Document Type : Research Paper


Civil Engineering Department, University of Garmian, Kalar City, Iraq


The objective of my research is to establish facts and determine their significance. A new ε-convergent piecewise uniform mesh has been produced, by deriving a hybrid technique to find out the extent of subdomains (τ) of the singular boundary layers that occur when solving some of the differential equation problems numerically, where ε, is set to multiply terms covering the highest derivatives in the differential equation, in which determinant is zero, these boundary layers are adjacent to the boundary of the domain, where the solution yields a very deep gradient. The mesh has been used with the difference scheme function code in the MATLAB program; specifically, PDEPE that is solving initial-boundary value problems pertained parabolic-elliptic PDEs. It was applied to solve multiple examples then comparing the maximum error of the solutions with its counterpart "uniform mesh" and proving its superiority. Results, solutions, and comparisons were exposed with concise explanatory MATLAB plots manifested in some necessary tables for comparative studies.


Main Subjects

  • I. Shishkin, 1989, Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer, USSR Computational Mathematics and Mathematical Physics. 29(4), 1-10.
  • J. H. Miller et al., 1998, Fitted mesh methods for problems with parabolic boundary layers, Mathematical Proceedings of the Royal Irish Academy, 98A (2), 173-190, Royal Irish Academy.
  • Erich Zauderer, 2006, Partial Differential equation of Applied Mathematics , Third Edition, Wiley Interscience, A John & Sons, INC., Publication.
  • Torsten Linb, 2010, Layer-Adapted Meshes For Reaction-Convection-Diffusion Problems, Springer.
  • [[1]] Louise Olsen-Kettle , 2011, Numerical solution of partial differential, equations, Lecturer book, ISBN: 978-1-74272-149-1, The University of Queensland School of Earth Sciences Centre for Geoscience Computing.
  • R.Branco, J.A.Ferreira, 2008, A singular perturbation of the heat equation with memory, Journal of Computational and Applied Mathematics, 218(2), PP 376-394.
  • Zhou, Shuang-Shuang, et al., 2021, Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, AIMS Mathematics 6.11 (2021): 12114-12132.
  • Ashurov, R. R., and O. T. Muhiddinova., 2021, Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator, Lobachevskii Journal of Mathematics 42.3 (2021): 517-525.
  • Goatin, Paola, and Alexandra Würth., 2023, The initial boundary value problem for second order traffic flow models with vacuum: existence of entropy weak solutions., Nonlinear Analysis 233 (2023): 113295.
  • Javidi, Mohammad, and Mahdi Saedshoar Heris., 2023, New numerical methods for solving the partial fractional differential equations with uniform and non-uniform meshes, The Journal of Supercomputing (2023): 1-32.
  • Yuan, Wenping, Hui Liang, and Yanping Chen., 2023, "On the convergence of piecewise polynomial collocation methods for variable-order space-fractional diffusion equations." Mathematics and Computers in Simulation 209 (2023): 102-117.
  • I. Shishkin, 1990, Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations. Second doctorial thesis, Keldysh Institute, Moscow, Russian.
  • B. Thomas, M. D. Weir, and J. Hass, 2006, Thomas' Calculus, Edition 12, Addison –Wesley, Pearson.
  • Martin Stynes, Eugene O’Riordan, 1997, A Uniformly Convergent Galerkin Method on a Shishkin Mesh for a Convection-Diffusion Problem, Journal of Mathematical Analysis and Applications, , 214(1), pp. 36-54.
  • R.D. and M. Berzins, 1990, A method for the spatial discretization of parabolic equations in one space variable, SIAM journal on Scientific and statistical computing, vol.11, pp. 1-32.
  • Jichun Li and Yi-Tung Chen, 2008, Computational Partial Differential Equation Using MATLAB, CRC Press, A Chapman & Hall Book, Page30.