Search Results (443)

In this investigation, the nonlinear dust-acoustic waves in the lunar terminator region are studied in a three-component complex plasma comprising Boltzmann-distributed electrons and ions and inertial, cold, negatively charged dust grains. The fluid model is reduced, via the reductive perturbation technique, to a planar Korteweg–de Vries (KdV) equation that governs the evolution of small-amplitude dust-acoustic structures in this environment. Hirota’s direct method is then employed to derive exact multiple-soliton solutions, which allow us to examine the parameter dependence of dust-acoustic solitons and to characterize their overtaking collisions. The analysis shows that the soliton polarity and amplitude are controlled by the equilibrium electron–ion density ratio and the electron-to-ion temperature ratio, and that multi-soliton interactions remain elastic, with only finite phase shifts after collision. In the second part of the study, the planar integer KdV model is generalized to a time-fractional KdV (FKdV) equation to incorporate nonlocal temporal memory effects in the dust-acoustic dynamics. This FKdV equation is analyzed using two analytical approximation schemes: the Tantawy technique, recently proposed as a direct and rapidly convergent approach to fractional evolution equations, and the new iterative method, a widely used high-accuracy scheme in the fractional literature. For both methods, higher-order approximations are constructed, and their absolute and global maximum residual errors are quantified. The results demonstrate that the Tantawy technique provides compact approximations with superior accuracy and stability compared with the new iterative method for the present FKdV-soliton problem. The combined integer- and fractional-analytic framework provides a physically transparent framework for understanding how nonlinearity, dispersion, and fractional memory jointly shape dust-acoustic solitary structures in the electrostatically complex lunar terminator plasma, which is of paramount interest for future lunar missions like Luna-25 and Luna-27. Full article

The solubility values of polyphenolic compounds in different extraction solvents are crucial for their recovery from natural matrices. Hansen solubility parameters (HSPs) stand out as a predictive tool for evaluating solute-solvent affinity and thus rational solvent selection for extraction processes. In this study, HSPs of trans-resveratrol and hesperetin were calculated using a semi-empirical method to assess the capability to predict the solubility behavior of both polyphenols in organic binary solvent mixtures. Experimental solubility of both polyphenols was determined in up to 21 monosolvents at 298.15 K and 0.1 MPa and used to classify them to iteratively calculate HSPs. Calculated HSPs were compared and discussed with literature values in terms of molecular interactions, demonstrating a fair agreement. Solubility of trans-resveratrol and hesperetin was then determined in methanol + MEK, ethanol + MEK, methanol + MiBK, ethanol + MiBK, and methanol + ethanol binary solvent mixtures. trans-Resveratrol achieved higher mole fraction solubility than hesperetin in all binary mixtures across the whole molar fraction range except in methanol + MiBK. Both compounds exhibited enhanced solubility in all alcohols + ketone binary mixtures, attributed to synergistic solvent effects. HSP analysis revealed a minimum Hansen distance between solute and solvent mixtures at compositions corresponding to the solubility maximum in synergistic systems. Additionally, calculated HSPs proved to effectively estimate the concentration at which this phenomenon occurs in all tested systems, reaching a robust correlation between maximum solubility and minimum Hansen distance. Overall, insights from this study underscore the effectiveness of experimentally derived HSPs in predicting the solubility behavior of polyphenols and seek to provide valuable guidance on solvent selection strategies for the recovery of bioactive compounds. Full article

Fractional differential equations provide a flexible framework for describing evolutionary processes in complex media, where nonlocality and memory effects play central roles, and classical integer-order models are frequently inadequate to capture these behaviors. In this work, we revisit the time-fractional Harry Dym (HD) evolution equation in the Caputo sense and construct high-precision analytical approximations using the recently developed Tantawy technique (TT). The method generates a rapidly convergent fractional-power series in time without resorting to perturbative assumptions, auxiliary decomposition polynomials, linearization procedures, or integral transforms, and it remains computationally economical even at high approximation orders. Closed, compact expressions are derived up to the fifth-order approximation and can be systematically extended, yielding excellent agreement with the known exact solution of the classical/integer HD model and with approximations obtained via the new iterative method. A detailed error analysis is carried out by computing absolute and maximum residual errors over the entire computational domain, demonstrating the accuracy, stability, and robustness of the TT for the HD-type fractional nonlinear evolution equation. From a physical perspective, the proposed framework offers a reliable tool for modeling nonlinear wave structures in dispersive media with significant memory and, more generally, for treating a broad class of fractional nonlinear wave equations arising in physics and engineering. Full article

This paper presents a comprehensive study on constructing exact and approximate solutions to Cauchy problems for time-fractional systems with variable coefficients. An innovative iterative approach is developed for solving functional equations with initial conditions, combining rigorous mathematical foundations with practical computational efficiency. The proposed technique effectively handles the nonlocal nature of fractional operators through a carefully designed iterative scheme that maintains simplicity while achieving high accuracy. It demonstrates particular strength in solving nonlinear systems with well-defined conditions and variable coefficients, where traditional methods often fail. Through systematic theoretical analysis and numerical validation, we establish the method’s convergence properties and computational advantages, showing its capability to generate both exact closed-form solutions, when available, and high-precision approximations otherwise. The approach remains computationally tractable even for complex cases where variable coefficients and memory effects of fractional systems present significant challenges to conventional solution approaches. Full article

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems. Full article

In this paper, we developed and constructed the Haar wavelet (HW) for the two-dimensional (2D) time-fractional neuronal dynamics model (TFNDM) with the dynamical electro-diffusion behaviour of ions in nerve cells. The Haar wavelet method is considered in space and the difference method in time for the time-fractional Riemann–Liouville (TFRL) derivative. The calculation CPU time of this proposed method is very short because the Haar matrix and Haar integral matrix are stored only once and used for each iteration. Moreover, the results show that the solution of the Haar wavelet method is good even when there are fewer grid points. Full article

With the increasing demands for autonomy and coverage efficiency in tasks such as security patrol and post-disaster exploration using mobile robots, achieving random, efficient, and complete coverage path planning has become a critical challenge. Traditional chaotic path planning methods, while capable of generating unpredictable trajectories, still have limitations in terms of randomness strength, traversal uniformity, and convergence coverage. To address this, this study proposes a complete-coverage random path planning method based on a novel four-dimensional fractal-fractional multi-scroll chaotic system. The main contributions of this research are as follows: First, by introducing additional state variables and fractal-fractional operators into the classical Chen system, a fractal-fractional chaotic system with a multi-scroll attractor structure is constructed. The output of this system is then mapped into robot angular velocity commands to achieve area coverage in unknown environments. Key findings include: the novel chaotic system possesses two positive Lyapunov exponents; Spectral Entropy (SE) and Complexity (CO) analyses indicate that when parameter B is fixed and the fractional order α increases, the dynamic complexity of the system significantly rises; in a 50 × 50 grid environment, the robot driven by this system achieved a coverage rate of 98.88% within 10,000 iterations, outperforming methods based on Lorenz, Chua systems, and random walks; ablation experiments further demonstrate that the combined effects of the fractal order β, fractional order α, and multi-scroll nonlinear terms are key to enhancing system complexity and coverage performance. The significance of this study lies in that it not only provides new ideas for constructing complex chaotic systems but also offers a reliable theoretical foundation and practical solution for mobile robots to perform efficient, random, and high-coverage autonomous inspection tasks in unknown regions. Full article

The existing methods for solving non-linear equations encounter convergence issues and computing constraints, especially when used in fractional-order or complex non-linear problems. This study develops a higher-order fractional technique for solving non-linear equations based on the Caputo fractional derivative. The proposed method uses a fractional framework to improve local convergence and stability while ensuring high efficiency in every iteration step. Local convergence analysis using generalized Taylor series expansion reveals that the order of the new fractional scheme for solving non-linear equations is $5¢+1$, where $¢\in$ (0,1] represents the Caputo fractional order, determining the memory depth of the Caputo fractional derivative. The performance of the method is further investigated using a variety of non-linear problems from engineering optimization and applied sciences, such as engineering control systems, computational chemistry, thermodynamics models, and operations research, such as inventory optimization. Analyzing the key performance metrics, such as dynamical analysis, percentage convergence, residual error, and computation time, confirms the advantages of the developed method over the state-of-the-art. This study provides a solid framework for higher-order fractional iterative approaches, paving the way for advanced applications of non-linear problems. Full article

The present study aims to integrate the Adomian decomposition method with the Sumudu transform for solving delay fractional partial differential equations. Integrating these two methods enhances accuracy and computational efficiency, providing an effective approach for handling delays in fractional-order models. The decomposition method effectively decomposes complex fractional differential equations into convergent series, while the Sumudu transform simplifies and transforms these equations, facilitating the analysis of delayed systems. Its effectiveness has been demonstrated through multiple examples. Full article

In this paper, we develop an accelerated three-step iterative scheme for the approximation of fixed points of contraction mappings in Banach spaces, with a particular focus on applications to fractional models. Strong convergence of the proposed iteration is established under standard contraction assumptions, together with stability and data dependence results. A refined rate of convergence analysis shows that the new scheme achieves a smaller effective contraction factor and converges faster than several classical two- and three-step iterative methods, including the Picard, Mann, Ishikawa, and S-iteration processes. The theoretical results are applied to Caputo-type fractional differential equations by reformulating the associated boundary value problems as fixed-point equations. Existence and uniqueness of solutions follow from the Banach contraction principle, while the accelerated convergence of the proposed iteration leads to improved numerical efficiency. Extensive numerical experiments, including fractional differential equations and nonlinear contraction mappings on the real line, are presented to validate the theoretical findings. The results demonstrate that the proposed three-step iteration provides an effective and reliable computational tool for fractional and non-local models. Full article

In this work, we propose a fractional Jacobian–based parallel two-stage iterative framework for the numerical solution of nonlinear systems arising from elliptic PDE discretizations. The core of the approach is a high-order fractional two-step scheme (S1), which combines a linear Newton-type correction with a quadratic fractional correction and incorporates a structured parallel interaction mechanism inspired by Weierstrass-type schemes. Under standard regularity assumptions, a rigorous local convergence analysis shows that the S1 scheme provides a high-order local correction mechanism, yielding a convergence order of $2\mu +3$ under suitable local accuracy conditions. To enhance robustness with respect to the choice of initial guesses, a safeguarded realization of the method, denoted by SBVM_*, is introduced. Since the safeguard mechanism may modify the local iteration map, convergence of SBVM_* is ensured under appropriate acceptance conditions, while its asymptotic behavior coincides with that of the S1 scheme once the safeguard becomes inactive. The dynamical behavior of the resulting iterative map is further investigated through bifurcation diagrams and Lyapunov exponent analysis, providing practical guidelines for parameter selection and enabling the identification of stable operating regimes while avoiding chaotic behavior. Extensive numerical experiments involving linear and nonlinear elliptic benchmark problems from engineering and biomedical applications demonstrate that SBVM_* achieves improved convergence behavior, enhanced numerical stability, and reduced computational cost relative to existing parallel solvers such as ELVM_* and ACVM_*. The proposed framework therefore provides an effective and scalable numerical approach for the solution of nonlinear elliptic models arising in biomedical and engineering contexts. Full article

This study concentrates on a higher-order Hadamard fractional differential equation defined on an unbounded interval, which is subject to integral and discrete boundary conditions. Through the employment of the upper and lower solution method combined with Banach’s contraction mapping principle, we have successfully established distinct iterative sequences for the targeted differential equation. To demonstrate the practical relevance of our theoretical findings, we provide a typical example. Full article

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation. Full article

Owing to their high flexibility, autonomous operation, and rapid deployment capability, unmanned aerial vehicles (UAVs) serve as effective aerial platforms for sensing and communication in remote and time-critical scenarios. However, their limited onboard energy budget poses a significant bottleneck for sustained operations. This paper investigates an energy-efficient UAV-assisted integrated sensing and communication (ISAC) system, aiming to maximize the sensing energy efficiency (SEE), defined as the ratio of the total radar estimation rate to the total energy consumption. Unlike prior works focused solely on rate maximization or fairness, our design jointly optimizes the UAV’s 3D trajectory, task scheduling, and power allocation under kinematic and coverage constraints to maximize the SEE. To solve the formulated non-convex fractional programming problem, we propose an efficient iterative algorithm based on the Dinkelbach method and block coordinate descent (BCD). Simulation results demonstrate that the proposed scheme achieves a superior trade-off between sensing performance and energy consumption. Full article

The Kudryashov–Sinelshchikov equation (KSE) is crucial in modeling pressure waves in liquids containing gas bubbles, capturing both nonlinear wave phenomena and dispersion effects. This article applies the reproducing kernel Hilbert space method (RKHSM) to find a numerical solution for the time-fractional KSE. We develop a numerical solution to the KSE using the RKHSM, which offers an efficient and accurate approach for solving nonlinear partial differential equations due to its smoothness and orthogonality properties. The key components of this method include the reproducing kernel (RK) theory, important Hilbert spaces, normal basis, orthogonalization, and homogenization. We construct an appropriate RK and derive an iterative solution that converges rapidly to the exact solution. The effectiveness of this approach is demonstrated through numerical simulations in which we analyze the behavior of pressure waves and compare the results with existing analytical and numerical solutions. The RKHSM consistently demonstrates highly accurate, rapid convergence, and remarkable stability across a wide range of problems. Thus, the RKHSM is a promising tool for studying wave propagation in bubbly liquids. Full article