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The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Migraine is a common neurological disease with a still-unclear etiology and pathogenesis. Patent foramen ovale (PFO) is a kind of congenital heart disease that leads to a right-to-left shunt (RLS). Although previous studies have shown that PFO has an effect on migraine, a clear conclusion about the link between PFO and migraine is lacking. We first summarized the PFO potential mechanisms associated with migraine, including microembolus-triggered cortical spreading depression (CSD), the vasoactive substance hypothesis, impaired cerebral autoregulation (CA), and a common genetic basis. Further, we analyzed the changes in brain structure and function in migraine patients and migraine patients with PFO. We found that in migraine patients with PFO, the presence of PFO may affect the structure of the cerebral cortex and the integrity of white matter, which is mainly locked in subcortical, deep white matter, and posterior circulation, and may lead to changes in brain function, such as cerebellum and colliculus, which are involved in the processing and transmission of pain. In summary, this paper provides neuroimaging evidence and new insights into the correlation between PFO and migraine, which will help to clarify the etiology and pathogenesis of migraine, and aid in the diagnosis and treatment of migraine in the future.