## WSEAS Transactions on Mathematics

Print ISSN: 1109-2769, E-ISSN: 2224-2880## Volume 23, 2024

### In-depth Study of Eigenvalues in a Boundary Value Problem for a given Partial *q*-Differential Equation

**Authors:**Elisabeta Peti, Enkeleda Zajmi Kotonaj, Vera Hoxha

**Abstract:**In this paper, we are focused on studying a boundary values problem of the second-order differential equation of Euler-type in the classical version of Calculus, given by the following expression: $$u{_{xx}}= f \left ( {\frac{αx+βy+a}{γx+δy+b}}\right )u{_{yy}},α,β,γ,δ,a,b∈\mathbb{R} $$ (1) and includes the above boundary conditions: $$u(0, y) = u(N, y) = 0. $$ (2) Firstly, we have proposed the construction of a new function $$f$$ with the intention of transforming the equation (1) into an Euler-type equation. Since all of these problems are too difficult to solve in Classical Calculus, this study aims to convert them into equations of this type for the ease of study in $$q$$-Calculus. Then, we proposed a transformation method for both the equation and the boundary conditions. Thus, the boundary value problem consists of a second-order partial differential equation and boundary conditions dependent on the eigenvalues. By using the procedure of $$q$$-difference over a time scale $$\mathbb{T}_{q}$$, we obtained a second-order Euler $$q$$-difference equation with Dirichlet boundary conditions. Also, we have analyzed the exact number of eigenvalues for all cases that arise from the study of our problem. Here, we have presented three theorems, two of which show the correct number of eigenvalues of two issues with eigenvalues derived from our principal problem. The last one shows a relation between two eigenvalues of a problem for $$\mathbb{T}_{q}^{\mathbb{N}_{0}}$$ and $$\mathbb{T}_{q}^{\mathbb{-N}_{0}}$$. We also have given some examples that prove the above conclusions.

**Keywords:**Time Scale, Quantum Calculus, $$q$$-Differential Equation, $$q$$−Difference Equation, Eigenvalues, Eigenfunctions

**Pages:**340-350

**DOI:**10.37394/23206.2024.23.37