We introduce Lipschitz classes \(\mbox{Lip}^M(\eta,2)\) and \(\mbox{Lip}^M(\eta,\infty)\) of functions associated with the canonical Sturm–Liouville operator \(\mathscr{L}^M\); and we prove a new versions of Titchmarsh and Boas-type theorems for the canonical Sturm–Liouville transform \(\mathscr{F}^M\). An application to the canonical Sturm–Liouville multipliers is given. Titchmarsh and Boas-type theorems for the canonical Fourier–Bessel transform and the canonical Fourier–Jacobi transform are special cases of this work.
An integrable version of SLIR (former SEIR) epidemiological model is introduced with the time dependence of the force of infection. The explicit formulas for special solutions are obtained for this model. From these formulas by classical analysis methods one can obtain the formulas for important metrics of spread of disease: maximum number of infectious people and their corresponding peak times. The usefulness of explicit formulas is illustrated by application to the spread of Covid-19. It is shown in the model example that the time dependence of the force of infection produces the bimodal dynamics of infectious people. Multimodal behavior of the spread of disease can be visualized in graphs of time behavior of numbers of Covid-19 infectious people (see https://www.worldometers.info/coronavirus). These multiple waves are explained biologically by different variants of Covid-19 virus. In previous SLIR models multimodal distribution of the number of infectious people appeared only in numerical simulations or under the assumption that there is an external periodic action.
Fractional ordinary differential equations modeling diffusion-like processes are important tools for describing systems where transport is unusual or depends on past behavior, such as in porous media, biological systems, or viscoelastic materials. Since exact solutions are rarely available, we rely on numerical methods to solve them. In this work, we introduce a spectral collocation method that uses Delannoy polynomials to approximate solutions of these equations involving the Caputo derivative. By building operational matrices for the Caputo derivative, the method directly constructs the algebraic system, which is then solved efficiently using Newton’s method. Tests show that the method is highly accurate, converges quickly, and is computationally efficient. These results suggest that Delannoy polynomials are a strong alternative to classical polynomial bases for spectral solutions of fractional differential problems.
In this paper, we present a numerical solution to the fractional differential equation governing water pollution. The Caputo derivative will be used to model water pollution. For the numerical solution, we propose a fractional numerical scheme and use it to generate figures with mathematical software. We analyze qualitative properties of the determination of equilibrium points and examine their stability. We have analyzed the influence of the order of the fractional derivative on modeling pollution problems.
Let \(G\) be a finite group. The product order relatively prime graph (PORP graph) \(\Gamma_{porp}(G)\) is the graph with vertex set \(V(\Gamma_{porp}(G))=G\) and two distinct vertices \(a\) and \(b\) are adjacent if \((o(a),o(ab))=1\) or \((o(b),o(ab))=1\). In this paper, we investigate the product order relatively prime graph associated with the dihedral group \(D_n\). We study the graph for two important classes of dihedral groups, namely \(D_{p^\alpha}\), where \(p\) is a prime and \(\alpha\ge1\), and \(D_{pq}\), where \(p\) and \(q\) are distinct odd primes. We determine the graph structure and establish several structural properties, including the degree sequence, total number of edges, independence number, clique number and structural decomposition. Furthermore, we investigate graph-theoretic properties such as planarity, bipartiteness, Eulericity, Hamiltonicity, and characterize the conditions under which the graph is complete.
