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The narrow escape problem arises in deriving the asymptotic expansion of thesolution of an inhomogeneous mixed Dirichlet-Neumann boundary value problem. Inthis paper, we mainly deal with narrow escape problem in a smooth domainconnected to a long neck-Dendritic spine shape domain, which has a certainsignificance in biology. Since the special geometry of dendritic spine, wedevelop a new model for solving this narrow escape problem which isNeumann-Robin Boundary Model. This model transform spine singular domain tosmooth spine head domain by inserting Robin boundary condition to theconnection part between spine head and neck. We rigorously find the high-orderasymptotic expansion of Neumann-Robin Boundary Model and apply it to thesolution of narrow escape problem in a dendritic spine shape domain. Ourresults show that the asymptotic expansion of the Neumann-Robin Boundary Modelcan be easily applied to the narrow escape problem for any smooth spine headdomain with straight spine neck. By numerical simulations, we show that thereis great agreement between the results of our Neumann-Robin Boundary Model andthe original escape problem. In this paper, we also get some results fornon-straight long spine neck case by considering curvature of spine neck.

A New Model for Solving Narrow Escape Problem in Domain with Long Neck