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Keywords = Goursat functions

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21 pages, 5020 KB  
Article
Influence of Heat Transfer on Stress Components in Metallic Plates Weakened by Multi-Curved Holes
by Faizah M. Alharbi and Nafeesa G. Alhendi
Axioms 2025, 14(5), 369; https://doi.org/10.3390/axioms14050369 - 14 May 2025
Cited by 1 | Viewed by 779
Abstract
This manuscript addresses an application study by employing a mathematical model of a thermoelastic plate weakened by multi-curved holes under the effect of stress forces in the presence of heat conduction. When the initial heat flow is directed to the plate system, complex [...] Read more.
This manuscript addresses an application study by employing a mathematical model of a thermoelastic plate weakened by multi-curved holes under the effect of stress forces in the presence of heat conduction. When the initial heat flow is directed to the plate system, complex variable procedures are used to compute the basic Goursat functions, taking into account the time-dependent variables through conformal mapping, which transfers the domain to the exterior of a unit circle. The problem reduces to a general form of a contact problem in two dimensions, which is called an integrodifferential equation of the second type with the Cauchy kernel. Additionally, different hole shapes are generated using Maple 2023. Computational simulations are performed to determine the normal and shear stress components in the presence and absence of heat effects at various times. Furthermore, numerical calculations of Goursat functions are carried out and graphically displayed for some specific materials. This investigation provides valuable information about industries, such as those regarding ceramic tile, glass, rubber, paint, ceramic pigment, and metal alloys. Full article
(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences, 2nd Edition)
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29 pages, 1790 KB  
Review
Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances
by Markus Scholle, Florian Marner and Philip H. Gaskell
Water 2020, 12(5), 1241; https://doi.org/10.3390/w12051241 - 27 Apr 2020
Cited by 18 | Viewed by 7768
Abstract
The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex [...] Read more.
The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature. Full article
(This article belongs to the Special Issue Physical and Mathematical Fluid Mechanics)
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12 pages, 623 KB  
Article
An Infinite Plate with a Curvilinear Hole and Flowing Heat
by A.S. Sabbah, M.A. Abdou and A.S. Ismail
Math. Comput. Appl. 2004, 9(2), 321-332; https://doi.org/10.3390/mca9020321 - 1 Aug 2004
Viewed by 1514
Abstract
In the present paper, we apply the complex variable method (Cauchy method) to derive exact expressions for the Goursat's functions for the boundary value problem of infinite plate weakened by a curvilinear hole. The hole is conformally mapped on the domain outside a [...] Read more.
In the present paper, we apply the complex variable method (Cauchy method) to derive exact expressions for the Goursat's functions for the boundary value problem of infinite plate weakened by a curvilinear hole. The hole is conformally mapped on the domain outside a unit circle by means of a general rational mapping function. Also the stress components, when an initial heat is uniformly flowing in the perpendicular direction of the hole, are obtained, Some applications are investigated. The interesting cases, when the shape of the hole takes different famous formulas are included as special cases, The work of many previous authors can be considered as special cases of this work. Full article
12 pages, 612 KB  
Article
Thermoelastic Problem for an Infinite Plate with a Curvilinear Hole Having Finite Poles
by M.A. Abdou and F.A. Salama
Math. Comput. Appl. 2004, 9(2), 237-248; https://doi.org/10.3390/mca9020237 - 1 Aug 2004
Viewed by 1450
Abstract
In the present paper, Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Goursat's functions for the boundary value problems of the infinite plate weakened by a hole having many poles and arbitrary shape which [...] Read more.
In the present paper, Muskhelishvili's complex variable method of solving two-dimensional elasticity problems has been applied to derive exact expressions for Goursat's functions for the boundary value problems of the infinite plate weakened by a hole having many poles and arbitrary shape which is conformally mapped n the dorhain outside a unit circle by means of general rational mapping function. Also the components of stress are obtained, when a stationary heat is flowing uniformly in the perpendicular direction of the hole. Some applications are considered. The interesting cases when the shape of the hole takes different shapes are included as special cases. Full article
8 pages, 298 KB  
Article
Goursat Problem for the Factorizable Hyperbolic Equation in Two Independent Variables
by I K. Johnpillai and F M. Mahomed
Math. Comput. Appl. 2003, 8(1), 55-62; https://doi.org/10.3390/mca8010055 - 1 Apr 2003
Cited by 2 | Viewed by 1550
Abstract
For the scalar linear hyperbolic partial differential equations (PDEs) in two independent variables to be factorizable, the Laplace invariants h or k must be zero. In this paper, we find the Riemann function for the Goursat problem using the Lie group theoretical method [...] Read more.
For the scalar linear hyperbolic partial differential equations (PDEs) in two independent variables to be factorizable, the Laplace invariants h or k must be zero. In this paper, we find the Riemann function for the Goursat problem using the Lie group theoretical method where the hyperbolic . equation involved is factorized. What emerges is that the ordinary differential equation (ODE) whose solution gives the Riemann function for the Goursat problem is factorizable. Finally, an example is given as application of the methods Full article
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