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Search Results (445)

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Keywords = Liouville equation

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20 pages, 547 KiB  
Article
An Efficient Spectral Method for a Class of Asymmetric Functional-Order Diffusion–Wave Equations Using Generalized Chelyshkov Wavelets
by Quan H. Do and Hoa T. B. Ngo
Symmetry 2025, 17(8), 1230; https://doi.org/10.3390/sym17081230 - 4 Aug 2025
Viewed by 47
Abstract
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 [...] Read more.
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
(This article belongs to the Topic Numerical Methods for Partial Differential Equations)
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13 pages, 268 KiB  
Article
A 4 × 4 Matrix Spectral Problem Involving Four Potentials and Its Combined Integrable Hierarchy
by Wen-Xiu Ma and Ya-Dong Zhong
Axioms 2025, 14(8), 594; https://doi.org/10.3390/axioms14080594 - 1 Aug 2025
Viewed by 90
Abstract
This paper introduces a specific matrix spectral problem involving four potentials and derives an associated soliton hierarchy using the zero-curvature formulation. The bi-Hamiltonian formulation is derived via the trace identity, thereby establishing the hierarchy’s Liouville integrability. This is exemplified through two systems: generalized [...] Read more.
This paper introduces a specific matrix spectral problem involving four potentials and derives an associated soliton hierarchy using the zero-curvature formulation. The bi-Hamiltonian formulation is derived via the trace identity, thereby establishing the hierarchy’s Liouville integrability. This is exemplified through two systems: generalized combined NLS-type equations and modified KdV-type equations. Owing to Liouville integrability, each member of the hierarchy admits a bi-Hamiltonian structure and, consequently, possesses infinitely many symmetries and conservation laws. Full article
17 pages, 333 KiB  
Article
Hille–Yosida-Type Theorem for Fractional Differential Equations with Dzhrbashyan–Nersesyan Derivative
by Vladimir E. Fedorov, Wei-Shih Du, Marko Kostić, Marina V. Plekhanova and Darya V. Melekhina
Fractal Fract. 2025, 9(8), 499; https://doi.org/10.3390/fractalfract9080499 - 30 Jul 2025
Viewed by 256
Abstract
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 [...] Read more.
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
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
33 pages, 403 KiB  
Article
Some Further Insight into the Sturm–Liouville Theory
by Salvatore De Gregorio, Lamberto Lamberti and Paolo De Gregorio
Mathematics 2025, 13(15), 2405; https://doi.org/10.3390/math13152405 - 26 Jul 2025
Viewed by 139
Abstract
Some classical texts on the Sturm–Liouville equation (p(x)y)q(x)y+λρ(x)y=0 are revised to highlight further properties of its solutions. Often, in the [...] Read more.
Some classical texts on the Sturm–Liouville equation (p(x)y)q(x)y+λρ(x)y=0 are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, ρ=const is assumed (and, further, ρ=1). Instead, here we preserve ρ(x) and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to yq(x)y+λρ(x)y=0. We show that many results are identical with those with λρq=const. This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex λ. We provide here many detailed properties of the solutions of the above equation, with ρ=ρ(x). The conclusion, as we may easily infer, is that, for large enough λ, locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of λ. The particular case of q(x)=αρ(x) is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated. Full article
27 pages, 929 KiB  
Article
A Stochastic Schrödinger Evolution System with Complex Potential Symmetry Using the Riemann–Liouville Fractional Derivative: Qualitative Behavior and Trajectory Controllability
by Dimplekumar Chalishajar, Ravikumar Kasinathan, Ramkumar Kasinathan, Dhanalakshmi Kasinathan and Himanshu Thaker
Symmetry 2025, 17(8), 1173; https://doi.org/10.3390/sym17081173 - 22 Jul 2025
Viewed by 180
Abstract
This work investigates fractional stochastic Schrödinger evolution equations in a Hilbert space, incorporating complex potential symmetry and Poisson jumps. We establish the existence of mild solutions via stochastic analysis, semigroup theory, and the Mönch fixed-point theorem. Sufficient conditions for exponential stability are derived, [...] Read more.
This work investigates fractional stochastic Schrödinger evolution equations in a Hilbert space, incorporating complex potential symmetry and Poisson jumps. We establish the existence of mild solutions via stochastic analysis, semigroup theory, and the Mönch fixed-point theorem. Sufficient conditions for exponential stability are derived, ensuring asymptotic decay. We further explore trajectory controllability, identifying conditions for guiding the system along prescribed paths. A numerical example is provided to validate the theoretical results. Full article
(This article belongs to the Special Issue Advances in Nonlinear Systems and Symmetry/Asymmetry)
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13 pages, 9670 KiB  
Article
Exact Solitary Wave Solutions and Sensitivity Analysis of the Fractional (3+1)D KdV–ZK Equation
by Asif Khan, Fehaid Salem Alshammari, Sadia Yasin and Beenish
Fractal Fract. 2025, 9(7), 476; https://doi.org/10.3390/fractalfract9070476 - 21 Jul 2025
Viewed by 293
Abstract
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 [...] Read more.
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
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23 pages, 1065 KiB  
Article
Modeling and Neural Network Approximation of Asymptotic Behavior for Delta Fractional Difference Equations with Mittag-Leffler Kernels
by Pshtiwan Othman Mohammed, Muteb R. Alharthi, Majeed Ahmad Yousif, Alina Alb Lupas and Shrooq Mohammed Azzo
Fractal Fract. 2025, 9(7), 452; https://doi.org/10.3390/fractalfract9070452 - 9 Jul 2025
Viewed by 348
Abstract
The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order 0.5 subject to a given initial condition. We establish [...] Read more.
The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order 0.5 subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article
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20 pages, 317 KiB  
Article
Linking Controllability to the Sturm–Liouville Problem in Ordinary Time-Varying Second-Order Differential Equations
by Manuel De la Sen
AppliedMath 2025, 5(3), 87; https://doi.org/10.3390/appliedmath5030087 - 8 Jul 2025
Viewed by 245
Abstract
This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. [...] Read more.
This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values. Full article
16 pages, 1929 KiB  
Article
Dynamical Behavior of Solitary Waves for the Space-Fractional Stochastic Regularized Long Wave Equation via Two Distinct Approaches
by Muneerah Al Nuwairan, Bashayr Almutairi and Anwar Aldhafeeri
Mathematics 2025, 13(13), 2193; https://doi.org/10.3390/math13132193 - 4 Jul 2025
Viewed by 200
Abstract
This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the [...] Read more.
This study investigates the influence of multiplicative noise—modeled by a Wiener process—and spatial-fractional derivatives on the dynamics of the space-fractional stochastic Regularized Long Wave equation. By employing a complete discriminant polynomial system, we derive novel classes of fractional stochastic solutions that capture the complex interplay between stochasticity and nonlocality. Additionally, the variational principle, derived by He’s semi-inverse method, is utilized, yielding additional exact solutions that are bright solitons, bright-like solitons, kinky bright solitons, and periodic structures. Graphical analyses are presented to clarify how variations in the fractional order and noise intensity affect essential solution features, such as amplitude, width, and smoothness, offering deeper insight into the behavior of such nonlinear stochastic systems. Full article
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36 pages, 4192 KiB  
Article
Fractional Calculus for Neutrosophic-Valued Functions and Its Application in an Inventory Lot-Sizing Problem
by Rakibul Haque, Mostafijur Rahaman, Adel Fahad Alrasheedi, Dimplekumar Chalishajar and Sankar Prasad Mondal
Fractal Fract. 2025, 9(7), 433; https://doi.org/10.3390/fractalfract9070433 - 30 Jun 2025
Cited by 1 | Viewed by 347
Abstract
Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of [...] Read more.
Past experiences and memory significantly contribute to self-learning and improved decision-making. These can assist decision-makers in refining their strategies for better outcomes. Fractional calculus is a tool that captures a system’s memory or past experience through its repeating patterns. In the realm of uncertainty, neutrosophic set theory demonstrates greater suitability, as it independently assesses membership, non-membership, and indeterminacy. In this article, we aim to extend the theory further by introducing fractional calculus for neutrosophic-valued functions. The proposed method is applied to an economic lot-sizing problem. Numerical simulations of the lot-sizing model suggest that strong memory employment with a memory index of 0.1 can lead to an increase in average profit in memory-independent phenomena with a memory index of 1 by approximately 44% to 49%. Additionally, the neutrosophic environment yields superior profitability results compared to both precise and imprecise settings. The synergy of fractional-order dynamics and neutrosophic uncertainty modeling paves the way for enhanced decision-making in complex, ambiguous environments. Full article
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25 pages, 325 KiB  
Review
Advances in Fractional Lyapunov-Type Inequalities: A Comprehensive Review
by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Foundations 2025, 5(2), 18; https://doi.org/10.3390/foundations5020018 - 27 May 2025
Viewed by 564
Abstract
In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, ψ-Riemann–Liouville, [...] Read more.
In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, ψ-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article
(This article belongs to the Section Mathematical Sciences)
14 pages, 287 KiB  
Article
Bicomplex k-Mittag-Leffler Functions with Two Parameters: Theory and Applications to Fractional Kinetic Equations
by Ahmed Bakhet, Shahid Hussain, Mohra Zayed and Mohamed Fathi
Fractal Fract. 2025, 9(6), 344; https://doi.org/10.3390/fractalfract9060344 - 26 May 2025
Viewed by 333
Abstract
In this paper, we aim to extend the bicomplex two-parameter Mittag-Leffler (M-L) function by introducing a new k-parameter. This results in the definition of the bicomplex k-M-L function with two parameters. This generalization offers more flexibility and broader applicability in modeling [...] Read more.
In this paper, we aim to extend the bicomplex two-parameter Mittag-Leffler (M-L) function by introducing a new k-parameter. This results in the definition of the bicomplex k-M-L function with two parameters. This generalization offers more flexibility and broader applicability in modeling complex fractional systems. We explore its key properties, develop new theorems, and establish the corresponding k-Riemann–Liouville fractional calculus within the bicomplex setting for the extended function. Furthermore, we solve several fractional differential equations using the bicomplex k-M-L function with two parameters. The results prove the enhanced flexibility and generality of the proposed function, particularly in deriving fractional kinetic equations, offering novel insights beyond existing bicomplex fractional models. Full article
18 pages, 320 KiB  
Article
Solvability of a Riemann–Liouville-Type Fractional-Impulsive Differential Equation with a Riemann–Stieltjes Integral Boundary Condition
by Keyu Zhang, Donal O’Regan and Jiafa Xu
Fractal Fract. 2025, 9(5), 323; https://doi.org/10.3390/fractalfract9050323 - 19 May 2025
Viewed by 363
Abstract
In this work, we address the solvability of a Riemann–Liouville-type fractional-impulsive integral boundary value problem. Under some conditions on the spectral radius corresponding to the related linear operator, we use fixed-point methods to obtain several existence theorems for our problem. In particular, we [...] Read more.
In this work, we address the solvability of a Riemann–Liouville-type fractional-impulsive integral boundary value problem. Under some conditions on the spectral radius corresponding to the related linear operator, we use fixed-point methods to obtain several existence theorems for our problem. In particular, we obtain the existence of multiple positive solutions via the Avery–Peterson fixed-point theorem. Note that our linear operator depends on the impulsive term and the integral boundary condition. Full article
22 pages, 2193 KiB  
Article
Novel Hybrid Function Operational Matrices of Fractional Integration: An Application for Solving Multi-Order Fractional Differential Equations
by Seshu Kumar Damarla and Madhusree Kundu
AppliedMath 2025, 5(2), 55; https://doi.org/10.3390/appliedmath5020055 - 10 May 2025
Viewed by 986
Abstract
Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder [...] Read more.
Although fractional calculus has evolved significantly since its origin in the 1695 correspondence between Leibniz and L’Hôpital, the numerical treatment of multi-order fractional differential equations remains a challenge. Existing methods are often either computationally expensive or reliant on complex operational frameworks that hinder their broader applicability. In the present study, a novel numerical algorithm is proposed based on orthogonal hybrid functions (HFs), which were constructed as linear combinations of piecewise constant sample-and-hold functions and piecewise linear triangular functions. These functions, belonging to the class of degree-1 orthogonal polynomials, were employed to obtain the numerical solution of multi-order fractional differential equations defined in the Caputo sense. A generalized one-shot operational matrix was derived to explicitly express the Riemann–Liouville fractional integral of HFs in terms of the HFs themselves. This allowed the original multi-order fractional differential equation to be transformed directly into a system of algebraic equations, thereby simplifying the solution process. The developed algorithm was then applied to a range of benchmark problems, including both linear and nonlinear multi-order FDEs with constant and variable coefficients. Numerical comparisons with well-established methods in the literature revealed that the proposed approach not only achieved higher accuracy but also significantly reduced computational effort, demonstrating its potential as a reliable and efficient numerical tool for fractional-order modeling. Full article
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18 pages, 1218 KiB  
Article
Modification to an Auxiliary Function Method for Solving Space-Fractional Stochastic Regularized Long-Wave Equation
by Muneerah Al Nuwairan and Adel Elmandouh
Fractal Fract. 2025, 9(5), 298; https://doi.org/10.3390/fractalfract9050298 - 4 May 2025
Cited by 2 | Viewed by 362
Abstract
This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function [...] Read more.
This study aims to explore the effect of spatial-fractional derivatives and the multiplicative standard Wiener process on the solutions of the stochastic fractional regularized long-wave equation (SFRLWE) and contribute to its analysis. We introduce a new systematic method that combines the auxiliary function method with the complete discriminant polynomial system. This method proves to be effective in discovering precise solutions for stochastic fractional partial differential equations (SFPDEs), including special cases. Applying this method to the SFRLWE yields new exact solutions, offering fresh insights. We investigated how noise affects stochastic solutions and discovered that more intense noise can result in flatter surfaces. We note that multiplicative noise can stabilize the solution, and we show how fractional derivatives influence the dynamics of noise. We found that the noise strength and fractional derivative affect the width, amplitude, and smoothness of the obtained solutions. Additionally, we conclude that multiplicative noise impacts and stabilizes the behavior of SFRLWE solutions. Full article
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