Mathematics

Research Interests:

I work on numerical methods for nonlinear singularly perturbed differential equations. A short explanation: singularly perturbed D.E's are equations in which the highest derivative is multiplied by an arbitrary small quantity (perturbation parameter). Singularly perturbed problems arise in various disciplines such as option pricing in financial mathematics, Computational fluid dynamics and Semiconductor design for example. Solutions typically exhibit steep gradients, inversely proportional to the perturbation parameter. The steep gradient may appear on the boundary or in the interior of the problem domain (boundary\interior layers). Software packages have `difficulties' with numerically solving these problems. They regularly return oscillatory solutions, exacerbated as the perturbation parameter vanishes. Thus we must construct numerical methods which converge independently of the perturbation parameter (parameter uniform methods). I am currently working on nonlinear singularly perturbed interior layer problems. For problems where the layer location is known e.g. at the boundary or in the interior (in the case of linear problems say), a usual approach is to apply a fine Shiskin mesh on the layer region in a numerical algorithm. However, in general, the location of the interior layer for nonlinear problems is unknown. Although approximations of the location may exist, these approximations are not immediately justifiable in a numerical algorithm. My current interest is in investigating the requirements for such approximations to be successfully used in proposed numerical algorithms.

Selected Peer Reviewed Journals

• O'Riordan E., Quinn J. 2012. Parameter-uniform Numerical Methods for some Singularly Perturbed Nonlinear Initial Value Problems. Numerical Algorithms, 61, 4, pp579-611.
• O'Riordan E., Quinn J. 2012. A Singularly Perturbed Convection Diffusion Turning Point Problem with an Interior Layer. Computational Methods In Applied Mathematics, 12, 2, pp206-220.
• O'Riordan, E., Quinn J. 2011. Paramete-uniform Numerical Methods for some Linear and Nonlinear Singularly Perturbed Convection Diffusion Boundary Turning Point Problems. Bit Numerical Mathematics, 51, 2, pp317-337.

Selected Non-peer Reviewed Journals

• Dr. Jason Quinn. .

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• O'Riordan E., Quinn J., Numerical Method for a Nonlinear Singularly Perturbed Interior Layer Problem, In: C. Clavero, J.L. Gracia, F. J. Lisbona ed.BAIL 2010 - Boundary and Interior Layers, Computational and Asymptotic Mathods, 05-JUL-10 - 09-JUL-10, Zaragoza, Spain, 187 - 195