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The helm graph H_n is the graph obtained from an n-wheel graph by adjoining a pendant edge at each node of the cycle. Helm graphs are graceful (Gallian 2018), with the odd ...

An elliptic partial differential equation given by del ^2psi+k^2psi=0, (1) where psi is a scalar function and del ^2 is the scalar Laplacian, or del ^2F+k^2F=0, (2) where F ...

In bipolar coordinates, the Helmholtz differential equation is not separable, but Laplace's equation is.

The Helmholtz differential equation is not separable in bispherical coordinates.

In two-dimensional Cartesian coordinates, attempt separation of variables by writing F(x,y)=X(x)Y(y), (1) then the Helmholtz differential equation becomes ...

In cylindrical coordinates, the scale factors are h_r=1, h_theta=r, h_z=1, so the Laplacian is given by del ...

As shown by Morse and Feshbach (1953), the Helmholtz differential equation is separable in confocal paraboloidal coordinates.

In conical coordinates, Laplace's equation can be written ...

In elliptic cylindrical coordinates, the scale factors are h_u=h_v=sqrt(sinh^2u+sin^2v), h_z=1, and the separation functions are f_1(u)=f_2(v)=f_3(z)=1, giving a Stäckel ...

As shown by Morse and Feshbach (1953) and Arfken (1970), the Helmholtz differential equation is separable in oblate spheroidal coordinates.