The Department of Mathematics at the College of Education for Pure Sciences organized a seminar titled "A Sequential Analytical Approach for Reconstructing Cortical Potential in Electroencephalography (EEG)."
The seminar, presented by Assistant Professor Dr. Faiza Lafta Hassan, explored a fully analytical method for solving the inverse problem in electroencephalography (EEG) using a two-layered, heterogeneous, circular-annular head model. The ill-posed Cauchy problem for the Laplace equation within a multi-layered annular domain is addressed—bypassing numerical domain discretization, optimization techniques, or regularization—by applying the Homotopy Perturbation Method (HPM) sequentially across each tissue layer.
The primary mathematical contribution is the derivation of a closed-form series representation (Theorem 3.1), which expresses the internal electrical potential entirely in terms of Cauchy data on the outer boundary and its angular derivatives. Furthermore, the study establishes conditional convergence within the Sobolev space $H^1$ and provides a quantitative estimate of the truncation error for analytical data whose domain of analyticity exceeds the depth of logarithmic continuity. Additionally, the transmission conditions for electrical conductivity are incorporated analytically. Five numerical experiments demonstrate the method's accuracy when utilizing data consisting of a finite number of Fourier modes and specific noise levels, though it is noted that unconditional stability is not claimed for all types of noisy data

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