Issues: volume-1 | issue-1 | July 2026
A MODIFIED NEWTON–KRYLOV APPROACH FOR SOLVING NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Abstract:
Nonlinear fractional differential equations (FDEs) arise routinely in viscoelasticity, anomalous diffusion, and
control models, yet their discrete counterparts produce Jacobians that are dense, ill-conditioned, and expensive
to invert as the fractional order departs from unity. This paper reports an empirical evaluation of a modified
Newton–Krylov (mNK) scheme in which the linear correction step is solved using a preconditioned GMRES
iteration built on a shifted Grünwald Letnikov operator, rather than a full Newton step recomputed at every
outer iteration. Five benchmarks nonlinear FDEs, spanning fractional orders α between 0.3 and 1.8 and drawn
from Caputo and Riemann–Liouville formulations, were discretised using an L1/L2 scheme and solved on mesh
sizes ranging from 64 to 2048 nodes. Outer-iteration counts, residual decay, wall-clock time, and memory
footprint were recorded and benchmarked against classical Newton, Newton GMRES without the fractional
preconditioner, and an Adomian decomposition baseline. The modified scheme reduced average outer iterations
by 34–47% and CPU time by roughly 40% relative to unpreconditioned Newton–Krylov, with the advantage
widening as α approached the diffusion-dominated regime near 1.8, where classical Newton frequently
stagnated. Error norms remained within 10⁻⁶ to 10⁻⁸ across all test cases. These findings support the abstract's
central claim that coupling a fractional-order-aware preconditioner with inexact Newton iteration yields a
numerically robust and computationally economical route to nonlinear FDE solutions, a conclusion the paper
returns to and substantiates further in its closing discussion.
Authors:
Damini Yadav1
ALGEBRAIC STRUCTURES IN GENERALIZED FIBONACCI SEQUENCES OVER FINITE COMMUTATIVE RINGS
Abstract:
The Fibonacci recurrence, ordinarily studied over the integers, acquires a considerably richer algebraic
character once it is reduced over a finite commutative ring. This paper examines periodicity, rank of apparition,
and structural behaviour of generalized Fibonacci-type sequences defined over Z_n across a spread of moduli
that include primes, prime powers, and composite integers with mixed factorisations. Building on the classical
notion of the Pisano period, the analysis is extended to a family of second-order linear recurrences G(k) = c1
G(k-1) + c2 G(k-2) mod n, parametrised by coefficients c1 and c2, to examine how variation in these
coefficients reshapes the period structure, particularly when c2 fails to be a unit of Z_n. Computational data
gathered across thirty moduli show that the period of the standard Fibonacci recurrence is invariant under
choice of initial seed whenever the seed vector is non-degenerate, a fact traceable to the companion-matrix
formulation of the recurrence over the ring. Prime-power lifting exhibits a broadly multiplicative pattern,
period(p^(k+1)) equal top times period(p^k) in the cases observed, subject to exceptions associated with WallSun-Sun-type behaviour. A Jacobsthal-type variant, in which the ring's zero divisors interact directly with the
recurrence coefficients, is shown to become only eventually periodic rather than purely periodic, a distinction
with no counterpart over fields. These findings position the generalized Fibonacci sequence as a diagnostic
device for probing unit structure, zero-divisor behaviour, and matrix invertibility within finite commutative
rings, with downstream relevance for pseudo-random sequence design and coding-theoretic constructions.
Authors:
Ishika1
TOPOLOGICAL INVARIANTS OF WEIGHTED SIMPLICIAL COMPLEXES AND THEIR APPLICATIONS IN DATA CLUSTERING
Abstract:
Clustering algorithms built on distance thresholds or centroid geometry often fail when data lies on curved
manifolds, contains multi-scale structure, or exhibits noise that distorts local density estimates. This paper
investigates whether topological invariants derived from weighted simplicial complexes - specifically Betti
numbers, persistence diagrams, and Euler characteristics computed across a filtration of edge weights - can
serve as a more stable basis for cluster identification than conventional distance-based methods. A weighted
Vietoris-Rips complex was constructed from five benchmark datasets spanning synthetic and real-world
domains, and persistent homology was computed at successive filtration values to extract zeroth and first-order
Betti numbers. These invariants were then used to define a topological clustering criterion, compared against kmeans, DBSCAN, and spectral clustering using adjusted Rand index and silhouette scores. Results indicate that
topology-based clustering achieved a mean adjusted Rand index of 0.81 across the five datasets, compared with
0.64 for k-means and 0.71 for DBSCAN, with the largest gains observed on datasets containing non-convex or
nested cluster shapes. Persistence diagrams also proved more resistant to Gaussian noise injection, retaining
stable Betti-1 counts up to a noise variance of 0.15 where distance-based methods degraded sharply. These
findings support the proposition, stated at the outset, that weighted simplicial complexes encode structural
information about data shape that persists across scale and noise in ways centroid- or density-based methods
cannot capture, and they suggest a practical pathway for incorporating topological data analysis into standard
clustering pipelines without prohibitive computational cost for datasets under approximately 2,000 points.
Authors:
Shraddha Pansari1
SPECTRAL GAP ESTIMATES FOR PERTURBED GRAPH LAPLACIANS ON NON-UNIFORM NETWORKS
Abstract:
The algebraic connectivity of a graph, the second-smallest eigenvalue of its Laplacian, governs how fast
information mixes and how resilient a network stays when edges are added or dropped. Most classical
perturbation bounds, built around Weyl's inequality, assume near-regular degree sequences, which is precisely
the assumption real networks tend to break. This paper asks a narrower and more testable question: how does
the spectral gap actually move under controlled edge rewiring once a network stops being uniform, and how
tight is the standard operator-norm bound when it does? Four network families of matched size, an Erdos-Renyi
baseline, a Barabasi-Albert scale-free graph, a Watts-Strogatz small-world graph, and a heavy-tailed
configuration model, were generated at three node scales and subjected to rewiring at five intensities. The
Laplacian spectrum was recomputed at every step and the observed change in the gap was checked against the
Weyl bound derived from the spectral norm of the perturbation matrix. Heterogeneous, high-variance degree
sequences absorbed rewiring with a shrinking gap and a bound that stayed loose, ratios below 0.01 in the
heavy-tailed case, while near-regular structures tracked the bound far more closely. A regression of gap
sensitivity on degree coefficient of variation returned a correlation of -0.68, indicating that heterogeneity itself,
not merely network size, predicts how conservative the classical estimate becomes. The results argue for
heterogeneity-aware correction terms in spectral stability theory rather than uniform worst-case bounds applied
indiscriminately across network types.
Authors:
BRIJ BHAN SINGH1
ON THE CONVERGENCE BEHAVIOR OF ADAPTIVE FINITE ELEMENT METHODS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS
Abstract:
Singularly perturbed boundary value problems remain a persistent challenge in numerical analysis because
solutions typically exhibit thin boundary or interior layers whose width scales with a small perturbation
parameter epsilon. Uniform meshes fail to resolve these layers without prohibitive refinement, motivating the
use of adaptive finite element methods (AFEM) driven by a posteriori error estimators. This paper investigates
the convergence behavior of a residual-based adaptive finite element scheme applied to a family of one- and
two-dimensional reaction-diffusion problems with boundary layers. A sequence of adaptively refined meshes
was generated using a Dorfler marking strategy combined with newest-vertex bisection, and the resulting error
decay was measured in the energy norm across a range of perturbation parameters spanning epsilon = 10^-2 to
10^-6. Five sets of computational data are reported, covering degrees-of-freedom versus error decay, effectivity
indices of the estimator, layer-resolution ratios, computational cost comparisons against uniform refinement,
and parameter-robustness trends. The results indicate that the adaptive scheme recovers close to optimal
convergence order regardless of epsilon, whereas uniform refinement degrades sharply as epsilon decreases.
The estimator-maintained effectivity indices between 0.85 and 1.30 across all tested regimes, confirming
reliability and efficiency in the sense of a posteriori error control.
Authors:
Uday Pratap1
