Application of zero eigenvalue for solving the potential, heat, and wave equations using a sequence of special functions
Summary: In the solution of boundary value problems, usually zero eigenvalue is ignored. This case also happens in calculating the eigenvalues of matrices, so that we would often like to find the nonzero solutions of the linear system \(AX=\lambda X\) when \(\lambda\neq0\). But \(\lambda=0\) implies that \(\text{det} A=0\) for \(X\neq0\) and then the rank of matrix \(A\) is reduced at least one degree. This comment can similarly be stated for boundary value problems. In other words, if at least one of the eigens of equations related to the main problem is considered zero, then one of the solutions will be specified in advance. By using this note, first we study a class of special functions and then apply it for the potential, heat, and wave equations in spherical coordinate. In this way, some practical examples are also given.
scientific article; zbMATH DE number 5822515