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In this article, we describe an approach for solving partial differentialequations with general boundary conditions imposed on arbitrarily shapedboundaries. A function that has a prescribed value on the domain in which adifferential equation is valid and smoothly but rapidly varying values on theboundary where boundary conditions are imposed is used to modify the originaldifferential equations. The mathematical derivations are straight forward, andgenerically applicable to a wide variety of partial differential equations. Todemonstrate the general applicability of the approach, we provide fourexamples: (1) the diffusion equation with both Neumann and Dirichlet boundaryconditions, (2) the diffusion equation with surface diffusion, (3) themechanical equilibrium equation, and (4) the equation for phase transformationwith additional boundaries. The solutions for a few of these cases arevalidated against corresponding analytical and semi-analytical solutions. Thepotential of the approach is demonstrated with five applications:surface-reaction diffusion kinetics with a complex geometry,Kirkendall-effect-induced deformation, thermal stress in a complex geometry,phase transformations affected by substrate surfaces, and a self-propellingdroplet.

Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries