AppliedMath, Volume 4, Issue 4 (December 2024) – 24 articles

Computing the solution of the linear Caputo fractional differential equation with variable coefficients cannot be obtained in closed form as in the integer-order case. However, to use ‘q’, the order of the fractional derivative, as a parameter for our mathematical model, we need to compute the solution of the equation explicitly and/or numerically. The traditional methods, such as the integrating factor or variation of parameters methods used in the integer-order case, cannot be directly applied because the product rule of the integer derivative does not hold for the Caputo fractional derivative. In this work, we present a series solution method to compute the solution of the linear Caputo fractional differential equation with variable coefficients. This provides an opportunity to compare its solution with the corresponding integer solution, namely $q=1.$ Additionally, we develop a series solution method using analytic functions in the space of $C^q$ continuous functions. We also apply this series solution method to nonlinear Caputo fractional differential equations where the nonlinearity is in the form $f(t,u)=u^2.$ We have provided numerical examples to show the application of our series solution method. Full article

Regular longitude-latitude grids are commonly used in reanalysis and climate prediction model datasets. However, this approach can disproportionately represent high-latitude regions if simple averaging is applied, leading to overestimation of their contribution. To explore the impact of Earth’s curvature on global warming and heat wave frequency, this study analyzed 450 years of surface temperature data (1850–2300) from a climate prediction model. When area weighting was applied, the global average temperature for the 1850–2300 period was found to be 8.2 °C warmer than in the unweighted case, due to the reduced influence of colder temperatures in high latitudes. Conversely, the global warming trend for the weighted case was 0.276 °C per decade, compared to 0.330 °C per decade for the unweighted case, reflecting a moderation of the polar amplification trend. While unweighted models projected a 317% increase in the frequency of global heat waves above 35 °C by 2300 compared to 1850, the weighted models suggested this frequency might be overestimated by up to 5.4%, particularly due to reduced weighting for subtropical deserts relative to tropical regions. These findings underscore the importance of accounting for Earth’s curvature in climate models to enhance the accuracy of climate change projections. Full article

This paper investigates the asymptotic eventual stability (AE-S) for nonlinear impulsive Caputo fractional differential equations (ICFDEs) with fixed impulse moments, employing auxiliary Lyapunov functions (ALF) which are specifically constructed as analogues of vector Lyapunov functions (VLF). A novel derivative tailored for VLF is introduced, offering a more robust framework than existing approaches based on scalar Lyapunov functions (SLF). Adequate conditions for AE-S involving ICFDEs are provided. We also used the predictor corrector method to implement a numerical solution for a given impulsive Caputo fractional differential equation. These findings extend and improve upon existing results, providing significant advancements in the stability analysis of systems with memory effects and impulsive dynamics. The study holds practical relevance for modeling and analyzing real-world systems, including control processes, biological systems, and economic dynamics where fractional-order behavior and impulses play a crucial role. Full article

This study investigates the three-dimensional, steady, laminar boundary-layer equations of a non-Newtonian fluid over a flat plate in the absence of body forces. The classical boundary-layer theory, introduced by Prandtl in 1904, suggests that fluid flows past a solid surface can be divided into two regions: a thin boundary layer near the surface, where steep velocity gradients and significant frictional effects dominate, and the outer region, where friction is negligible. Within the boundary layer, the velocity increases sharply from zero at the surface to the freestream value at the outer edge. The boundary-layer approximation significantly simplifies the Navier–Stokes equations within the boundary layer, while outside this layer, the flow is considered inviscid, resulting in even simpler equations. The viscoelastic properties of the fluid are modeled using the Rivlin–Ericksen tensors. Lie group analysis is applied to reduce the resulting third-order nonlinear system of partial differential equations to a system of ordinary differential equations. Finally, we determine the admissible forms of the freestream velocities in the x- and z-directions. Full article

We study a separable Hilbert space of smooth curves taking values in the Segal–Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential operators of the first and second order. We show how to build a Hilbert basis for this space. We study these first- and second-order partial derivation non-self-adjoint operators defined on this space, showing that these operators are defined on dense subspaces of the initial space of smooth curves; we determine their respective adjoints, compute their respective commutators, determine their eigenvalues and, under some normalisation conditions on the eigenvectors, we present examples of a discrete set of eigenvalues. Using these derivation operators, we study a Schrödinger-type equation, building particular solutions given by their representation as smooth curves on the Segal–Bergmann space, and we show the existence of general solutions using an Fourier–Hilbert base of the space of smooth curves. We point out the existence of self-adjoint operators in the space of smooth curves that are obtained by the composition of the partial derivation operators with multiplication operators, showing that these operators admit simple sequences of eigenvalues and eigenvectors. We present two applications of the Schrödinger-type equation studied. In the first one, we consider a wave associated with an object having the mass of an electron, showing that two waves, when considered as having only a free real space variable, are entangled, in the sense that the probability densities in the real variable are almost perfectly correlated. In the second application, after postulating that a usual package of information may have a mass of the order of magnitude of the neutron’s mass attributed to it—and so well into the domain of possible quantisation—we explore some consequences of the model. Full article

A local and a semi-local convergence analysis are presented for the Kurchatov-type method to solve numerically nonlinear equations in a Banach space. The method depends on a real parameter. By specializing the parameter, we obtain methods already studied in the literature under different types of conditions, such us Newton’s, and Steffensen’s, and Kurchatov’s methods, the Secant method, and other methods. This study is carried out under generalized conditions for first-order divided differences, as well as first-order derivatives. Both in the local case and in the semi-local case, the error estimates, the radii of the region of convergence, and the regions of the solution’s uniqueness are determined. A numerical majorizing sequence is constructed for studying semi-local convergence. The approach of restricted convergence regions is used to develop a convergence analysis of the considered method. The new approach allows a comparison of the convergence of different methods under a uniform set of conditions. In particular, the assumption of generalized continuity used to control the divided difference provides more precise knowledge on the location of the solution as well as tighter error estimates. Moreover, the generality of the approach makes it useful for studying other methods in an analogous way. Numerical examples demonstrate the applicability of our theoretical results. Full article

High-dimensional datasets, where the number of features far exceeds the number of observations, present significant challenges in feature selection and model performance. This study proposes a novel two-stage feature-selection approach that integrates Artificial Bee Colony (ABC) optimization with Adaptive Least Absolute Shrinkage and Selection Operator (AD_LASSO). The initial stage reduces dimensionality while effectively dealing with complex, high-dimensional search spaces by using ABC to conduct a global search for the ideal subset of features. The second stage applies AD_LASSO, refining the selected features by eliminating redundant features and enhancing model interpretability. The proposed ABC-ADLASSO method was compared with the AD_LASSO, LASSO, stepwise, and LARS methods under different simulation settings in high-dimensional data and various real datasets. According to the results obtained from simulations and applications on various real datasets, ABC-ADLASSO has shown significantly superior performance in terms of accuracy, precision, and overall model performance, particularly in scenarios with high correlation and a large number of features compared to the other methods evaluated. This two-stage approach offers robust feature selection and improves predictive accuracy, making it an effective tool for analyzing high-dimensional data. Full article

For automobile and aerospace engineers, implementing Hall currents and thermal radiation in cooling systems helps increase the performance and durability of an engine. In the case of solar energy systems, the effectiveness of heat exchangers and solar collectors can be enhanced by the best use of hybrid nanofluids and the implementation of a Hall current, thermophoresis, Brownian motion, a heat source/sink, and thermal radiation in a time-dependent hybrid nanofluid flow over a disk for a Bayesian regularization ANN backpropagation algorithm. In the current physical model of Cobalt ferrite $CoF{e}_2{O}_4$ and aluminum oxide $A{l}_2{O}_3$ mixed with water, a new category of the nanofluid is called the hybrid nanofluid. The study uses MATLAB bvp4c to unravel such intricate relations, transforming PDEs into ODEs. This analysis enables the numerical solution of several BVPs that govern the system of the given problem. Hall currents resulting from the interaction between magnetic fields and the electrically conducting nanofluid, and thermal radiation as an energy transfer mechanism operating through absorption and emission, are central factors for controlling these fluids for use in various fields. The graphical interpretation assists in demonstrating the character of new parameters. The heat source/sink parameter is advantageous to thermal layering, but using a high Schmidt number limits the mass transfer. Additionally, a backpropagation technique with Bayesian regularization is intended for solving ordinary differential equations. Training state, performance, error histograms, and regression demonstration are used to analyze the output of the neural network. In addition to this, there is a decrease in the fluid velocity as magnetic parameter values decrease and a rise in the fluid temperature while the disk is spinning. Thermal radiation adds another level to the thermal behavior by altering how the hybrid nanofluid receives, emits, and allows heat to pass through it. Full article

This study introduces an analysis concerning spacetime perturbations within the context of quantum foam models. Under the framework of Sobolev spaces, $H^s\left({\mathbb{R}}^n\right)$, we establish the existence and uniqueness of solutions to a linearized wave equation considering the harmonic gauge condition. Energy estimates are derived, demonstrating the conservation of both standard and higher-order energy functionals, which ensures the stability and regularity of metric perturbations over time. In addition, a spectral analysis of the d’Alembertian operator is conducted through Fourier transform techniques. Explicit calculations of Sobolev norms further confirm that these norms remain uniformly bounded, reinforcing the stability of solutions in the Sobolev space framework. Full article

This paper aims to study the modified Gardner (mG) equation that was proposed in the literature a short while ago. We first construct conserved vectors of the mG equation by invoking three different techniques; namely the method of multiplier, Noether’s theorem, and the conservation theorem owing to Ibragimov. Thereafter, we present exact solutions to the mG equation by invoking a complete discrimination system for the fifth degree polynomial. Finally, we simulate collisions of solitons for the mG equation. Full article

Collective migration underlies key developmental and disease processes in vertebrates. Mathematical models describing collective migration can shed light on emergent patterns arising from simple mechanisms. In this paper, a mathematical model for collective migration is given for arbitrary numbers and types of individuals using principles outlined as a set of assumptions, such as the assumed preference for individuals to be “close but not too close" to others. The model is then specified to the case of two species with arbitrary numbers of individuals in each species. A particular form of signal response is used that may be parameterized based on experiments involving two or three agents. In its simplest form, the model describes two species of individuals that emit distinct signals, distinguishes between them, and exhibits responses unique to the type by moving according to signal gradients in various planar regions, a situation described as "mixotaxis". Beyond this simple form, initial conditions and boundary conditions are altered to simulate specific, additional in vitro as well as in vivo dynamics. The behaviors that were specifically accounted for include motility, directed migration, and a functional response to a signal. Ultimately, the paper’s results highlight the ability of a single framework for signal and response to account for patterns seen in multi-species systems, in particular the emergent self-organization seen in the embryonic development of placodal cells, which display chase-and-run behavior, flocking behavior, herding behavior, and the splitting of a herd, depending on initial conditions. Numerical experiments focus around the primary example of neural crest and placodal cell “chase-and-run” and “flocking” behaviors; the model reproduces the separation of placodal cells into distinct clumps, as described in the literature for neural crest and placodal cell development. This model was developed to describe a heterogeneous environment and can be expanded to capture other biological systems with one or more distinct species. Full article

Particulate matter smaller than 2.5 μm (PM2.5) in Madrid is a critical concern due to its impacts on public health. This study employs advanced methodologies, including the CRISP-DM model and hybrid Prophet–Long Short-Term Memory (LSTM), to analyze historical data from monitoring stations and predict future PM2.5 levels. The results reveal a decreasing trend in PM2.5 levels from 2019 to mid-2024, suggesting the effectiveness of policies implemented by the Madrid City Council. However, the observed interannual fluctuations and peaks indicate the need for continuous policy adjustments to address specific events and seasonal variations. The comparison of local policies and those of the European Union underscores the importance of greater coherence and alignment to optimize the outcomes. Predictions made with the Prophet–LSTM model provide a solid foundation for planning and decision making, enabling urban managers to design more effective strategies. This study not only provides a detailed understanding of pollution patterns, but also emphasizes the need for adaptive environmental policies and citizen participation to improve air quality. The findings of this work can be of great assistance to environmental policymakers, providing a basis for future research and actions to improve air quality in Madrid. The hybrid Prophet–LSTM model effectively captured both seasonal trends and pollution spikes in PM2.5 levels. The predictions indicated a general downward trend in PM2.5 concentrations across most districts in Madrid, with significant reductions observed in areas such as Chamartín and Arganzuela. This hybrid approach improves the accuracy of long-term PM2.5 predictions by effectively capturing both short-term and long-term dependencies, making it a robust solution for air quality management in complex urban environments, like Madrid. The results suggest that the environmental policies implemented by the Madrid City Council are having a positive impact on air quality. Full article

This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the $(x,t)$ plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation. Full article

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out. Full article

The computer network has fundamentally transformed modern interactions, enabling the effortless transmission of multimedia data. However, the openness of these networks necessitates heightened attention to the security and confidentiality of multimedia content. Digital images, being a crucial component of multimedia communications, require robust protection measures, as their security has become a global concern. Traditional color image encryption/decryption algorithms, such as DES, IDEA, and AES, are unsuitable for image encryption due to the diverse storage formats of images, highlighting the urgent need for innovative encryption techniques. Chaos-based cryptosystems have emerged as a prominent research focus due to their properties of randomness, high sensitivity to initial conditions, and unpredictability. These algorithms typically operate in two phases: shuffling and replacement. During the shuffling phase, the positions of the pixels are altered using chaotic sequences or matrix transformations, which are simple to implement and enhance encryption. However, since only the pixel positions are modified and not the pixel values, the encrypted image’s histogram remains identical to the original, making it vulnerable to statistical attacks. In the replacement phase, chaotic sequences alter the pixel values. This research introduces a novel encryption technique for color images (RGB type) based on DNA subsequence operations to secure these images, which often contain critical information, from potential cyber-attacks. The suggested method includes two main components: a high-speed permutation process and adaptive diffusion. When implemented in the MATLAB software environment, the approach yielded promising results, such as NPCR values exceeding 98.9% and UACI values at around 32.9%, demonstrating its effectiveness in key cryptographic parameters. Security analyses, including histograms and Chi-square tests, were initially conducted, with passing Chi-square test outcomes for all channels; the correlation coefficient between adjacent pixels was also calculated. Additionally, entropy values were computed, achieving a minimum entropy of 7.0, indicating a high level of randomness. The method was tested on specific images, such as all-black and all-white images, and evaluated for resistance to noise and occlusion attacks. Finally, a comparison of the proposed algorithm’s NPCR and UAC values with those of existing methods demonstrated its superior performance and suitability. Full article

This study investigates the role of calcium ions in the release of action potentials by comparing two models based on the framework: the standard HH model and a HH + Ca model that incorporates calcium ion channels. Purkinje cells’ responses to four types of electrical current stimuli—constant direct current, step current, square wave current, and sine current—were simulated to analyze the impact of calcium on action potential characteristics. The results indicate that, under the constant direct current stimulation, the action potential firing frequency of both models increased with the escalating current intensity, while the delay time of the first action potential decreased. However, when the current intensity exceeded a specific threshold, the peak amplitude of the action potential gradually diminished. The HH + Ca model exhibited a longer delay in the first action potential compared to the HH model but maintained an action potential release under stronger currents. In response to the step current, both models showed an increased action potential frequency with a higher current, but the HH + Ca model generated subthreshold oscillations under weak currents. With the square wave current, the action potential frequency increased, though the HH + Ca model experienced suppression under high-frequency weak currents. Under the sine current, the action potential frequency rose, with the HH + Ca model showing less depression near the sine peak due to calcium’s role in modulating membrane potential. These findings suggest that calcium ions contribute to a more stable action potential release under varying stimuli. Full article

In the present study, we analyze the price time series behavior of selected vegetable products, using complex network analysis in two approaches: (a) correlation complex networks and (b) visibility complex networks based on transformed time series. Additionally, we apply time variability methods, including Hurst exponent and Hjorth parameter analysis. We have chosen products available throughout the year from the Central Market of Thessaloniki (Greece) as a case study. To the best of our knowledge, this kind of study is applied for the first time, both as a type of analysis and on the given dataset. Our aim was to investigate alternative ways of classifying products into groups that could be useful for management and policy issues. The results show that the formed groups present similarities related to their use as plates as well as price variation mode and variability depending on the type of analysis performed. The results could be of interest to government policies in various directions, such as products to develop greater stability, identify fluctuating prices, etc. This work could be extended in the future by including data from other central markets as well as with data with missing data, as is the case for products not available throughout the year. Full article

The objective of this study is to model astrophysical systems using the nonlinear Fokker–Planck equation, with the Adomian method chosen for its iterative and precise solutions in this context, applying boundary conditions relevant to data from the Rossi X-ray Timing Explorer (RXTE). The results include analysis of 156 X-ray intensity distributions from X-ray binaries (XRBs), exhibiting long-tail profiles consistent with Tsallis q-Gaussian distributions. The corresponding q-values align with the principles of Tsallis thermostatistics. Various diffusion hypotheses—classical, linear, nonlinear, and anomalous—are examined, with q-values further supporting Tsallis thermostatistics. Adjustments in the parameter α (related to the order of fractional temporal derivation) reveal the extent of the memory effect, strongly correlating with fractal properties in the diffusive process. Extending this research to other XRBs is both possible and recommended to generalize the characteristics of X-ray scattering and electromagnetic waves at different frequencies originating from similar astronomical objects. Full article

The mathematical modeling of multicellular systems is an important branch of biophysics, which focuses on how the system properties emerge from the elementary interaction between the constituent elements. Recently, mathematical structures have been proposed within the thermostatted kinetic theory for the modeling of complex living systems and have been profitably employed for the modeling of various complex biological systems at the cellular scale. This paper deals with a class of generalized thermostatted kinetic theory frameworks that can stand in as background paradigms for the derivation of specific models in biophysics. Specifically, the fundamental homogeneous thermostatted kinetic theory structures of the recent literature are recovered and generalized in order to take into consideration further phenomena in biology. The generalizations concern the conservative, the nonconservative, and the mutative interactions between the inner system and the outer environment. In order to sustain the strength of the new structures, some specific models of the literature are reset into the style of the new frameworks of the thermostatted kinetic theory. The selected models deal with breast cancer, genetic mutations, immune system response, and skin fibrosis. Future research directions from the theoretical and modeling viewpoints are discussed in the whole paper and are mainly devoted to the well-posedness in the Hadamard sense of the related initial boundary value problems, to the spatial–velocity dynamics and to the derivation of macroscopic-scale dynamics. Full article

For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or the minimality conditions. Some simulations of complex Helmholtz equations are performed to exhibit the performance of the proposed two-block iterative schemes endowed with optimal values of parameters. The primary novelty and major contribution of this paper lies in using the orthogonality condition of residual vectors to optimize the iterative process. The proposed method might fill a gap in the current literature, where existing iterative methods either lack explicit parameter optimization or struggle with high wave numbers and large damping constants in the complex Helmholtz equation. The two-block splitting iterative scheme provides an efficient and convergent solution, even in challenging cases. Full article

A mixed graph has both edges and directed edges (or “arcs”). A complete mixed graph on v vertices, denoted $M_v$, has, for every pair of vertices u and v, an edge $\{u,v\}$, an arc $(u,v)$, and an arc $(v,u)$. A decomposition of the complete mixed graph on v vertices into a partial orientation of a three-cycle with one edge and two arcs (of which there are three types) is a mixed triple system of order v. Necessary and sufficient conditions for the existence of a mixed triple system of order v are well known. In this work packings and coverings of the complete mixed graph with mixed triples are considered. Necessary conditions are given for each of the three relevant mixed triples, and these conditions are shown to be sufficient for two of the relevant mixed triples. For the third mixed triple, a conjecture is given concerning the sufficient conditions. Applications of triple systems in general are discussed, as well as possible applications of mixed graphs, mixed triple systems, and packings and coverings with mixed triples. Full article

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order $\mathsf{\beta }$ is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of $\mathsf{\beta }$ and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order $\mathsf{\beta }$ becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution $\mathrm{a}\mathrm{n}\mathrm{d}$ observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation. Full article

Swarm Intelligence (SI) represents a paradigm shift in artificial intelligence, leveraging the collective behavior of decentralized, self-organized systems to solve complex problems. This study provides a comprehensive review of SI, focusing on its application to multi-robot systems. We explore foundational concepts, diverse SI algorithms, and their practical implementations by synthesizing insights from various reputable sources. The review highlights how principles derived from natural swarms, such as those of ants, bees, and birds, can be harnessed to enhance the efficiency, robustness, and scalability of multi-robot systems. We explore key advancements, ongoing challenges, and potential future directions. Through this extensive examination, we aim to provide a foundational understanding and a detailed taxonomy of SI research, paving the way for further innovation and development in theoretical and applied contexts. Full article