Disciplines
Ordinary Differential Equations and Applied Dynamics | Partial Differential Equations
Abstract (300 words maximum)
We investigate the properties of the eigenvalues of the fractal Laplacian. We begin by defining the fractal Laplacian operator in one dimension and formulate the corresponding Dirichlet eigenvalue problem. Analytical solutions are obtained for specific fractal parameters, and computational results illustrate the structure of eigenvalues and their associated eigenfunctions. We extend our analysis to two dimensions using separation of variables. Our findings contribute to a deeper understanding of how fractal geometry affects the spectral characteristics of differential operators.
Academic department under which the project should be listed
CSM - Mathematics
Primary Investigator (PI) Name
Eric Stachura
Included in
Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons
Properties of Eigenvalues of the Fractal Laplacian
We investigate the properties of the eigenvalues of the fractal Laplacian. We begin by defining the fractal Laplacian operator in one dimension and formulate the corresponding Dirichlet eigenvalue problem. Analytical solutions are obtained for specific fractal parameters, and computational results illustrate the structure of eigenvalues and their associated eigenfunctions. We extend our analysis to two dimensions using separation of variables. Our findings contribute to a deeper understanding of how fractal geometry affects the spectral characteristics of differential operators.