Conditions for the existence of polynomial solutions of certain second-order differential equations have recently been investigated by several authors. In this paper a new algorithmic procedure is given to determine necessary and sufficient conditions for a differential equation with polynomial coefficients containing parameters to admit polynomial solutions and to compute these solutions. The ...

Maxwell’s equations admit both retarded and advanced solutions, but only the retarded solutions seem to characterize the radiative phenomena that actually occur in nature. The reason for this, however, is not usually explained in electrodynamics textbooks. We justify the use of the retarded solutions by considering a simple toy model, which allows us to avoid the technical complexities of elect...

We obtain an explicit representation for quasiradial γ-harmonic functions, which shows that these functions have essentially algebraic nature. In particular, we give a complete description of all γ which admit algebraic quasiradial solutions. Unlike the cases γ = ∞ and γ = 1, only finitely many algebraic solutions is shown to exist for any fixed |γ| > 1. Moreover, there is a special extremal se...

A modification is proposed to the equations of linear elasticity as used to deform Euler and Navier-Stokes meshes. In particular it is seen that the equations do not admit rigid body rotations as solutions, and it is shown how these solutions may be recovered by modifying the constitutive law. The result is significantly more robust to general deformations, and combined with incremental applica...

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although ...

We determine the Killing spinors for a class of magnetic brane solutions with Minkowski worldvolume of the theory of AdS Einstein Maxwell theories in d dimensions. We also obtain curved magnetic brane solutions with Ricci-flat worldvolumes. If we demand that the curved brane solution admits Killing spinors, then its worldvolume must admit parallel spinors. Classes of Ricci-flat worldvolumes adm...

This paper considers a general form of the porous medium equation with nonlinear source term: ut = (D(u)ux)x + F(u), n = 1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions...