Search Results (482)

This work analyzes the aero-elastic oscillations of cantilever beams reinforced by carbon nanotubes (CNTs). Four different distributions of single-walled CNTs are assumed as the reinforcing phase, in the thickness direction of the polymeric matrix. A modified third-order piston theory is used as an accurate tool to model the supersonic air flow, rather than a first-order piston theory. The nonlinear dynamic equation governing the problem accounts for Von Kármán-type nonlinearities, and it is derived from Hamilton’s principle. Then, the Galerkin decomposition technique is adopted to discretize the nonlinear partial differential equation into a nonlinear ordinary differential equation. This is solved analytically according to a multiple time scale method. A comprehensive parametric analysis was conducted to assess the influence of CNT volume fraction, beam slenderness, Mach number, and thickness ratio on the fundamental frequency and lateral dynamic deflection. Results indicate that FG-X reinforcement yields the highest frequency response and lateral deflection, followed by UD and FG-A patterns, whereas FG-O consistently exhibits the lowest performance metrics. An increase in CNT volume fraction and a reduction in slenderness ratio enhance the system’s stiffness and frequency response up to a critical threshold, beyond which a damped beating phenomenon emerges. Moreover, higher Mach numbers and greater thickness ratios significantly amplify both frequency response and lateral deflections, although damping rates tend to decrease. These findings provide valuable insights into the optimization of CNTR composite structures for advanced aeroelastic applications under supersonic conditions, as useful for many engineering applications. Full article

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

Asymmetric functional-order (variable-order) fractional diffusion–wave equations (FO-FDWEs) introduce considerable computational challenges, as the fractional order of the derivatives can vary spatially or temporally. To overcome these challenges, a novel spectral method employing generalized fractional-order Chelyshkov wavelets (FO-CWs) is developed to efficiently solve such equations. In this approach, the Riemann–Liouville fractional integral operator of variable order is evaluated in closed form via a regularized incomplete Beta function, enabling the transformation of the governing equation into a system of algebraic equations. This wavelet-based spectral scheme attains extremely high accuracy, yielding significantly lower errors than existing numerical techniques. In particular, numerical results show that the proposed method achieves notably improved accuracy compared to existing methods under the same number of basis functions. Its strong convergence properties allow high precision to be achieved with relatively few wavelet basis functions, leading to efficient computations. The method’s accuracy and efficiency are demonstrated on several practical diffusion–wave examples, indicating its suitability for real-world applications. Furthermore, it readily applies to a wide class of fractional partial differential equations (FPDEs) with spatially or temporally varying order, demonstrating versatility for diverse applications. Full article

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

We develop novel extensions in the theory of weighted Lorentz spaces. In particular, we generalize classical results by introducing variable-exponent Lorentz spaces, establish sharp constants and quantitative bounds for maximal operators, and extend the framework to encompass fractional maximal operators. Moreover, we analyze endpoint cases through the study of oscillation operators and reveal new connections with weighted Hardy spaces. These results provide a unifying approach that not only refines existing inequalities but also opens new avenues in harmonic analysis and partial differential equations. Full article

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved $(G^{\prime }/G)$ expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering. Full article

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

The implementation of chaotic behavior and a sensitivity assessment of the newly developed M-fractional Kuralay-II equation are the foremost objectives of the present study. This equation has significant possibilities in control systems, electrical circuits, seismic wave propagation, economic dynamics, groundwater flow, image and signal denoising, complex biological systems, optical fibers, plasma physics, population dynamics, and modern technology. These applications demonstrate the versatility and advantageousness of the stated model for complex systems in various scientific and engineering disciplines. One more essential objective of the present research is to find closed-form wave solutions of the assumed equation based on the $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach. The results achieved are in exponential, rational, and trigonometric function forms. Our findings are more novel and also have an exclusive feature in comparison with the existing results. These discoveries substantially expand our understanding of nonlinear wave dynamics in various physical contexts in industry. By simply selecting suitable values of the parameters, three-dimensional (3D), contour, and two-dimensional (2D) illustrations are produced displaying the diagrammatic propagation of the constructed wave solutions that yield the singular periodic, anti-kink, kink, and singular kink-shape solitons. Future improvements to the model may also benefit from what has been obtained as well. The various assortments of solutions are provided by the described procedure. Finally, the framework proposed in this investigation addresses additional fractional nonlinear partial differential equations in mathematical physics and engineering with excellent reliability, quality of effectiveness, and ease of application. Full article

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article

This study introduces a novel analytical technique known as the double local fractional Yang–Laplace transform method ($LF{L}_{\zeta }^2$) and rigorously investigates its foundational properties, including linearity, differentiation, and convolution. The proposed method is formulated via double local fractional integrals, enabling a robust mechanism for addressing local fractional partial differential equations defined on fractal domains, particularly Cantor sets. Through a series of illustrative examples, we demonstrate the applicability and efficacy of the $LF{L}_{\zeta }^2$ transform in solving complex local fractional partial differential equation models. Special emphasis is placed on the local fractional Laplace equation, the linear local fractional Klein–Gordon equation, and other models, wherein the method reveals significant computational and analytical advantages. The results substantiate the method’s potential as a powerful tool for broader classes of problems governed by local fractional dynamics on fractal geometries. Full article

In this article, we present the extended simple equations method (SEsM) for finding exact solutions to systems of fractional nonlinear partial differential equations (FNPDEs). The expansions made to the original SEsM algorithm are implemented in several directions: (1) In constructing analytical solutions: exact solutions to FNPDE systems are presented by simple or complex composite functions, including combinations of solutions to two or more different simple equations with distinct independent variables (corresponding to different wave velocities); (2) in selecting appropriate fractional derivatives and appropriate wave transformations: the choice of the type of fractional derivatives for each system of FNPDEs depends on the physical nature of the modeled real process. Based on this choice, the range of applicable wave transformations that are used to reduce FNPDEs to nonlinear ODEs has been expanded. It includes not only various forms of fractional traveling wave transformations but also standard traveling wave transformations. Based on these methodological enhancements, a generalized SEsM algorithm has been developed to derive exact solutions of systems of FNPDEs. This algorithm provides multiple options at each step, enabling the user to select the most appropriate variant depending on the expected wave dynamics in the modeled physical context. Two specific variants of the generalized SEsM algorithm have been applied to obtain exact solutions to two time-fractional shallow-water-like systems. For generating these exact solutions, it is assumed that each system variable in the studied models exhibits multi-wave behavior, which is expressed as a superposition of two waves propagating at different velocities. As a result, numerous novel multi-wave solutions are derived, involving combinations of hyperbolic-like, elliptic-like, and trigonometric-like functions. The obtained analytical solutions can provide valuable qualitative insights into complex wave dynamics in generalized spatio-temporal dynamical systems, with relevance to areas such as ocean current modeling, multiphase fluid dynamics and geophysical fluid modeling. Full article