Parameter Uniformly Convergent Numerical Schemes for Time-Fractional Singularly Perturbed Convection-Diffusion Equations

Habtamu, Getachew

dc.contributor.author	Habtamu, Getachew
dc.date.accessioned	2026-08-18T10:11:00Z
dc.date.available	2026-08-18T10:11:00Z
dc.date.issued	2025-12
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/17020
dc.description.abstract	In this dissertation, different parameter uniformly convergent numerical methods are proposed to solve time-fractional singularly perturbed convection-diffusion equations with/without delay in the time variable. The time-fractional derivative is given in the sense of Caputo with order α ∈ (0, 1). These types of model problems have wide applications in various branches of science and engineering, which are characterized by a small perturbation parameter ϵ (0 < ϵ << 1) that multiplies the diffusion term. The solution exhibits a regular boundary layer near the right side of the spatial domain as the perturbation parameter ϵ → 0. This means that the solution varies rapidly in the boundary region and behaves smoothly away from the layer region. Due to this multiscale property of the solution, standard numerical methods give oscillatory solutions on a uniform mesh, which are not typical properties of the solution of the problems, unless the step sizes must be smaller than the perturbation parameter, which is computationally expensive. To overcome this difficulty, parameter uniformly convergent numerical methods are needed. Hence, the aim of this dissertation is to develop simple, more accurate, and parameter uniformly convergent numerical methods for the mathematical problems considered. The L1, L2 − 1σ, and Crank-Nicolson schemes are used to discretize the model problems in the temporal direction, and the fitted operator, fitted mesh, and hybrid methods are used to discretize the model equations in the spatial direction. The matrix inversion method is used to solve the resulting linear system. The stability and convergence analysis of the developed methods are the immediate consequence of the discrete maximum principle. The parameter uniform convergence analysis of the developed methods is discussed in the maximum norm. To validate the developed methods, different numerical examples are considered. The results are presented with various graphs and tables. The results support the theoretical findings. Moreover, comparisons are made with some of the existing methods in the literature, and it is observed that the proposed schemes in this dissertation outperform in terms of accuracy and order of convergence. Finally, the suggested numerical methods are simple, accurate, and parameter uniformly convergent.	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Parameter Uniformly Convergent Numerical Schemes for Time-Fractional Singularly Perturbed Convection-Diffusion Equations	en_US
dc.type	Dissartation	en_US