Fractal Fract., Volume 9, Issue 2 (February 2025) – 76 articles

In this manuscript, we study a class of equations with two different Riemann–Liouville-type orders of nabla difference operators. We show some new and fundamental properties of the related Green’s function. Depending on the values of the orders of the operators, we split our research into two main cases, and for each one of them, we obtain suitable conditions under which we prove that the considered problem possesses a positive solution. We consider the latter to be the main novelty in this work. Our main tool in both cases of our study is Guo–Krasnoselskii’s fixed point theorem. In the end, we give particular examples in order to offer a concrete demonstration of our new theoretical findings, as well as some possible future work in this direction. Full article

This paper investigates a new class of fractional stochastic differential systems with non-Gaussian processes and Poisson jumps. Firstly, we examine the solvability results for the considered system. Furthermore, new stability results for the proposed system are derived. The findings are established through the application of Grönwall’s inequality, the successive approximation method, and the corollary of the Bihari inequality. Finally, the validity of the results is proved through an example. Full article

In this article, we prove a new Milne-type inequality involving Hadamard fractional integrals for functions with $GA$-convex first derivatives. The limits of the error estimates involve incomplete gamma and confluent hypergeometric functions. The results of this study open the door to further investigation of this subject, as well as extensions to other forms of generalized convexity, weighted formulas, and higher dimensions. Full article

Freeze–thaw action is a key factor in the deterioration of the dynamic mechanical behavior of rocks in cold regions. This study used yellow sandstone, which is prevalent in the seasonally cold region of Xinjiang, China. The yellow sandstone samples were subjected to various temperatures and a range of freeze–thaw cycles. Impact mechanical tests were performed using a Split Hopkinson Pressure Bar (SHPB) system on the treated samples. The effects of freezing temperature and changes in impact load on the mechanical properties of frozen–thawed sandstone were examined. Additionally, the damage fractal characteristics of the sandstone were analyzed using fractal theory. The results indicate that as the freezing temperature decreases, the stress–strain curves of frozen–thawed specimens exhibit a clear initial compaction stage. The dynamic strength of the specimens decreases with lower freezing temperatures and shows a logarithmic relationship with the loading strain rate; however, the dynamic deformation modulus exhibits no significant correlation with the strain rate. The fractal dimension is positively correlated with the strain rate, indicating that lower freezing temperatures correspond to a higher rate of increase in the fractal dimension. These findings offer valuable insights into the damage deterioration characteristics of frozen–thawed rocks under varying temperature conditions. Full article

This manuscript aims to establish the existence, uniqueness, and stability of solutions for Langevin fractional differential equations involving the generalized Liouville-Caputo derivative. Using a novel approach, we derive existence and uniqueness results through fixed-point theorems, extending and generalizing several existing findings in the literature. To demonstrate the applicability of our results, we provide a practical example that validates the theoretical framework. Full article

As mining progresses, complexities arise, leading to potential coal-rock gas dynamic disasters triggered by mining disturbances. These dynamic phenomena are influenced by factors such as loading mode, coal properties, and the presence of gas. To gain a comprehensive understanding of the mechanical properties, deformation, and failure characteristics of gas-containing coal under cyclic loading, we conducted uniaxial compression tests. These tests varied in loading frequencies, amplitudes, and durations. By analyzing the peak stress variation of gas-containing coal and utilizing digital image correlation (DIC) technology, we captured the deformation characteristics of the loaded coal surface. Following the tests, we examined the fragmentation degree of gas-containing coal under different cyclic loading using fractal theory. This involved screening and crushing samples to assess the impact of varying loading on coal fragmentation. The results showed that peak stress is positively correlated with loading frequency and negatively correlated with loading amplitude and the number of cycles. Cyclic loading significantly affects the surface deformation morphology of gas-containing coal, and there is a correlation between the stress level of the coal sample and its surface deformation, with the formation and development of cracks corresponding to the stress level. Fractal theory can analyze the crushing characteristics of materials and quantitatively characterize their degree of crushing, and the fractal dimension is closely related to the mode of cyclic loading and comprehensively reflects various experimental factors. The results of our study aim to provide insights that can guide the prevention and control of coal mine dynamic disasters. Full article

This research is based on the artificial intelligence approach for the error and regression analysis of contaminants, nutrients, and oxygen level in water bodies using a Caputo’s difference model. The model is composed of four subgroups including contaminant concentration (which is denoted by C), the temperature of the fluid T, oxygen concentration O, and nutrients N. ${\xi }_C,{\xi }_T,{\xi }_O,{\xi }_N$ are assumed as diffusion constants for the respective classes. The fractional-order difference model is investigated for the existence and uniqueness of solutions as well as Hyers–Ulam stability, subjected to certain assumptions. The computational results demonstrate that the maximum contaminant concentrations reach $0.01046$ mg/L for ${\xi }_C=0.1$ and ${\mathcal{W}}_R$ = 0.1, resulting in nutrient levels as low as $4.9969$ mg/L. The model predicts that increased pollutant loads increase local temperatures to ${20.009}^{ \circ }\mathrm{C}$. Furthermore, an inverse correlation between reaction rates and contaminant concentrations is also observed, whereby an increase in ${\mathcal{W}}_R$ from $0.1$ to $0.2$ reduces concentrations to $0.0038327$ mg/L. Full article

The co-evolutionary research of complex network propagation dynamics systems has gradually become a hot topic in domestic and international research in recent years. This article reviews the research progress of epidemic dynamics systems and public opinion dynamics systems, providing a theoretical basis and knowledge reserve for the co-evolutionary research of the “epidemic–opinion” system. Firstly, following the path of process complexity, this article points out the similarities in mathematical modeling between the two types of systems from a dynamic perspective, as well as the latest research progress. Based on this, the article fully considers the common complex network attributes of these two types of systems, and from the perspective of the increasing complexity of networks, it sorts out the relevant research progress of the “epidemic–opinion” system and the necessity of its co-evolutionary research. Finally, from the perspective of complex engineering systems, the article looks forward to the difficulties and problems that may be encountered in the co-evolutionary research process. From the two dimensions of process complexity and network complexity, the latest research progress is summarized, while key issues and potential difficulties in the next step of co-evolutionary research for the “epidemic–opinion” system are pointed out, providing a reference and inspiration for relevant researchers. Full article

This study investigates the surface characteristics of printing papers using fractal geometry, focusing on surface roughness and surface friction as independent properties. The fractal dimension (FD) was analyzed using the power spectral density method, which provided a more distinct characterization of paper surfaces compared to the variogram method. Surface roughness and friction were measured using a stylus-based contact profilometer, and mean absolute deviation (MAD) parameters, such as the mean absolute deviation of surface roughness (RMAD) and friction, were calculated to capture surface variability. The results revealed that while conventional parameters, such as roughness average (Ra) and the average coefficient of friction, are highly sensitive to measurement conditions, MAD-based parameters demonstrate greater robustness and stability. For instance, the regression equation for RMAD vs. Ra showed a strong correlation, with an R² value close to 1.0. However, the slopes were significantly less than one. Furthermore, FD exhibited weak correlations with surface roughness and friction, with R² values of 0.342 and 0.016, respectively, highlighting its unique ability to characterize autocorrelation or complexity of surface. Additionally, the effects of coating on paper surfaces were evaluated, revealing reduced flocculation of surface profiles but a 5% increase in FD, indicating enhanced surface complexity. These findings underscore the complementary role of FD in providing a comprehensive understanding of surface properties, with potential applications in quality control and the design of paper products. Full article

In this paper, the finite-approximate controllability for a class of fractional composite relaxation equations with different nonlocal conditions is discussed. Firstly, under the condition that the nonlocal term is compact, the existence of mild solutions to the equations is obtained by employing resolvent theory, the variational method, and Schauder’s fixed-point theorem. Moreover, under the assumption that the corresponding linear equation is approximately controllable, the fractional composite relaxation equation with the nonlocal condition is derived to be finite-approximately controllable. Furthermore, the existence of mild solutions and the finite-approximate controllability to the equations are considered for the weaker nonlocal problem. Finally, the example of nonlocal problem is provided to verify the feasibility of the results in this paper. Full article

In the context of advancements in deep resource development and underground space utilisation, deep underground engineering faces the challenge of investigating the mechanical behaviour of rocks under high-stress conditions. The present study is based on a gold mine, and the bulk ore taken from the mine perimeter rock was processed into two sets of specimens containing semicircular arched roadways with half and full penetrations. The tests were carried out using a true triaxial rock test system. The results indicate that the true triaxial stress–strain curve included stages such as compression density, linear elasticity, yielding, and destructive destabilisation following the peak; the yield point was more pronounced than that in uniaxial and conventional triaxial tests; and the peak stress and strain of the semi-excavation were higher than those of the full excavation. Furthermore, full excavation led to greater deformation along the σ₃ direction. The acoustic emission energy showed a sudden increase during the unloading stage, then fluctuated and increased with increasing stress until significant destabilisation occurred. Additionally, increased burial stress in the half-excavation decreased the proportion of tension cracks and shear cracks. Conversely, in semi-excavation, the proportion of tensile cracks decreased, while that of shear cracks increased. However, the opposite was observed in full excavation. In terms of fractal dimension, semi-excavation fragmentation due to stress concentration followed a power distribution, while the mass fragmentation in full excavation followed a random distribution due to uniform stress release. Furthermore, the specimen strength was positively correlated with fragmentation degree, and primary defects also influenced this degree. This study provides a crucial foundation for predicting and preventing rock explosions in deep underground engineering. Full article

Background: This study addresses the challenge of predicting data sequences characterized by a mix of partial linearity and partial nonlinearity. Traditional forecasting models often struggle to accurately capture the complex patterns of change within the data. Methods: To this end, this study introduces a novel polynomial-driven discrete grey power model (PFDPGM(1,1)) that includes time perturbation parameters, enabling a flexible representation of complex variation patterns in the data. The model aims to determine the accumulation order, nonlinear power exponent, time perturbation parameter, and polynomial degree to minimize the fitting error under various criteria. The estimation of unknown parameters is carried out by leveraging a hybrid optimization algorithm, which integrates Particle Swarm Optimization (PSO) and the Grey Wolf Optimization (GWO) algorithm. Results: To validate the effectiveness of the proposed model, the annual total renewable energy consumption in the BRICS countries is used as a case study. The results demonstrate that the newly constructed polynomial-driven discrete grey power model can adaptively fit and accurately predict data series with diverse trend change characteristics. Conclusions: This study has achieved a significant breakthrough by successfully developing a new forecasting model. This model is capable of handling data sequences with mixed trends effectively. As a result, it provides a new tool for predicting complex data change patterns. Full article

We explore the existence of solutions for a Hadamard fractional differential equation, subject to nonlocal boundary conditions, which contain Hadamard fractional derivatives and Riemann–Stieltjes integrals. This problem is a resonant one in the sense that the corresponding homogeneous boundary-value problem has nontrivial solutions. In the proof of the main result, we use the Mawhin continuation theorem. Full article

This study aims to investigate the behavior of viscoelastic materials exhibiting complex mechanical behavior characterized by both elastic and viscous properties. They are widely used in various engineering applications, such as structural components, transportation systems, energy storage devices, microelectromechanical systems (MEMS), and earthquake research and detection. Accurate modeling of viscoelastic behavior is crucial for predicting its performance under dynamic loading conditions. In this study, we modify the equations governing the thermoelastic resistance to describe the thermal variables of a thermoelastic micro-beam supported by a two-parameter Pasternak viscoelastic foundation by using a fractional Moore–Gibson–Thompson (MGT) model in the context of non-locality. The temperature, bending displacement, and moment were computed and graphically displayed using the Laplace transform method. Different theoretical approaches have been compared in order to explain how the phase delay affects physical phenomena. Numerical results show that the wave fluctuations of variables in thermoelastic micro-beams are slightly smaller for the studied model and that the speed of these plane waves depends on fractional and non-local parameters. Full article

Influenced by the hydrogeological structure and other factors, the change in groundwater depth shows seasonal fluctuation characteristics. Human activities have disrupted the long-term stable pattern of groundwater change, which makes the short-term prediction of groundwater depth important. To cope with the emergence of short-term groundwater prediction scenarios, for the first time, a discrete grey seasonal model with fractional order accumulation is proposed in this paper (FDGSM(1,1)). First, the DGM(1,1) model, which has a relative advantage over fluctuating data, was chosen as the basis for the transformation of the proposed model. Then, the fractional order accumulation operator is used to reduce the seasonal fluctuations in the data series. Finally, grey seasonal variables are introduced to construct the time response function. The proposed model has the basic properties of the traditional grey forecasting model, which is proven to be stable and seasonal. Additionally, the prediction performance of the proposed model is verified in a real scenario of Handan groundwater. This paper expands the seasonal prediction field of the grey prediction model, enriches the research system of the grey system theory and fractional order, and has a positive influence on the short-term prediction of groundwater depth. Full article

The Yangtze River Basin serves as a vital ecological barrier in China, with its water conservation function playing a critical role in maintaining regional ecological balance and water resource security. This study takes the Minjiang River Basin (MRB) as a case study, employing fractal theory in combination with the InVEST model and the SWAT-BiLSTM model to conduct an in-depth analysis of the spatiotemporal patterns of regional water conservation. The research aims to uncover the relationship between the spatiotemporal dynamics of watershed water conservation capacity and its ecosystem service functions, providing a scientific basis for watershed ecological protection and management. Firstly, fractal theory is introduced to quantify the complexity and spatial heterogeneity of natural factors such as terrain, vegetation, and precipitation in the Minjiang River Basin. Using the InVEST model, the study evaluates the water conservation service functions of the research area, identifying key water conservation zones and their spatiotemporal variations. Additionally, the SWAT-BiLSTM model is employed to simulate the hydrological processes of the basin, particularly the impact of nonlinear meteorological variables on hydrological responses, aiming to enhance the accuracy and reliability of model predictions. At the annual scale, it achieved NSE and R² values of 0.85 during calibration and 0.90 during validation. At the seasonal scale, these values increased to 0.91 and 0.93, and at the monthly scale, reached 0.94 and 0.93. The model showed low errors (RMSE, RSR, RB). The findings indicate significant spatial differences in the water conservation capacity of the Minjiang River Basin, with the upper and middle mountainous regions serving as the primary water conservation areas, whereas the downstream plains exhibit relatively lower capacity. Precipitation, terrain slope, and vegetation cover are identified as the main natural factors affecting water conservation functions, with changes in vegetation cover having a notable regulatory effect on water conservation capacity. Fractal dimension analysis reveals a distinct spatial complexity in the ecosystem structure of the study area, which partially explains the geographical distribution characteristics of water conservation functions. Furthermore, simulation results based on the SWAT-BiLSTM model show an increasingly significant impact of climate change and human activities on the water conservation functions of the Minjiang River Basin. The frequent occurrence of extreme climate events, in particular, disrupts the hydrological processes of the basin, posing greater challenges for water resource management. Model validation demonstrates that the SWAT model integrated with BiLSTM achieves high accuracy in capturing complex hydrological processes, thereby better supporting decision-makers in formulating scientific water resource management strategies. Full article

Memristor-based fractional-order chaotic systems can record information from the past, present, and future, and describe the real world more accurately than integer-order systems. This paper proposes a novel memristor model and verifies its characteristics through the pinched loop (PHL) method. Subsequently, a new fractional-order memristive Hopfield neural network (4D-FOMHNN) is introduced to simulate induced current, accompanied by Caputo’s definition of fractional order. An Adomian decomposition method (ADM) is employed for system solution. By varying the parameters and order of the 4D-FOMHNN, rich dynamic behaviors including transient chaos, chaos, and coexistence attractors are observed using methods such as bifurcation diagrams and Lyapunov exponent analysis. Finally, the proposed FOMHNN system is implemented on a field-programmable gate array (FPGA), and the oscilloscope observation results are consistent with the MATLAB numerical simulation results, which further validate the theoretical analysis of the FOMHNN system and provide a theoretical basis for its application in the field of encryption. Full article

In three periods of 3.25 years each, and at the same six different heights of a basin geomorphology, measurements were made, in the form of a time series, of urban meteorological variables (MV) (temperature, relative humidity, wind speed magnitude) and pollutants (P) (PM₁₀, PM_2.5, and CO). It is verified that each time series has a fractal dimension, and the value of its maximum Kolmogorov entropy is determined. These values generate two entropic surfaces according to measurement periods: one for urban meteorology and another for pollutants. The calculation of the gradient to each entropic surface multiplied by the average temperature of the period according to the measurement location gives, approximately, the average entropic force for each location. Combining these results with an analysis of the ratio between urban meteorological entropies and pollutant entropies, it is shown that in a basin morphology the entropic forces associated with pollutants are dominant, a source of heat, and there is a high probability that they produce extreme events. This condition also favors anomalous subdiffusion. Full article

Fractional differential equations (FDEs) are employed to describe the physical universe. This article investigates the attractivity of solutions for FDEs and Ulam–Hyers–Rassias stability, involving the $\Psi$-Hilfer fractional derivative. Important results are presented using Krasnoselskii’s fixed point theorem, which provides a framework for analyzing the stability and attractivity of solutions. Novel results on the attractiveness of solutions to nonlinear FDEs in Banach spaces are derived, and the existence of solutions, stability properties, and behavior of system equilibria are examined. The application of $\Psi$-Hilfer fractional derivatives in modeling financial crises is explored, and a financial crisis model using $\Psi$-Hilfer fractional derivatives is proposed, providing more general and global results. Furthermore, we also perform a numerical analysis to validate our theoretical findings. Full article

The functional significance of RSNs is examined via simultaneous EEG-fMRI studies on the basis of the relation of RSNs with different frequency bands of EEG and EEG-based microstate analysis. In this study, we try to identify RSNs from microstates of cortical surface maps of the BOLD signal. In addition, the scale-free dynamics of these map sequences were also examined. The structural and resting state functional MRI images were acquired on a 3T scanner with three different fMRI acquisition protocols from seven subjects. Microstate segmentations from EEG, fMRI, and simulated data were evaluated. Wavelet-based fractal analysis was performed on map sequence time series and the Hurst exponent (H) was calculated. By using HRF-deconvolved fMRI time series, the effect of the HRF (hemodynamic response function) on fMRI-derived microstates was tested. The fMRI map sequence has a system with a memory system smaller than 16 s. When the HRF was deconvolved, the duration of the memory of the system was reduced to 4 s. On the other hand, the results of simulation data indicated that these systems are specific to the resting state BOLD signal. Similar to EEG microstates, fMRI also has microstates and both of them have scale-free dynamics. fMRI microstate dynamics have two different components, one is related to the HRF and the other is independent of the HRF. The significance of fMRI microstates and their relation with RSNs need to be further studied. Full article

Resonant DC-DC converters are a class of strongly nonlinear systems with rich nonlinear phenomena. In order to describe the dynamic behavior of resonant DC-DC converters more accurately, the nonlinear dynamic behavior of fractional order (FO) resonant DC-DC converters is studied deeply, based on the fractional order nature of inductance and capacitance. Firstly, a Sigmoid function state model of the fractional order resonant converter is established and integrated with phase shift control. A discrete model of the converter is established by using an estimation correction algorithm. Secondly, the mathematical and equivalent circuit models of the fractional order converter are constructed in MATLAB. The circuit simulations and the experimental results verified the correctness of the Sigmoid function model. Thirdly, the effect of circuit parameters on the converter’s nonlinear dynamics is analyzed using bifurcation diagrams, time-domain waveforms, and phase diagrams. Finally, an experimental platform is established to validate the theoretical analysis. The results demonstrate that increasing the proportional coefficient and load resistance destabilizes the system, leading to rich nonlinear phenomena such as bifurcation and chaos. Compared to integer order converters, fractional order converters offer a broader stable operating range. Fractional order models can more accurately reflect the nonlinear dynamic characteristics of resonant DC-DC converters. Full article

In this paper, we discuss the positive solutions to a class of semipositone boundary value problems of fractional differential equations. The nonlinearity $f(t,\mathfrak{x})$ may be singular at $t=0,1$ and satisfies $f(t,\mathfrak{x})\ge -\mathfrak{a}\left(t\right)\mathfrak{x}-\mathfrak{R}\left(t\right)$. We derive some new properties of the Green’s function of the auxiliary problems, and discover the multiplicity and existence of the positive solutions by utilizing the fixed point index theory. Two examples are illustrated to validate the main results. Full article

A novel time delay Lotka–Volterra (TDLV) model was developed by extending the concept of time delay from integer order to fractional order. The TDLV model was constructed to simulate the dynamics of aboveground biomass per individual of three dominant herbaceous plant species (Leymus chinensis, Agropyron cristatum, and Stipa grandis) in the typical grasslands of Inner Mongolia. Comparative analysis indicated that the TDLV model outperforms candidate models, such as Logistic, GM(1,1), GM(1,N), DGM(2,1), and Lotka–Volterra model, in terms of all fitting criteria. The results demonstrate that interspecies competition exhibits clear feedback and suppression effects, with Leymus chinensis playing a central role in regulating community dynamics. The system is locally stable and eventually converges to an equilibrium point, though Stipa grandis maintains relatively low biomass, requiring further monitoring. Time delays are prevalent in the system, influencing dynamic processes and causing damping oscillations as populations approach equilibrium. Full article

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article

This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is useful when studying one-dimensional nonlinear waves involving damping effect, and is used in fluid dynamics, among other applications. Two new mathematical methods that can be used to obtain an approximate solution to this complex non-linear problem are the natural residual power series method and the new iteration transform method. Therefore, it can be deduced that the Caputo operator aids in modeling of the fractional derivatives, as it provides a better description of the physical realities. Thus, the objective of the present work is to advance the knowledge accumulated on the behavior of solutions to the damped Burgers’ equation, as well as to check the applicability of the proposed approaches to other nonlinear fractional partial differential equations. Full article

In order to break the bottleneck of the integer-order transfer function in vehicle ISD (inerter-spring-damper) suspension design, a positive real synthesis design method of vehicle mechatronic ISD suspension based on the fractional-order biquadratic transfer function is proposed. The emergence of the fractional-order components disrupts the equivalence relationship between the passivity of components and the positive realness of integer-order transfer functions in traditional networks. In this paper, the positive real condition of the fractional-order biquadratic transfer function is given. Then, a quarter dynamic model of the vehicle mechatronic ISD suspension is established, and the parameters of the fractional-order biquadratic transfer function and vehicle suspension are obtained by an NSGA-II multi-objective genetic algorithm. Moreover, the structure of the external circuit and the parameters of the electrical components are obtained by the fractional-order passive network synthesis theory. The simulation results show that under the condition of random road input and vehicle speed of 20 m/s, the root-mean-square (RMS) value of the vehicle body acceleration and the dynamic tire load of the fractional-order ISD suspension are reduced by 7.98% and 18.75% compared with the traditional passive suspension, while under the same condition, the integer-order ISD suspension can only reduce by 5.34% and 16.07%, respectively. The results show that employing a fractional-order biquadratic electrical network in the vehicle mechatronic ISD suspension enhances vibration isolation performance compared with the suspension using an integer-order biquadratic electrical network. Full article

This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of at least one solution is proven using Schaefer’s fixed-point theorem, while uniqueness is established via Banach’s fixed-point theorem. Stability is examined through the lens of Ulam–Hyers (U-H) stability. Finally, we illustrate the application of our theoretical findings with a numerical example. Full article

This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “p”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications. Full article

In this short note, we consider fractal interpolation in the Banach space $V^{\theta }\left(I\right)$ of convex Lipschitz functions defined on a compact interval $I\subset \mathbb{R}$. To this end, we define an appropriate iterated function system and exhibit the associated Read–Bajraktarević operator T. We derive conditions for which T becomes a Ratkotch contraction on a closed subspace of $V^{\theta }\left(I\right)$, thus establishing the existence of fractal functions of class $V^{\theta }\left(I\right)$. An example illustrates the theoretical findings. Full article

This paper presents a contemporary introduction of fractional calculus for Type 2 interval-valued functions. Type 2 interval uncertainty involves interval uncertainty with the goal of more assembled perception with reference to impreciseness. In this paper, a Riemann–Liouville fractional-order integral is constructed in Type 2 interval delineated vague encompassment. The exploration of fractional calculus is continued with the manifestation of Riemann–Liouville and Caputo fractional derivatives in the cited phenomenon. In addition, Type 2 interval Laplace transformation is proposed in this text. Conclusively, a mathematical model regarding economic lot maintenance is analyzed as a conceivable implementation of this theoretical advancement. Full article