This study presents a high-order numerical scheme for solving the generalized fractional-order Burgers-Huxley equation, a nonlinear evolutionary PDE combining integer-order spatial derivatives with temporal fractional derivatives. The model captures essential features of reaction-diffusion-convection systems and neural pulse propagation dynamics. We develop a fourth-order compact finite difference method for spatial discretization coupled with a Grünwald-Letnikov approximation for the fractional time derivative. The resulting nonlinear system is solved using an efficient iterative algorithm. Rigorous analysis demonstrates that the proposed method is unconditionally stable and achieves $\mathcal{O}(\tau^3 + h^4)$ convergence, where $\tau$ and $h$ represent temporal and spatial step sizes, respectively. Numerical experiments confirm the theoretical results and illustrate the scheme's accuracy through multiple test cases, demonstrating its effectiveness in handling this class of fractional PDEs.

A recent definition of a fractional derivative operator with a generalized exponential kernel and its singular kernel extension have been developed. This new operator has features that differ from those of existing operators such as the Mittag-Leffler kernel fractional derivative operator. In particular, since non-singular kernel fractional derivative operators impose data limitations, leading to restrictions on the initialization of the associated fractional differential equations, significant difficulties arise in obtaining their solutions. This paper highlights how singular kernel extensions of non-singular kernel operators address the associated initialization problem. It focuses on providing approximate analytical solutions to the time-fractional Burgers’ equation associated with extensions of non-singular kernel fractional derivatives. We study the unique existence of solutions to the considered problems. We further develop the homotopy analysis transform method to handle the time-fractional Burgers’ equation incorporating the generalized exponential kernel and the Mittag-Leffler kernel fractional derivatives. Moreover, we perform numerical comparisons between the dynamic behavior of the studied equations when utilizing the considered extensions of non-singular kernel fractional derivatives.

This paper introduces an efficient meshless method for solving 2D nonlinear Volterra integro-differential equations with tempered fractional kernels. It uses a compact radial basis function (RBF)-based partition of unity approach for spatial discretization, transforming the original problem into a semi-discrete system. Hermite-Birkhoff RBF interpolation is employed within partition of unity patches, using both smooth and discontinuous weight functions to construct the global solution. Discontinuous weights yield a sparse global matrix, enhancing computational efficiency. The method also leverages the scaling property of polyharmonic spline kernels for robust local approximations. For time discretization, a convolution quadrature rule with a second-order backward differentiation formula is used. Numerical experiments, including problems without exact solutions, confirm the method's accuracy, robustness, and efficiency.

Kawasaki disease (KD) is an acute, self-limited pediatric vasculitis of unknown etiology and is one of the leading causes of acquired coronary artery complications in children. Endothelial dysfunction, vascular endothelial growth factor (VEGF) activity, adhesion molecule/chemokine activation, and inflammatory cytokine responses play important roles in its pathogenesis. This paper presents a delay differential equation model with stochastic perturbations to study lesion-level inflammatory mechanisms involved in Kawasaki disease pathogenesis. The model describes interactions among healthy endothelial cells, vascular endothelial growth factor (VEGF), adhesion molecules/chemokines, and inflammatory cytokine activity. Mathematically, endothelial-cell injury promotes VEGF production, VEGF contributes to adhesion molecule and chemokine activation, and the combined adhesion molecule/chemokine activity stimulates inflammatory cytokine production after a time delay. The variables are interpreted as aggregated biological activities, not as individual molecular species. The model is not designed to represent the acute, subacute, and convalescent clinical phases of Kawasaki disease separately, and coronary artery inflammation is not included as an independent state variable. Instead, endothelial dysfunction and inflammatory cytokine activity are used as indirect mechanistic indicators of vascular inflammatory progression. The model is shown to preserve positivity and boundedness under suitable dissipativity assumptions. Equilibrium points and an inflammatory feedback threshold quantity are discussed, and local stability is analyzed through the characteristic equations of the delayed system. Reported incidence data from 2020–2025 are used only as qualitative motivation for considering variability and delayed biological responses. A stochastic extension is then formulated to represent random biological and environmental fluctuations, and a stochastic nonstandard finite difference scheme is proposed to preserve positivity and boundedness in numerical simulations. The results provide a mathematical framework for studying delayed stochastic inflammatory interactions in Kawasaki disease, while highlighting that explicit modeling of clinical phases and coronary artery involvement remains an important direction for future work.

Objective The aim of current research is to present the numerical performances of the cancer treatment model based on chemotherapy and stem cells using one of the heuristic computing neural network procedures. The cancer treatment model through chemotherapy and stem cells is categorized into stem cells, affected cells, tumor cells, and chemotherapy-based concentration drug. Method A process of artificial neural network is applied using the hybrid optimization of global and local search schemes, which are taken as genetic algorithm (GA) and an active set (AS). An error-based fitness function is designed by using the differential model and then optimized by the hybridization of both global and local search schemes. GA is applied to exploit the global result and give a primary guess to the AS that further improves the results locally. AS is rooted in the GA, where GA produces new populaces and AS optimizes the fitness function for every individual. The hybridization of these two schemes is used iteratively for purifying the results. Ten numbers of neurons and log-sigmoid activation functions has been used to solve the cancer treatment model based on chemotherapy and stem cells. Results For the correctness of the stochastic solver, the obtained numerical results have been compared with any traditional scheme. Moreover, the reliability and capability of the scheme are performed through the absolute error around 10-05 to 10-07 along with different statistical approaches for solving the mathematical model. Novelty The proposed artificial neural network structure along with the hybrid optimization of global and local search schemes has never been implemented before to solve the cancer treatment model based on chemotherapy and stem cells.

The concept of fixed points is a fundamental tool for analyzing nonlinear phenomena. This study investigates the existence and stability of solutions to boundary value problems involving Caputo-Hadamard fractional differential equations. Under appropriate conditions, M & ouml;nch's fixed point theorem is applied to rigorously establish the existence of solutions. The stability of these solutions is then studied independently using Ulam-Hyers and Ulam-Hyers-Rassias concepts to assess their sensitivity to small perturbations. Additionally, successive approximation methods are employed to efficiently obtain approximate solutions, and the convergence of the numerical scheme is examined. Several illustrative examples, supported by tables and figures, demonstrate the theoretical and numerical contributions of the study. This work highlights a novel and systematic analysis combining rigorous existence results with a detailed stability study and numerical approximation for Caputo-Hadamard fractional boundary value problems.

The Tribonacci sequence, an extension of the Fibonacci sequence, exhibits rapid growth and a distinctive recursive structure. It plays a significant role in algebra, combinatorics, number theory, dynamical systems, and fractal geometry. Moreover, its applications extend to computer science for modeling recursive algorithms, cryptographic structures and to the natural sciences, such as population dynamics and biological growth patterns. In this study, we construct new sequence spaces based on the fusion of Tribonacci and lacunary sequences, generalized through modulus functions under appropriate conditions. We focus particularly on psi-Tribonacci lacunary statistical convergence of order alpha and psi-strong Tribonacci lacunary summability of order alpha. To strengthen our theoretical contributions, we integrate Neural Networks as computational tools for analyzing the convergence behavior and structural relationships among these sequence spaces. Several illustrative examples, supported by numerical simulations and figures, demonstrate the practical interpretation and effectiveness of the proposed concepts. Furthermore, refinements in lacunary sequences allow us to establish deeper interrelations, contributing to a more comprehensive understanding of these spaces and their applications in both mathematical theory and computational frameworks.

In this paper we propose a new perspective related to Lucas sequence with statistical and summability, some considerations have been given in connection with their applications of signal processing. We define the notions of λ-Lucas statistically convergent and strongly λ-Lucas summability through modulus functions. The study of investigates on the Lucas transform and its embedding diagram into the representative form of Lucas numbers is studied in the context of convergence and summability problems. Furthermore, we provide inclusions as well as principles of equivalence, which lead to conditions guaranteeing the uniqueness of restrictions and generalize statistical convergence theory. Numerical simulations on a wide variety of signals such as noisy sinusoidal signals and blurred images demonstrate the flexibility of the proposed approach. These experiments show a steady decay to zero, which implies considerable noise immunity. In addition, by providing alternative perspectives of assessing signal quality, compression strategies and filtering techniques, those methods help in distinguishing the relevant signal information from background noise. Bringing together theoretical results and practical real-world signal analysis examples, the paper introduces classical and recent contributions to summability theory as well as new signal-processing procedures.

The dynamics of cancer, such as tumor growth and the existence of cancer stem cells, pose a persistent challenge to contemporary medicine. With consideration of cancer stem cells, we present a fractional-order model to explain an intricate relationship between cellular crowding and tumor growth and offer more accurate depictions of actual biological processes. Also, by using the fixed-point theorem, we examine the possibility of finding solutions to our model and their uniqueness. Further, we use the Laplace residual power series approach to obtain certain analytical solutions and arrive at explicit solutions that shed light on the dynamics of the given system and the development of tumors. Furthermore, we show the influence of fractional derivative dynamics and cancer stem cells on tumor growth, as the fractional derivative enriches our knowledge of the complex underlying mechanisms of tumor progression that, in turn, have consequences for the determination of targeted therapies. In addition, the main contribution of this work lies in expanding the literature on the mathematical modeling of tumors, as it emphasizes the critical importance of the impact of fractional derivative dynamics on the complex interactions in tumor microenvironments.

Mathematical modeling is geared toward establishing models based on actual problems reflecting different aspects of the real world along with its reciprocal dynamics and interactions through mathematics whilst addressing universal notions, which foregrounds their uniqueness enabling the conceptualization, mechanization and automation of intellectual activity. This very essence sets out the solutions for real-life challenges in line with the results commensurately devised and calculated, which suggests that the core of a mathematical model lies in the factuality that it represents a dynamic simulation, rather than a reductionist and fixed way of reasoning. Hence, mathematical modeling entails flexibility in employing the pertinent knowledge of mathematics and keen observation as well as analysis of different phenomena and problems in life so that the optimally applicable mathematical model can be extracted. Complexity can be measured since there notably exist a number of complexity measures with complex systems possessing many degrees of freedom. Nonetheless, the mere description or measurement of complexity would not be sufficient to understand complex systems. Besides this, the property of self-organization draws profoundly from the source of qualitative innovation in complex systems. Derived from nonlinear system theory, the treatment of chaotic along with the advent and advancements of nonlinear dynamics in physics has made it interlaced with progresses made in computer technologies and the computing power for handling nonlinearity numerically. These have highlighted further evidence regarding the existence and prevalence of complex and unstable phenomena with nonlinear systems having a few degrees of freedom exhibiting structural, static, mechanic and dynamical instabilities which are concerned with implicit metaphysical and methodological convictions, rendering them highly valued owing to constituting the nomological nucleus of pattern formation, processes of growth processes, phase transitions and self-organization. Fractals as objects that infinitely replicate their shape within their structure appear the same independently regardless of their form or their magnification level. Thus, fractals can continually be magnified, containing other fractals of the same form, shape and characteristic. Across these strands, fractal analysis is resorted to for explaining complex systems manifesting self-similarity across different scales whose multiplicity structure and behavior of fractals ensure the decomposition of time series data into different levels of granularity. This merit is particularly significant in certain fields that involve quantitative trends in which the pertinent data are inclined to exhibit similar patterns regardless of the time scale. Moreover, multiple nonlinear systems demonstrate phenomena in which fluctuations are inclined to enhance synchronization and periodic behaviors of the system. In this respect, fractal-fractional and wavelet methods are among those employed as techniques for characterizing complex and dynamic patterns in various domains to be able to detect specificity, regularity, self-similarity, singularity and significant attributes, which could all designate facilitating functions. Neural networks as the simplified models of the biological nervous system are comprised of highly interconnected network of a large number of processing elements or neurons in a human brain-inspired architecture. Furthermore, machines' ability of replicating the capacities of living systems, in particular of human intelligence is one of the most notable achievements in the context of emerging technologies. Being able to recognize objects and make decisions through many of the perceptual and cognitive abilities of live systems inspires Artificial Intelligence (AI)-based technologies whose potential could be utilized optimally in the biological world, including medical and clinical research and applications, bioengineering, biomedicine, genetics, and so forth, having facilitating functions in the early identification of the disease and its precise treatment based on personalized medicine targeted at individual patients. All these developments and opportunities signify the possibility of applying different AI-based mathematical modeling techniques and computing systems to various applications of numerous domains, considering the future of AI in the sphere of complex systems, nonlinear dynamics, singularity and chaos. Computational complexity characterizes the class of computational problems relying on their inherent difficulty and relating those classes to one another. A problem is inherently difficult if its solution requires significant resources irrespective of what algorithm is utilized. This initiation is formalized by computational complexity through mathematical models of computation for probing the problems and quantifying the number of resources, such as time and storage toward their solution. Levels of complexity are assigned to different collections of problems, so an understanding of the theoretical framework entails a substantial background and expertise in theoretical computer science considering the intriguing conjectures about such levels of complexity. Complexity science and systems science point toward bridging the gap between micro-level analysis and holistic perspectives, providing the means required for investigating interconnected, nonlinear and adaptive phenomena. Within such systems, fractals and wavelet methods are employed for identifying self-similar patterns, singularities and regularities that stem from otherwise irregular data. Consequently, the theoretical reflections on how all these processes are modeled, merging all together the advanced mathematical modeling, methods, optimization, analyses, computational and emerging technologies are addressed in our special issue which elaborates on and showcases the implications of solution-oriented applicable approaches in real-world systems and other related domains. Thus, our special issue aims at extrapolating new mathematical theoretical directions in view of the sound groundwork of pure (theoretical) mathematics in conjunction with mathematical modeling, analyses and applications that could be justified from a decision-theoretic viewpoint of constructively based solution and proof procedures enhanced by exponential growth of data volume, algorithmizing, computer affordances, hardware and storage capacities in science, medicine, biology, engineering through applied sciences, and many more realms pondering on the computability of the universe.

In this study, we develop a novel stage-structured fractional-order prey-predator model that captures more realistic ecological dynamics by incorporating the effects of juvenile hunting behavior-particularly those arising from structural habitat complexity and shoot bombardment. These factors influence the viability and survival of juvenile predators, leading to potential deviations in population predictions. To account for the inherent memory and hereditary properties in ecological interactions, we employ fractional-order derivatives, thereby extending the classical prey-predator framework. The proposed model accounts for maturation delays, stage-specific interactions, and differential survival rates, offering a more comprehensive understanding of population stabilization across multiple life stages. Our analysis highlights how structural complexity in the habitat critically affects juvenile hunting efficiency and the associated maladroit costs, such as increased mortality due to unsuccessful hunting attempts during key developmental periods. Both analytical and numerical techniques are employed to investigate the stability and bifurcation behavior of the system under varying ecological parameters. The results underscore the importance of habitat features and predator limitations in shaping population dynamics and ecological resilience. These insights contribute to the broader understanding of biodiversity conservation and the strategic management of ecosystems under natural and anthropogenic pressures.

This study presents a fractional-order mathematical model to investigate the transmission dynamics of tuberculosis (TB), including both pulmonary and multidrug-resistant cases. The model incorporates immunization as a control strategy, considering the potential future application of vaccines for infants and adults. Additionally, the model introduces a quarantine class to address the specific challenges of multidrug-resistant tuberculosis (TB). To capture the complex dynamics of the disease, fractal-fractional-order derivatives are employed, allowing for a more accurate depiction of the memory and nonlocal effects within the population. The model's equilibrium points, including the disease-free and endemic states, are derived and analyzed. A sensitivity analysis is conducted to assess the influence of key parameters on the basic reproduction number. Exploring the existence of solutions is achieved through fixed-point theory and its applications. We establish certain conditions for global asymptotical stability and use the next-generation matrix method to develop local stability analysis. The model's global stability is further investigated using the Lyapunov function method. Revisiting sensitivity analysis of the model parameters provides a comprehensive evaluation. Subsequently, the chaos of the TB epidemic model is investigated through the feedback control approach. In terms of computation, we generate a numerical scheme for the fractal-fractional tuberculosis (TB) model with the use of Newton polynomial. The graphical findings utilizing the numerical simulation (MATLAB version 18) are shown and briefly discussed. Utilizing available data, we graphically present the results, employing various fractal-fractional-orders to gain insight into the mechanisms underlying the dynamics with a chaotic behavior approach. The findings suggest that vaccination and quarantine interventions play significant roles in managing the spread of both drug-sensitive and drug-resistant tuberculosis (TB).

Rumor spreading has become a critical social issue with the widespread use of social media platforms. This study develops a stochastic fractional delay differential equation (SFDDE) model to describe rumor propagation in a population divided into four compartments: susceptible [Formula: see text], spreaders [Formula: see text], counter-rumor spreaders [Formula: see text], and stiflers [Formula: see text]. The proposed model ensures nonnegativity and boundedness of solutions for nonnegative initial conditions. Rigorous analytical investigations establish the local and global stability of both the Rumor-Free Equilibrium (RFE) and the Rumor-Present Equilibrium (RPE), with the reproduction number identified as a key threshold parameter. Supported by classical stability theorems, the model's positivity, boundedness, local and global dynamics, and sensitivity around the reproduction number are systematically examined. Furthermore, the Generalized Nonstandard Finite Difference (GL-NSFD) method is employed to obtain accurate and dynamically consistent numerical approximations, demonstrating the model's reliability and efficiency through simulations and graphical validation.

This paper presents a novel fractional-order mathematical model of myocardial infarction in women who are users of combined oral contraceptive pill and who also develop comorbidity due to various reasons. The system of equations incorporate Caputo fractional derivative to capture memory effects of the model. Existence and uniqueness of solution of the mathematical model is derived. Numerical simulations were rigorously conducted on the model with varying fractional order namely, 0.3, 0.5 and 0.8 using Euler's method. The numerical results thus obtained are simulated by Adam's method for 200 days period. The output from these simulations form the dataset of the Bayesian regularization neural network with dataset split for training, testing and validating the computational model. Bayesian regularization is incorporated to handle overfitting efficiently. Root Mean Square Errors are computed for all three fractional orders respectively. Regression analysis is conducted which yielded perfect correlation (R = 1) across the all datasets. The combined mathematical and computational analysis form a strong layout in myocardial infarction risk prediction, diagnosis and treatment in young women.

The analysis and investigation of Crimean-Congo Hemorrhagic Fever transmission throughout the population are the goals of this study. Using Atangana-Baleanu's-Caputo (ABC) sense concept, the mathematical model is transformed into fractional order for ongoing surveillance of Crimean-Congo hemorrhagic fever. To comprehend the stable state, a qualitative and quantitative investigation of the suggested fractional-order system is conducted. The system's sensitivity analysis is examined to see how sensitive certain parameters are and to see how quickly various factors change and effect the spread of illness. Using flip bifurcation, the local stability of the system with the impacts of both people and animals is confirmed and assessed. The global stability of the system is also examined with Lyapunov first derivative functions. The existence and positivity of the global derivative are shown to be confirmed using Lipschitz criteria for the rate of effects according to their sub-compartments and linear growth on a global scale. The sophisticated tool Atangana-Toufik scheme for various fractional values is used to create solutions for the fractional-order system together with error analysis verification. The real behavior of the Crimean-Congo Hemorrhagic Fever spread and ongoing surveillance are observed using simulations. This kind of research will help to comprehend the epidemic and will have ramifications for real data as well as future control methods.

In this paper, we investigated the Shehu transform and the formable integral transform, both belonging to the class of Laplace-type integral transforms. The formable integral transform represents a novel and effective method for solving both ordinary and partial differential equations. We use these transforms to solve some differential equations that involve various types of derivatives, including classical, Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives, investigating their complex dynamics and revealing the corresponding transfer functions. By extending the capabilities of Laplace and Sumudu integral transforms, these innovative integral transforms offer efficient solutions for classical as well as fractional differential equations, significantly expanding the range of solvable mathematical models. The derived transfer functions not only serve as powerful analytical tools but also demonstrate remarkable versatility in addressing diverse mathematical models, thus emphasizing their significant impact on mathematical analysis and applications.