Numerical methods for nonlinear singulary perturbed differential equations
Quinn, Jason (2012) Numerical methods for nonlinear singulary perturbed differential equations. PhD thesis, Dublin City University.
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In this thesis, parameter-uniform numerical methods are constructed and analysed for nonlinear singularly perturbed ordinary differential equations. In the case of first order problems, several classes of nonlinear problems are examined and various types of initial layers are identified. In the case of second order boundary value problems, singularly perturbed quasilinear problems of convection-diffusion type are studied. Problems with boundary turning points and problems with internal layers are examined. For all problems, the numerical methods consist of monotone nonlinear finite difference operators and appropriate piecewiseuniformShishkin meshes. The transition points in the Shishkin meshes are constructed based on sharp parameter-explicit bounds on the singular components of the continuous solution. Existence and uniqueness of both the continuous and discrete solutions are established using the method of upper and lower solutions. Numerical results are presented to both illustrate the theoretical error bounds and to display the performance of the numerical methods in practice.
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