Integral Equation Methods in Solving Problems of Elasticity and Potential Theory: A Unified Approach
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The employment of integral equation approaches has resulted in a revolution in the analytical and numerical treatment of boundary value concerns in the domains of elasticity and potential theory. This revolution has contributed to the advancement of these sciences. This research's objective is to offer a comprehensive account of the creation and use of integral formulations for the aim of resolving conventional concerns in a variety of fields of study. The article begins with basic similarities between linear elasticity and potential theory, and then on to cover major formulations within the field, such as Somigliana's identity and boundary integral approaches. An emphasis is placed on the benefits of reducing domain issues to boundary-only formulations, which provide computational efficiency, manage complicated geometries, and naturally integrate boundary conditions. These advantages are highlighted in the following sentence. There is a review of the significant contributions made by Betti, Somigliana, and Muskhelishvili, which is then followed by contemporary numerical implementations. In the last section of the paper, a discussion of difficulties and potential future paths is presented, with particular emphasis on nonlinear and three-dimensional situations.
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