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Udita N. Katugampola

Publications and source records attributed to Udita N. Katugampola.

12 recordsLinked to original sources
arXiv2026

On the Gap Structure of Generalized Stirling Numbers

Katugampola's 2015 study of generalized fractional differential operators produced triangular arrays of integer coefficients indexed by a fractional order r and by dimensions n and k, but no combinatorial interpretation has been established for any fractional order. We give the first such interpretation, with two main results: (i) a complete combinatorial interpretation for r = 1/2 and n = 1,2,3, and (ii) a rigorous proof that this interpretation cannot extend to n >= 4 within the same framework. For n = 1,2,3, we show that the coefficients for r = 1/2 count binary sequences satisfying two conditions: they contain at least one symbol B, and they have gap <= 1, where the gap is the distance between the first and last occurrence of B. Each sequence is assigned a type k by a parity-dependent rule involving the gap value, and exhaustive enumeration matches Katugampola's coefficients exactly. We then prove an obstruction theorem showing that the gap <= 1 condition forces any such model to produce at most two distinct types per row, whereas Katugampola's array requires at least three types for every n >= 4. Thus the gap <= 1 binary-sequence interpretation works if and only if n = 1,2,3. Our results turn a computational observation into a rigorous impossibility theorem and provide guidance for future attempts to obtain complete combinatorial interpretations of fractional-calculus coefficients.

math.CO↗
arXiv2020

Finding Eigenvectors: Fast and Nontraditional Approach

Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first find the eigenvalues and then find the corresponding eigenvectors. We show that we do not need the second step when diagonalizing matrices with a spectrum, $\left|σ(A)\right|\leq 2$ since those vectors already appear as nonzero columns of the $\textit{eigenmatrices}$, a term defined in this work. We further generalize this for matrices with $\left|σ(A)\right|> 2$ and show that eigenvectors lie in the column spaces of eigenmatrices of the complementary eigenvalues, an approach without using the classical Gauss-Jordan elimination of rows of a matrix. We introduce two major results, namely, the $\textit{2-Spectrum Lemma}$ and the $\textit{Eigenmatrix Theorem}$. As a conjecture, we further generalize the Jordan canonical forms for a new class of generalized eigenvectors that are produced by repeated multiples of certain eigenmatrices. We also provide several shortcut formulas to find eigenvectors that does not use echelon forms. The method discussed in this work may be summarized with the mnemonic "Find your puppy at your neighbors'!" argument, where puppy is the eigenvector and the neighbors are the complementary eigenmatrices.

math.HO↗
arXiv2016

New fractional integral unifying six existing fractional integrals

In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erdélyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^ρ\mathcal{I}^{α, β}_{a+;η, κ}f\right)(x)=\frac{ρ^{1-β}x^κ}{Γ(α)}\int_a^x \frac{τ^{ρη+ρ-1}}{(x^ρ-τ^ρ)^{1-α}}f(τ)\text{d}τ, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erdélyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.

math.CA↗
arXiv2016

Applications of fractional calculus in solving Abel-type integral equations: Surface-volume reaction problem

In this paper we consider a class of partial integro-differential equations of fractional order, motivated by an equation which arises as a result of modeling surface-volume reactions in optical biosensors. We solve these equations by employing techniques from fractional calculus; several examples are discussed. Furthermore, for the first time, we encounter an order of the fractional derivative other than $\frac{1}{2}$ in an applied problem. Hence, in this paper we explore the applicability of fractional calculus in real-world applications, further strengthening the true nature of fractional calculus.

math.CA↗
arXiv2016

Hermite-Hadamard and Hermite-Hadamard-Fejér type Inequalities for Generalized Fractional Integrals

In this paper we obtain the Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for fractional integrals which generalize the two familiar fractional integrals namely, the Riemann-Liouville and the Hadamard fractional integrals into a single form. We prove that, in most cases, we obtain the Riemann--Liouville and the Hadamard equivalence just by taking limits when a parameter $ρ\rightarrow 1$ and $ρ\rightarrow 0^+$, respectively.

math.CA↗
arXiv2016

Existence and Uniqueness results for a class of Generalized Fractional Differential Equations

The author (Bull. Math. Anal. App. 6(4)(2014):1-15), introduced a new fractional derivative, \[{}^ρ\mathcal{D}_a^αf (x) = \frac{ρ^{α-n+1}}{Γ({n-α})} \, \bigg(x^{1-ρ} \,\frac{d}{dx}\bigg)^n \int^x_a \frac{τ^{ρ-1} f(τ)}{(x^ρ- τ^ρ)^{α-n+1}}\, dτ\] which generalizes two familiar fractional derivatives, namely, the Riemann-Liouville and the Hadamard fractional derivatives to a single form. In this paper, we derive the existence and uniqueness results for a generalized fractional differential equation governed by the fractional derivative in question.

math.CA↗
arXiv2014

A New Fractional Derivative with Classical Properties

We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value Theorem. The definition, \[ D^α(f)(t) = \lim_{ε\rightarrow 0} \frac{f(te^{εt^{-α}}) - f(t)}ε, \] is the most natural generalization that uses the limit approach. For $0\leq α< 1$, it generalizes the classical calculus properties of polynomials. Furthermore, if $α= 1$, the definition is equivalent to the classical definition of the first order derivative of the function $f$. Furthermore, it is noted that there are $α-$differentiable functions which are not differentiable.

math.CA↗
arXiv2014

Mellin Transforms of the Generalized Fractional Integrals and Derivatives

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized $δ_{r,m}$ operators with generalized Stirling numbers and Lah numbers. For example, we show that $δ_{1,1}$ corresponds to the Stirling numbers of the $2^{nd}$ kind and $δ_{2,1}$ corresponds to the unsigned Lah numbers. Further, we show that the two operators $δ_{r,m}$ and $δ_{m,r}$, $r,m\in\mathbb{N}$, generate the same sequence given by the recurrence relation \[ S(n,k)=\sum_{i=0}^r \big(m+(m-r)(n-2)+k-i-1\big)_{r-i}\binom{r}{i} S(n-1,k-i), \;\; 0< k\leq n, \] with $S(0,0)=1$ and $S(n,0)=S(n,k)=0$ for $n>0$ and $1+min\{r,m\}(n-1) < k $ or $k\leq 0$. Finally, we define a new class of sequences for $r \in \{\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, ...\}$ and in turn show that $δ_{\frac{1}{2},1}$ corresponds to the generalized Laguerre polynomials.

math.CA↗
arXiv2014

A New Approach to Generalized Fractional Derivatives

The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, \[ \big({}^ρ\mathcal{I}^α_{a+}f\big)(x) = \frac{ρ^{1- α}}{Γ(α)} \int^x_a \frac{τ^{ρ-1} f(τ) }{(x^ρ- τ^ρ)^{1-α}}\, dτ, \] which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.

math.CA↗
arXiv2010

A New Technique for Text Data Compression

In this paper we use ternary representation of numbers for compressing text data. We use a binary map for ternary digits and introduce a way to use the binary 11-pair, which has never been use for coding data before, and we futher use 4-Digits ternary representation of alphabet with lowercase and uppercase with some extra symbols that are most commonly used in day to day life. We find a way to minimize the length of the bits string, which is only possible in ternary representation thus drastically reducing the length of the code. We also find some connection between this technique of coding dat and Fibonacci numbers.

cs.CR↗
arXiv2010

New Approach To A Generalized Fractional Integral

The paper presents a new formula for the fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form, which when a parameter fixed at different values, produces the above integrals as special cases. Conditions are given for such a generalized fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. Finally, we give a general definition of the Fractional derivatives.

math.CA↗