Title: Numerically Solving Partial Differential Equations Using Series Solutions, and Least Squares Methods

Authors: James E. Brown

DOI: 10.33599/nasampe/s.24.0184

Abstract: "In 2008, Dr Frank Abdi and the author began collaboration on the above topic. This method has similarities to classical series solution techniques. Select an appropriate series solution for the PDE, then differentiate the solution as needed for insertion into the partial differential equation (PDE) and boundary conditions (BCs). Sample the PDE and the BCs, at sufficiently many points to insure a unique solution. Each sampling point provides an equation, in terms of the series coefficients. Assemble the equations into a set of overdetermined equations. Finally, solve the equations using a Least Squares solver. "

References: [1] Mortari, Daniele., ”Least Squares Solution of Linear Differential Equations”, Proceedings of 27th AAS/AIAA Space Flight Mechanics Meeting Conference. San Antonia, Texas, USA, 5-9 February (2017). [2] Ibrahim, Salisu., “Discrete Least Square Method for Solving Differential Equations”, Advances and Applications in Discrete Mathematics, (2022): volume 30, pages 87-102 [3] Kazemi, Katayoun., “Solving Differential Equations with Least Square and Collocation Methods”., UNLV Theses, Dissertations, Professional Papers and Capstones (2015). [4] Eason, Ernest., “A Review of Least Squares Methods for Solving Partial Differential Equations”., International Journal for Numerical Methods in Engineering, volume 10, pages 1021-1046 (1976). [5] Timoshenko, Stephen., ‘Theory of Elasticity”, page 282, third edition., McGraw-Hill, 1970. [6] Haukaas, Terje.,”3D Elasticity Theory”., civil-terje_sites.olt.ubc.ca.,Jan27,2020, page 6. The University of British Columbia, Vancouver. [7] Howland, R. C. J., “On The Stresses in the Neighborhood of a Circular Hole in a Strip Under Tension’., Philosophical Translations of the Royal Society of London., June 20, 1929., Series A, volume 229, pages 49-86. [8] Timoshenko, Steven., ‘Theory of Elasticity”, pages 66 and 76-77, third edition., McGraw-Hill, 1970.

Conference: SAMPE 2024

Publication Date: 2024/05/20

SKU: TP24-0000000184

Pages: 15

Price: $30.00