Ford, NevilleRoberts, JasonRoberts, Jane N.2022-07-272022-07-272021-09-01Roberts, J. N. (2021). A new perspective on the numerical and analytical treatment of a certain singular Volterra integral equation [Unpublished doctoral thesis]. University of Chester.http://hdl.handle.net/10034/627044In this thesis, the focus of our attention is on a certain linear Volterra integral equation with singular kernel. The equation is of great interest due to the fact that, under certain conditions, it possesses an in finite family of solutions, out of which only one has C1-continuity. Numerous previous studies have been conducted and a variety of solution methods proposed. However, the emphasis has invariably been on determining just the differentiable solution. Thus, a significant gap in the research relating to this equation was identified and, therefore, our main objective here was to develop an effective solution method that allows us to approximate any chosen solution out of the infinite solution set. To this end, we converted the original integral equation into a singular differential form. Then, by applying a combination of analytical results from functional and real analysis, measure theory and the theory of Lebesgue integration, we reduced the problem to that of solving a regular initial value problem. Numerical methods were then applied and our experimental results proved that our method was highly effective, producing very accurate approximations to the true solution in a comparative study. Therefore, we feel our work here makes a significant contribution in this field of study, both from a theoretical viewpoint, as during the course of our research we established a direct relationship between the non-smooth solutions of the integral equation and the weak solutions of our differential scheme, and in practice. Integral equations of this form arise in the study of heat conduction, diffusion and in thermodynamics. Therefore, another of our aims was to construct a method that could readily be applied in 'real world' modelling. Thus, as traditional models most often present as differential equations and, furthermore, as our method significantly simplifies the process of computing the solutions, we believe we have achieved this objective. Hence, in the final chapter, we highlight some of the ways in which our method could be adopted in order to help solve some of today's most challenging problems.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Volterra integral equationThesis or dissertation2023-02-04Recommended 6 month embargoThe full-text may be used and/or reproduced in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-profit purposes provided that: - A full bibliographic reference is made to the original source - A link is made to the metadata record in ChesterRep - The full-text is not changed in any way - The full-text must not be sold in any format or medium without the formal permission of the copyright holders. - For more information please email researchsupport.lis@chester.ac.uk