Laplace inversion, the transient temperatures are calculated to make comparisons of thermal responses determined by Fourier's law, Cattaneo–Vernotte (C–V) equation, and G–K model. The numerical results display the specific thermal behaviors of G–K model in the cracked medium and demonstrate the G–K model's capabilities in eliminating the unrealistic phenomena accompanied by C–V equation. Our research would contribute to achieving a better understanding of the transient heat conduction in small-sized systems or composites at the macroscopic scale. [ABSTRACT FROM AUTHOR]

The Dirichlet problem for a multidimensional elliptic system in the half-space is considered. With the help of the Fourier transform, the problem of the solvability of the problem is reduced to the study of a second-order partial differential equation. [ABSTRACT FROM AUTHOR]

For the fourth-order partial differential equation proposed by Fushchich for mathematical description of heat and mass transfer processes, the class of uniqueness and the class of correctness of the Cauchy problem have been established. Using the Fourier transform, a solution to the Cauchy problem is found in an explicit analytical form. It is shown that the new equation does not improve the properties of the classical heat conduction equation, but it allows one to describe heat transfer processes taking into account relaxation effects within the framework of the Galileo-invariant mathematical theory of transfer. [ABSTRACT FROM AUTHOR]

New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

This paper studies thermo-hydro-mechanical (T-H-M) coupled responses of layered transversely isotropic (TI) saturated media under moving thermal and mechanical loads. Firstly, governing equations including dynamic balance equations, T-H-M coupled constitutive equations, and generalized energy equations are established based on the thermodynamics and poroelastic mechanics. Next, the Fourier transform technology is utilized to transform complex coupled partial differential equations into ordinary ones in the wavenumber domain. With the help of a moving coordinate system, the key matrix equation of the moving load problem is derived. Then, combining with boundary, external load and interlayer continuity conditions, the extended precise integration method (PIM) is used to derive the solution in the wavenumber domain. After using the Gauss-Legendre numerical integration technique, the physical solution is obtained by programming on the MATLAB platform. Finally, numerical examples are designed to verify the proposed method, and further explore the influences of thermal loads, TI permeability and stratification on the T-H-M response. [ABSTRACT FROM AUTHOR]