An Initial-Boundary Value Problem With New Technique of Piecewise Uniform Mesh

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