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Journal articles on the topic 'Fractional partial derivatives'

Author: Grafiati

Published: 7 June 2025

Last updated: 2 August 2025

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1

Chii-Huei, Yu. "Evaluating the Fractional Partial Derivatives of Some Two-Variables Fractional Functions." International Journal of Computer Science and Information Technology Research 12, no. 4 (2024): 5–11. https://doi.org/10.5281/zenodo.13959773.

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<strong>Abstract:</strong> In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional derivative and a new multiplication of fractional analytic functions, we can find the fractional partial derivatives of some two-variables fractional functions. In fact, our result is a generalization of classical calculus result. <strong>Keywords:</strong> Jumarie’s modified R-L fractional derivative, new multiplication, fractional analytic functions, fractional partial derivatives, two-variables fractional functions. <strong>Title:</strong> Evaluating the Fractional Partial De

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2

Li, Changpin, Deliang Qian, and YangQuan Chen. "On Riemann-Liouville and Caputo Derivatives." Discrete Dynamics in Nature and Society 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/562494.

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Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in sc

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3

Hu, Langhua, Duan Chen, and Guo-Wei Wei. "High-order fractional partial differential equation transform for molecular surface construction." Computational and Mathematical Biophysics 1 (December 20, 2012): 1–25. http://dx.doi.org/10.2478/mlbmb-2012-0001.

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AbstractFractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourie

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Kirianova, Ludmila Vladimirovna. "The Boundary Value Problem with Stationary Inhomogeneities for a Hyperbolic-Type Equation with a Fractional Derivative." Axioms 11, no. 5 (2022): 207. http://dx.doi.org/10.3390/axioms11050207.

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The paper presents an analytical solution of a partial differential equation of hyperbolic-type, containing both second-order partial derivatives and fractional derivatives of order below the second. Examples of applying the solution of a boundary value problem with stationary inhomogeneities for a hyperbolic-type equation with a fractional derivative in modeling the behavior of polymer concrete under the action of loads are considered.

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Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan. "Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations." Fractal and Fractional 7, no. 1 (2023): 80. http://dx.doi.org/10.3390/fractalfract7010080.

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In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditio

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6

Elbeleze, Asma Ali, Adem Kılıçman, and Bachok M. Taib. "Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/543848.

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We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

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Ali, Khalid K., F. E. Abd Elbary, Mohamed S. Abdel-Wahed, M. A. Elsisy, and Mourad S. Semary. "Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method." International Journal of Differential Equations 2023 (December 27, 2023): 1–19. http://dx.doi.org/10.1155/2023/1240970.

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The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent

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8

Lazopoulos, Konstantinos A. "On Λ-Fractional Differential Equations". Foundations 2, № 3 (2022): 726–45. http://dx.doi.org/10.3390/foundations2030050.

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Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional deriv

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9

Zainal, Nor Hafizah, and Adem Kılıçman. "Solving Fractional Partial Differential Equations with Corrected Fourier Series Method." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/958931.

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The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

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10

Alharthi, Nadiyah Hussain, Abdon Atangana, and Badr S. Alkahtani. "Numerical analysis of some partial differential equations with fractal-fractional derivative." AIMS Mathematics 8, no. 1 (2022): 2240–56. http://dx.doi.org/10.3934/math.2023116.

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<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation ha

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11

Abdallah, Mohamed Ahmed, and Khaled Abdalla Ishag. "Fractional Reduced Differential Transform Method for Solving Mutualism Model with Fractional Diffusion." International Journal of Analysis and Applications 21 (April 10, 2023): 33. http://dx.doi.org/10.28924/2291-8639-21-2023-33.

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This study presents the fractional reduced differential transform method for a nonlinear mutualism model with fractional diffusion. The fractional derivatives are described by Caputo's fractional operator. In this method, the solution is considered as the sum of an infinite series. Which converges rapidly to the exact solution. The method eliminates the need to use Adomian's polynomials to calculate the nonlinear terms. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with the ordinary derivative order index α=1 for the nonlinear

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12

Özkan, Ozan, and Ali Kurt. "Exact solutions of fractional partial differential equation systems with conformable derivative." Filomat 33, no. 5 (2019): 1313–22. http://dx.doi.org/10.2298/fil1905313o.

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Main goal of this paper is to have the new exact solutions of some fractional partial differential equation systems (FPDES) in conformable sense. The definition of conformable fractional derivative (CFD) is similar to the limit based definition of known derivative. This derivative obeys both rules which other popular derivatives do not satisfy such as derivative of the quotient of two functions, the derivative product of two functions, chain rule and etc. By using conformable derivative it is seen that the solution procedure for (PDES) is simpler and more efficient.

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13

Aljhani, Sami, Mohd Salmi Md Noorani, Khaled M. Saad, and A. K. Alomari. "Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott." Journal of Function Spaces 2021 (July 19, 2021): 1–12. http://dx.doi.org/10.1155/2021/2544688.

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A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homot

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14

Saeed, Umer, and Muhammad Umair. "A modified method for solving non-linear time and space fractional partial differential equations." Engineering Computations 36, no. 7 (2019): 2162–78. http://dx.doi.org/10.1108/ec-01-2019-0011.

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Purpose The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain. Design/methodology/approach The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations. Findings The fractional derivative of Lagrange polynomial is a big hurdle

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15

Turov, M. M., V. E. Fedorov, and B. T. Kien. "Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives." Bulletin of Irkutsk State University. Series Mathematics 38 (2021): 36–53. http://dx.doi.org/10.26516/1997-7670.2021.38.36.

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The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fraction

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16

Oduselu-Hassan, Oladayo Emmanuel, and David Oluwatomi Ojada. "Numerical solutions of fractional conformable derivative using a generalized Kudryashov method." Science World Journal 19, no. 4 (2025): 994–97. https://doi.org/10.4314/swj.v19i4.12.

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This paper addresses the numerical solutions of fractional differential equations (FDEs) using the Generalized Kudryashov Method (GKM) in the context of the conformable fractional derivative. Fractional calculus, particularly the conformable derivative, provides a versatile framework for modeling systems exhibiting memory and hereditary properties commonly found in complex physical phenomena. Traditional integer-order derivatives lack the capability to accurately represent such dynamics, which fractional derivatives effectively handle. The conformable derivative, a recent addition to fractiona

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17

Fedorov, Vladimir E., Mikhail M. Turov, and Bui Trong Kien. "A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators." Axioms 11, no. 3 (2022): 96. http://dx.doi.org/10.3390/axioms11030096.

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The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials o

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18

Thabet, Hayman, Subhash Kendre, and Subhash Unhale. "Numerical Analysis of Iterative Fractional Partial Integro-Differential Equations." Journal of Mathematics 2022 (May 17, 2022): 1–14. http://dx.doi.org/10.1155/2022/8781186.

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Many nonlinear phenomena are modeled in terms of differential and integral equations. However, modeling nonlinear phenomena with fractional derivatives provides a better understanding of processes having memory effects. In this paper, we introduce an effective model of iterative fractional partial integro-differential equations (FPIDEs) with memory terms subject to initial conditions in a Banach space. The convergence, existence, uniqueness, and error analysis are introduced as new theorems. Moreover, an extension of the successive approximations method (SAM) is established to solve FPIDEs in

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19

Touchent, Kamal Ait, Zakia Hammouch, and Toufik Mekkaoui. "A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives." Applied Mathematics and Nonlinear Sciences 5, no. 2 (2020): 35–48. http://dx.doi.org/10.2478/amns.2020.2.00012.

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AbstractIn this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new num

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20

Topsakal, Muammer, and Filiz TaŞcan. "Exact Travelling Wave Solutions for Space-Time Fractional Klein-Gordon Equation and (2+1)-Dimensional Time-Fractional Zoomeron Equation via Auxiliary Equation Method." Applied Mathematics and Nonlinear Sciences 5, no. 1 (2020): 437–46. http://dx.doi.org/10.2478/amns.2020.1.00041.

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AbstractIn this paper, the new exact solutions of nonlinear conformable fractional partial differential equations(CFPDEs) are achieved by using auxiliary equation method for the nonlinear space-time fractional Klein-Gordon equation and the (2+1)-dimensional time-fractional Zoomeron equation. The technique is easily applicable which can be applied successfully to get the solutions for different types of nonlinear CFPDEs. The conformable fractional derivative(CFD) definitions are used to cope with the fractional derivatives.

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21

Alsidrani, Fahad, Adem Kılıçman, and Norazak Senu. "On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives." Axioms 12, no. 9 (2023): 901. http://dx.doi.org/10.3390/axioms12090901.

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This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable ν. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To conf

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22

Ali, Amjid, Teruya Minamoto, Rasool Shah, and Kamsing Nonlaopon. "A novel numerical method for solution of fractional partial differential equations involving the $ \psi $-Caputo fractional derivative." AIMS Mathematics 8, no. 1 (2022): 2137–53. http://dx.doi.org/10.3934/math.2023110.

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<abstract><p>In this study, the $ \psi $-Haar wavelets operational matrix of integration is derived and used to solve linear $ \psi $-fractional partial differential equations ($ \psi $-FPDEs) with the fractional derivative defined in terms of the $ \psi $-Caputo operator. We approximate the highest order fractional partial derivative of the solution of linear $ \psi $-FPDE using Haar wavelets. By combining the operational matrix and $ \psi $-fractional integration, we approximate the solution and its other $ \psi $-fractional partial derivatives. Then substituting these approximat

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23

Alqhtani, Manal, Khaled M. Saad, Rasool Shah, Wajaree Weera, and Waleed M. Hamanah. "Analysis of the Fractional-Order Local Poisson Equation in Fractal Porous Media." Symmetry 14, no. 7 (2022): 1323. http://dx.doi.org/10.3390/sym14071323.

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This paper investigates the fractional local Poisson equation using the homotopy perturbation transformation method. The Poisson equation discusses the potential area due to a provided charge with the possibility of area identified, and one can then determine the electrostatic or gravitational area in the fractal domain. Elliptic partial differential equations are frequently used in the modeling of electromagnetic mechanisms. The Poisson equation is investigated in this work in the context of a fractional local derivative. To deal with the fractional local Poisson equation, some illustrative p

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24

Xu, Bo, Yufeng Zhang, and Sheng Zhang. "Fractional Rogue Waves with Translational Coordination, Steep Crest, and Modified Asymmetry." Complexity 2021 (April 20, 2021): 1–14. http://dx.doi.org/10.1155/2021/6669087.

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To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger (NLS) equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It is shown that the obtained first- and second-

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25

Osman, Mawia, Almegdad Almahi, Omer Abdalrhman Omer, Altyeb Mohammed Mustafa, and Sarmad A. Altaie. "Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations." Fractal and Fractional 6, no. 11 (2022): 646. http://dx.doi.org/10.3390/fractalfract6110646.

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In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variatio

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26

Kirane, Mokhtar, and Berikbol T. Torebek. "Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations." Fractional Calculus and Applied Analysis 22, no. 2 (2019): 358–78. http://dx.doi.org/10.1515/fca-2019-0022.

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Abstract In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.

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27

Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.83.

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Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical i

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Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.83.

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Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical i

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29

Owoyemi, Abiodun Ezekiel, Ira Sumiati, Endang Rusyaman, and Sukono Sukono. "Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation." International Journal of Quantitative Research and Modeling 1, no. 4 (2020): 194–207. http://dx.doi.org/10.46336/ijqrm.v1i4.91.

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Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical i

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30

Neirameh, Ahmad, Mostafa Eslami, and Mostafa Mehdipoor. "New types of soliton solutions for space-time fractional cubic nonlinear Schrodinger equation." Boletim da Sociedade Paranaense de Matemática 39, no. 2 (2021): 121–31. http://dx.doi.org/10.5269/bspm.33548.

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New definitions for traveling wave transformation and using of new conformable fractional derivative for converting fractional nonlinear evolution equations into the ordinary differential equations are presented in this study. For this aim we consider the time and space fractional derivatives cubic nonlinear Schrodinger equation. Then by using of the efficient and powerful method the exact traveling wave solutions of this equation are obtained. The new definition introduces a promising tool for solving many space-time fractional partial differential equations.

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31

Ford, Neville J., Kamal Pal, and Yubin Yan. "An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations." Computational Methods in Applied Mathematics 15, no. 4 (2015): 497–514. http://dx.doi.org/10.1515/cmam-2015-0022.

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AbstractWe introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order ${O(\Delta x^{3- \alpha })}$, ${1&lt;\alpha &lt;2}$. A shifted implicit finite difference method is ap

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Samad, Abdul, Imran Siddique, and Fahd Jarad. "Meshfree numerical integration for some challenging multi-term fractional order PDEs." AIMS Mathematics 7, no. 8 (2022): 14249–69. http://dx.doi.org/10.3934/math.2022785.

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<abstract><p>Fractional partial differential equations (PDEs) have key role in many physical, chemical, biological and economic problems. Different numerical techniques have been adopted to deal the multi-term FPDEs. In this article, the meshfree numerical scheme, Radial basis function (RBF) is discussed for some time-space fractional PDEs. The meshfree RBF method base on the Gaussian function and is used to test the numerical results of the time-space fractional PDE problems. Riesz fractional derivative and Grünwald-Letnikov fractional derivative techniques are used to deal the sp

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Khater, Mostafa M. A., Raghda A. M. Attia, and Dianchen Lu. "Explicit Lump Solitary Wave of Certain Interesting (3+1)-Dimensional Waves in Physics via Some Recent Traveling Wave Methods." Entropy 21, no. 4 (2019): 397. http://dx.doi.org/10.3390/e21040397.

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This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev’e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fra

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Saadeh, Rania, Ahmad Qazza, Abdelilah Kamal H. Sedeeg, and Ghassan Abufoudeh. "Exploration of a New Approach Related to Atangana-Baleanu Derivatives for Solving Fractional Partial Differential Equations." International Journal of Mathematical, Engineering and Management Sciences 10, no. 4 (2025): 896–912. https://doi.org/10.33889/ijmems.2025.10.4.043.

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This paper explores the application of fractional calculus to solve fractional partial differential equations (FPDEs) using the Sawi transform in combination with the Atangana-Baleanu fractional derivative. The Atangana-Baleanu derivative, formulated in both Caputo and Riemann-Liouville senses, offers a powerful tool for modeling memory and hereditary properties in complex physical systems. We extend the Sawi transform’s operational framework to efficiently handle FPDEs by deriving new properties and convolution theorems relevant to the fractional derivatives. The combination of the Sawi trans

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35

Gurefe, Yusuf. "The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative." Revista Mexicana de Física 66, no. 6 Nov-Dec (2020): 771. http://dx.doi.org/10.31349/revmexfis.66.771.

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In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important result

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36

Khalid, Thwiba A., Fatima Alnoor, Ebtesam Babeker, Ehssan Ahmed, and Alaa Mustafa. "Legendre Polynomials and Techniques for Collocation in the Computation of Variable-Order Fractional Advection-Dispersion Equations." International Journal of Analysis and Applications 22 (October 15, 2024): 185. http://dx.doi.org/10.28924/2291-8639-22-2024-185.

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The paper discusses a numerical approach to solving complicated partial differential equations, with a particular emphasis on fractional advection-dispersion equations of space-time variable order. With the use of fractional derivative matrices, Legendre polynomials, and numerical examples and comparisons, it surpasses current methods by utilizing spectral collocation techniques. It resolves equations involving spatial and time variables that are variable-order fractional advection–dispersion (VOFADE). Legendre polynomials serve as basis functions in this method, whereas Legendre operational m

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Dehghan, Mehdi, Jalil Manafian, and Abbas Saadatmandi. "Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions." Zeitschrift für Naturforschung A 65, no. 11 (2010): 935–49. http://dx.doi.org/10.1515/zna-2010-1106.

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In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of th

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38

Pashaei, Ronak, Mohammad Sadegh Asgari, and Amir Pishkoo. "Conformable Derivatives in Laplace Equation and Fractional Fourier Series Solution." International Annals of Science 9, no. 1 (2019): 1–7. http://dx.doi.org/10.21467/ias.9.1.1-7.

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In this paper the solution of conformable Laplace equation, \frac{\partial^{\alpha}u(x,y)}{\partial x^{\alpha}}+ \frac{\partial^{\alpha}u(x,y)}{\partial y^{\alpha}}=0, where 1 < α ≤ 2 has been deduced by using fractional fourier series and separation of variables method. For special cases α =2 (Laplace's equation), α=1.9, and α=1.8 conformable fractional fourier coefficients have been calculated. To calculate coefficients, integrals are of type "conformable fractional integral".

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39

Kostić, Marko. "Multidimensional Fractional Calculus: Theory and Applications." Axioms 13, no. 9 (2024): 623. http://dx.doi.org/10.3390/axioms13090623.

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In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex character

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40

Özarslan, Mehmet Ali, and Arran Fernandez. "On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators." Fractal and Fractional 5, no. 2 (2021): 45. http://dx.doi.org/10.3390/fractalfract5020045.

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Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed pa

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Lisha, N. M., and A. G. Vijayakumar. "Analytical Investigation of the Heat Transfer Effects of Non-Newtonian Hybrid Nanofluid in MHD Flow Past an Upright Plate Using the Caputo Fractional Order Derivative." Symmetry 15, no. 2 (2023): 399. http://dx.doi.org/10.3390/sym15020399.

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The objective of this paper is to examine the augmentation of the heat transfer rate utilizing graphene (Gr) and multi-walled carbon nanotubes (MWCNTs) as nanoparticles, and water as a host fluid in magnetohydrodynamics (MHD) flow through an upright plate using Caputo fractional derivatives with a Brinkman model on the convective Casson hybrid nanofluid flow. The performance of hybrid nanofluids is examined with various shapes of nanoparticles. The Caputo fractional derivative is utilized to describe the governing fractional partial differential equations with initial and boundary conditions o

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42

Sitnik, Sergey M., Vladimir E. Fedorov, Nikolay V. Filin, and Viktor A. Polunin. "On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral." Mathematics 10, no. 16 (2022): 2979. http://dx.doi.org/10.3390/math10162979.

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Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their propert

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Osman, Mawia, Muhammad Marwan, Syed Omar Shah, et al. "Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives." Fractal and Fractional 7, no. 12 (2023): 851. http://dx.doi.org/10.3390/fractalfract7120851.

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In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration meth

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Al-Smadi, Mohammed, Asad Freihat, Hammad Khalil, Shaher Momani, and Rahmat Ali Khan. "Numerical Multistep Approach for Solving Fractional Partial Differential Equations." International Journal of Computational Methods 14, no. 03 (2017): 1750029. http://dx.doi.org/10.1142/s0219876217500293.

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In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense

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Bogatyreva, F. T. "On the correctness of initial problems for the fractional diffusion equation." ADYGHE INTERNATIONAL SCIENTIFIC JOURNAL 23, no. 4 (2023): 16–22. http://dx.doi.org/10.47928/1726-9946-2023-23-4-16-22.

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The paper studies a second-order parabolic partial differential equation with fractional differentiation with respect to a time variable. The fractional differentiation operator is a linear combination of the Riemann-Liouville and Gerasimov-Caputo fractional derivatives. It is shown that the distribution of orders of fractional derivatives, included in the equation affects the correctness of the initial problems for the equation under consideration.

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García-Sandoval, Juan Paulo. "On representation and interpretation of Fractional calculus and fractional order systems." Fractional Calculus and Applied Analysis 22, no. 2 (2019): 522–37. http://dx.doi.org/10.1515/fca-2019-0031.

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Abstract In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials, and allows to transform fractional order systems in FOPDE

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Al-Momani, Monther, Ali Jaradat, Baha' Abughazaleh, and Abdulkarim Farah. "Solving Partial Differential Equations via the Conformable Double ARA-Sawi Transform." European Journal of Pure and Applied Mathematics 18, no. 2 (2025): 6099. https://doi.org/10.29020/nybg.ejpam.v18i2.6099.

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This paper presents a new method the conformable double ARA-Sawi transform to solve fractional partial differential equations that arise in physics and engineering. These equations often involve derivatives based on conformable calculus, which generalizes classical derivatives to fractional orders. We explore the core properties of the transform, such as its linearity and interaction with fractional derivatives, and establish conditions for its applicability. To demonstrate its practical use, we apply the method to solve two key equations: the conformable Klein-Gordon equation, which models wa

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Uddin, Salah, M. Mohamad, M. A. H. Mohamad, et al. "Caputo-Fabrizio Time Fractional Derivative Applied to Visco Elastic MHD Fluid Flow in the Porous Medium." International Journal of Engineering & Technology 7, no. 4.30 (2018): 533. http://dx.doi.org/10.14419/ijet.v7i4.30.28171.

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In this paper the laminar fluid flow in the axially symmetric porous cylindrical channel subjected to the magnetic field was studied. Fluidmodel was non-Newtonian and visco elastic. The effects of magnetic field and pressure gradient on the fluid velocity were studied by using a new trend of fractional derivative without singular kernel. The governing equations consisted of fractional partial differential equations based on the Caputo-Fabrizio new time-fractional derivatives NFDt. Velocity profiles for various fractional parameter a, Hartmann number, permeability parameter and elasticity were

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Aminikhah, Hossein, and Nasrin Malekzadeh. "An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative." Scientific World Journal 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/687695.

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A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in th

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Lukashchuk, Stanislav Yu. "On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems." Mathematics 10, no. 13 (2022): 2319. http://dx.doi.org/10.3390/math10132319.

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The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie poi

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