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15 pages, 422 KiB

Open AccessArticle

New Results on the Stability and Existence of Langevin Fractional Differential Equations with Boundary Conditions

by Rahman Ullah Khan, Maria Samreen, Gohar Ali and Ioan-Lucian Popa

Fractal Fract. 2025, 9(2), 127; https://doi.org/10.3390/fractalfract9020127 - 18 Feb 2025

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 [...] Read more.

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

(This article belongs to the Section General Mathematics, Analysis)

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12 pages, 501 KiB

Open AccessArticle

Concentrated Couple in a Plane and in a Half-Plane in the Framework of Fractional Nonlocal Elasticity

by Yuriy Povstenko, Tamara Kyrylych, Bożena Woźna-Szcześniak, Ireneusz Szcześniak and Andrzej Yatsko

Appl. Sci. 2025, 15(4), 2048; https://doi.org/10.3390/app15042048 - 15 Feb 2025

Viewed by 280

Abstract

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to [...] Read more.

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to elasticity problems for a concentrated couple in a plane and on the boundary of a half-plane within framework of a new theory of nonlocal elasticity, where the nonlocal kernel is the Green’s function of the Cauchy problem for the fractional diffusion equation. The obtained solutions are free from nonphysical singularities that appear in the classical local elasticity solutions. Full article

(This article belongs to the Special Issue Deformation and Fracture Behaviors of Materials)

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21 pages, 1488 KiB

Open AccessArticle

Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods

by Azzh Saad Alshehry and Rasool Shah

Fractal Fract. 2025, 9(2), 107; https://doi.org/10.3390/fractalfract9020107 - 10 Feb 2025

Viewed by 437

Abstract

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 [...] Read more.

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

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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24 pages, 619 KiB

Open AccessArticle

Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives

by F. Gassem, Mohammed Almalahi, Osman Osman, Blgys Muflh, Khaled Aldwoah, Alwaleed Kamel and Nidal Eljaneid

Fractal Fract. 2025, 9(2), 104; https://doi.org/10.3390/fractalfract9020104 - 8 Feb 2025

Viewed by 374

Abstract

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. [...] Read more.

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

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function, 2nd Edition)

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23 pages, 707 KiB

Open AccessArticle

Novel Fractional Boole’s-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis

by Wali Haider, Abdul Mateen, Hüseyin Budak, Asia Shehzadi and Loredana Ciurdariu

Mathematics 2025, 13(4), 551; https://doi.org/10.3390/math13040551 - 7 Feb 2025

Viewed by 342

Abstract

The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) [...] Read more.

The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Boole’s-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole’s-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole’s formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations. Full article

(This article belongs to the Section E: Applied Mathematics)

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35 pages, 2025 KiB

Open AccessArticle

Fractional Calculus for Type 2 Interval-Valued Functions

by Mostafijur Rahaman, Dimplekumar Chalishajar, Kamal Hossain Gazi, Shariful Alam, Soheil Salahshour and Sankar Prasad Mondal

Fractal Fract. 2025, 9(2), 102; https://doi.org/10.3390/fractalfract9020102 - 5 Feb 2025

Viewed by 388

Abstract

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 [...] Read more.

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

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

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31 pages, 817 KiB

Open AccessArticle

Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model

by Mohamed S. Algolam, Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan and Saeed Islam

Fractal Fract. 2025, 9(2), 92; https://doi.org/10.3390/fractalfract9020092 - 1 Feb 2025

Viewed by 410

Abstract

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a [...] Read more.

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a tunable power parameter within a non-singular kernel, enabling a nuanced representation of memory effects not achievable with traditional fixed-kernel derivatives. This flexible framework is analyzed using fixed-point theory, rigorously establishing the existence and uniqueness of solutions for four symmetric cases under specific conditions. Furthermore, we demonstrate the Hyers–Ulam stability, confirming the robustness of these solutions against small perturbations. The versatility and generalizability of this framework is underscored by its application to an epidemiological model of transmission of Hepatitis B Virus (HBV) and numerical simulations for all four symmetric cases. This study presents findings in both theoretical and applied aspects of fractional calculus, introducing an alternative framework for modeling complex systems with memory processes, offering opportunities for more sophisticated and accurate models and new avenues for research in fractional calculus and its applications. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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19 pages, 389 KiB

Open AccessFeature PaperArticle

On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability

by Ekaterina Madamlieva and Mihail Konstantinov

Mathematics 2025, 13(3), 484; https://doi.org/10.3390/math13030484 - 31 Jan 2025

Viewed by 488

Abstract

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. [...] Read more.

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. The present research seeks to reinvigorate interest in such delays within sophisticated frameworks of differential equations, particularly those involving fractional calculus. The primary objectives are to thoroughly examine neutral-type fractional differential equations with iterated delays and provide novel insights into their existence and uniqueness by applying Bielecki’s and Chebyshev’s norms for solution constraints analysis. Additionally, this work establishes Hyers–Ulam–Mittag–Leffler stability for these equations. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 3rd Edition)

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Plot of <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>Γ</mo> <mo>(</mo> <mi>z</mi> <mo>)</mo> <mo>Γ</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mi>z</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>16</mn> <mo>]</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> Full article ">Figure 2
Plot of <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>Γ</mo> <mo>(</mo> <mi>α</mi> <mo>)</mo> <mo>Γ</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>α</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> Full article ">Figure 3
Plot of the solutions <math display="inline"><semantics> <mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>x</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of Example 2. Full article ">

18 pages, 497 KiB

Open AccessArticle

Strict Stability of Fractional Differential Equations with a Caputo Fractional Derivative with Respect to Another Function

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Mathematics 2025, 13(3), 452; https://doi.org/10.3390/math13030452 - 29 Jan 2025

Viewed by 532

Abstract

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider [...] Read more.

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

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19 pages, 894 KiB

Open AccessArticle

Fixed/Preassigned Time Synchronization of Impulsive Fractional-Order Reaction–Diffusion Bidirectional Associative Memory (BAM) Neural Networks

by Rouzimaimaiti Mahemuti, Abdujelil Abdurahman and Ahmadjan Muhammadhaji

Fractal Fract. 2025, 9(2), 88; https://doi.org/10.3390/fractalfract9020088 - 28 Jan 2025

Viewed by 466

Abstract

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, [...] Read more.

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, this study presents certain characteristics of fractional-order calculus and several lemmas pertaining to the stability of general impulsive nonlinear systems, specifically focusing on FXT and PDT stability. Subsequently, we utilize a novel controller and Lyapunov functions to establish new sufficient criteria for achieving FXT and PDT synchronizations. Finally, a numerical simulation is presented to ascertain the theoretical dependency. Full article

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12 pages, 3493 KiB

Open AccessArticle

On a Preloaded Compliance System of Fractional Order: Numerical Integration

by Marius-F. Danca

Fractal Fract. 2025, 9(2), 84; https://doi.org/10.3390/fractalfract9020084 - 26 Jan 2025

Viewed by 415

Abstract

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, [...] Read more.

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

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21 pages, 631 KiB

Open AccessArticle

Fractional Mathieu Equation with Two Fractional Derivatives and Some Applications

by Ahmed Salem, Hunida Malaikah and Naif Alsobhi

Fractal Fract. 2025, 9(2), 80; https://doi.org/10.3390/fractalfract9020080 - 24 Jan 2025

Viewed by 544

Abstract

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains [...] Read more.

The importance of this research comes from the several applications of the Mathieu equation and its generalizations in many scientific fields. Two models of fractional Mathieu equations are provided using Katugampola fractional derivatives in the sense of Riemann-Liouville and Caputo. Each model contains two fractional derivatives with unique fractional orders, periodic forcing of the cosine stiffness coefficient, and many extensions and generalizations. The Banach contraction principle is used to prove that each model under consideration has a unique solution. Our results are applied to four real-life problems: the nonlinear Mathieu equation for parametric damping and the Duffing oscillator, the quadratically damped Mathieu equation, the fractional Mathieu equation’s transition curves, and the tempered fractional model of the linearly damped ion motion with an octopole. Full article

(This article belongs to the Section General Mathematics, Analysis)

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25 pages, 437 KiB

Open AccessArticle

Hermite–Hadamard-Type Inequalities for Harmonically Convex Functions via Proportional Caputo-Hybrid Operators with Applications

by Saad Ihsan Butt, Muhammad Umar, Dawood Khan, Youngsoo Seol and Sanja Tipurić-Spužević

Fractal Fract. 2025, 9(2), 77; https://doi.org/10.3390/fractalfract9020077 - 24 Jan 2025

Viewed by 520

Abstract

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically [...] Read more.

In this paper, we aim to establish new inequalities of Hermite–Hadamard (H.H) type for harmonically convex functions using proportional Caputo-Hybrid (P.C.H) fractional operators. Parameterized by

α

, these operators offer a unique flexibility: setting

α = 1

recovers the classical inequalities for harmonically convex functions, while setting

α = 0

yields inequalities for differentiable harmonically convex functions. This framework allows us to unify classical and fractional cases within a single operator. To validate the theoretical results, we provide several illustrative examples supported by graphical representations, marking the first use of such visualizations for inequalities derived via P.C.H operators. Additionally, we demonstrate practical applications of the results by deriving new fractional-order recurrence relations for the modified Bessel function of type-1, which are useful in mathematical modeling, engineering, and physics. The findings contribute to the growing body of research in fractional inequalities and harmonic convexity, paving the way for further exploration of generalized convexities and higher-order fractional operators. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

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16 pages, 399 KiB

Open AccessArticle

Fractional Calculus of Piecewise Continuous Functions

by Manuel Duarte Ortigueira

Fractal Fract. 2025, 9(2), 75; https://doi.org/10.3390/fractalfract9020075 - 24 Jan 2025

Viewed by 668

Abstract

The fractional derivative computation of piecewise continuous functions is treated with generality. It is shown why some applications give incorrect results and why Caputo derivative give strange results. Some examples are described. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

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31 pages, 17297 KiB

Open AccessArticle

Construction of the Closed Form Wave Solutions for TFSMCH and (1 + 1) Dimensional TFDMBBM Equations via the EMSE Technique

by Md. Asaduzzaman and Farhana Jesmin

Fractal Fract. 2025, 9(2), 72; https://doi.org/10.3390/fractalfract9020072 - 24 Jan 2025

Viewed by 900

Abstract

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of [...] Read more.

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of diffusion in the creation of shapes in liquid beads arising in plasma physics and fluid flow and to estimate the external long waves in nonlinear dispersive media. These equations are also used to characterize various types of waves, such as hydromagnetic waves, acoustic waves, and acoustic gravity waves. Here, we utilize the Caputo-type fractional order derivative to fractionalize the considered FONLEEs. Some trigonometric and hyperbolic trigonometric functions have been used to represent the obtained closed form traveling wave solutions. Furthermore, here, we reveal that the EMSE technique is a suitable, significant, and dominant mathematical tool for finding the exact traveling wave solutions for various FONLEEs in mathematical physics. We draw some 3D, 2D, and contour plots to describe the various wave behaviors and analyze the physical consequence of the attained solutions. Finally, we make a numerical comparison of our obtained solutions and other analogous solutions obtained using various techniques. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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