The new proof broadly consists of three steps: derive the macroscopic theory from the mesoscopic one; derive the mesoscopic theory from the microscopic one; and then stitch them together in a single ...

Unlike polynomials, this form results in unique graphs that feature ... These intercepts are also known as the solutions or roots of the equation. The \(y\)-intercept: the point where the graph ...

The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified ...

Finding exact analytic solutions to the partial equations is one of the most challenging problems in mathematical physics. Generally speaking, the exact solution to many categories of such equations ...