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Journal articles on the topic 'Fractional integral equation'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Aghili, Arman. "Non-homogeneous impulsive time fractional heat conduction equation." Journal of Numerical Analysis and Approximation Theory 52, no. 1 (2023): 22–33. http://dx.doi.org/10.33993/jnaat521-1316.

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This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a math

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2

TARASOV, VASILY E. "THE FRACTIONAL CHAPMAN–KOLMOGOROV EQUATION." Modern Physics Letters B 21, no. 04 (2007): 163–74. http://dx.doi.org/10.1142/s0217984907012712.

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The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived.

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3

Mondal, Subhabrata, and B. N. Mandal. "Solution of Abel Integral Equation Using Differential Transform Method." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (2018): 7521–32. http://dx.doi.org/10.24297/jam.v14i1.7172.

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The application of fractional differential transform method, developed for differential equations of fractional order, are extended to derive exact analytical solutions of fractional order Abel integral equations. The fractional integrations are described in the Riemann-Liouville sense and fractional derivatives are described in the Caputo sense. Abel integral equation occurs in the mathematical modeling of various problems in physics, astrophysics, solid mechanics and applied sciences. An analytic technique for solving Abel integral equation of first kind by the proposed method is introduced

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4

Kosmakova, M. T., S. A. Iskakov, and L. Zh Kasymova. "To solving the fractionally loaded heat equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 101, no. 1 (2021): 65–77. http://dx.doi.org/10.31489/2021m1/65-77.

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In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral

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5

Zhou, Mi, Hengtai Wang, Zhenghua Xu, and Lu Zhang. "The Study of Fractional Quadratic Integral Equations Involves General Fractional Integrals." Fractal and Fractional 9, no. 4 (2025): 249. https://doi.org/10.3390/fractalfract9040249.

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This paper investigates the well-posedness of analytical solutions to fractional quadratic differential equations, which involve generalized fractional integrals with respect to other functions. The nonlinear components f and h depend on spatial variables and the general fractional integral, respectively. We use the operator equation T1ωT2ω+T3ω=ω to investigate the existence of solutions, marking the first study of its kind. Using an auxiliary function and Boyd and Wang’s fixed-point theorem, the uniqueness and continuous dependence of the solution are obtained. In particular, we applied nonli

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6

Pandir, Yusuf, Yusuf Gurefe, and Emine Misirli. "The Extended Trial Equation Method for Some Time Fractional Differential Equations." Discrete Dynamics in Nature and Society 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/491359.

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Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractionalKn,nequation according to the complete discrimination system for polynomial method. As a resul

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Qayyum, Mubashir, Sidra Afzal, and Efaza Ahmad. "Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates." Journal of Mathematics 2023 (March 8, 2023): 1–12. http://dx.doi.org/10.1155/2023/5517617.

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In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as α = 1 . A modified

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8

Hammad, Hasanen A., Hassen Aydi, and Manuel De la Sen. "Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions." Complexity 2021 (February 22, 2021): 1–13. http://dx.doi.org/10.1155/2021/5730853.

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This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.

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9

Yépez-Martínez, Huitzilin, Ivan O. Sosa, and Juan M. Reyes. "Feng’s First Integral Method Applied to the ZKBBM and the Generalized Fisher Space-Time Fractional Equations." Journal of Applied Mathematics 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/191545.

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The fractional derivatives in the sense of the modified Riemann-Liouville derivative and Feng’s first integral method are employed to obtain the exact solutions of the nonlinear space-time fractional ZKBBM equation and the nonlinear space-time fractional generalized Fisher equation. The power of this manageable method is presented by applying it to the above equations. Our approach provides first integrals in polynomial form with high accuracy. Exact analytical solutions are obtained through establishing first integrals. The present method is efficient and reliable, and it can be used as an al

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10

Berdyshev, A. S., E. T. Karimov, and N. Akhtaeva. "Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation." International Journal of Differential Equations 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/268465.

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Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential opera

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11

Liu, Jian-Gen, Xiao-Jun Yang, and Yi-Ying Feng. "Analytical solutions of some integral fractional differential–difference equations." Modern Physics Letters B 34, no. 01 (2019): 2050009. http://dx.doi.org/10.1142/s0217984920500098.

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The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as

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12

Alesemi, Meshari. "An Innovative Approach to Nonlinear Fractional Shock Wave Equations Using Two Numerical Methods." Mathematics 11, no. 5 (2023): 1253. http://dx.doi.org/10.3390/math11051253.

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In this research, we propose a combined approach to solving nonlinear fractional shock wave equations using an Elzaki transform, the homotopy perturbation method, and the Adomian decomposition method. The nonlinear fractional shock wave equation is first transformed into an equivalent integral equation using the Elzaki transform. The homotopy perturbation method and Adomian decomposition method are then utilized to approximate the solution of the integral equation. To evaluate the effectiveness of the proposed method, we conduct several numerical experiments and compare the results with existi

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13

Aghili, Arman. "Solution to linear KdV and nonLinear space fractional PDEs." Boletim da Sociedade Paranaense de Matemática 39, no. 2 (2021): 63–73. http://dx.doi.org/10.5269/bspm.40360.

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In this work, the author will briefly discuss applications of the Fourier and Laplace transforms in the solution of certain singular integral equations and evaluation of integrals. By combining integral transforms and operational methods we get more powerful analytical tool for solving a wide class of linear or even non- linear fractional differential or fractional partial differential equation. Numerous examples and exercises occur throughout the paper.

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Yaslan, Handan Çerdik. "Approximate solution of the conformable integro-differential equations." Miskolc Mathematical Notes 26, no. 1 (2025): 547. https://doi.org/10.18514/mmn.2025.4540.

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In this paper, fractional linear and nonlinear integro-differential equations are solved by using an iteration method. Fractional derivative and fractional integral are considered in the conformable sense. The conformable integro-differential equation is converted to a conformable integral equation. Then, the conformable integral equation leads to an iteration sequence, the limit of which is a solution of the conformable integro-differential equation. In addition, stability and convergence analysis of the presented method are investigated. The applicability of the presented method is also show

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15

Albuohimad, Basim. "Exact solution of some fractional integral differential equations by using S-transform." Journal of Interdisciplinary Mathematics 28, no. 3-A (2025): 823–30. https://doi.org/10.47974/jim-1979.

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In this work, we studied fractional differential equations (FDE), which are the differentials of non-integer order. We also discussed the integral differential equations (IDE). By connecting (FDE) and (IDE), a fractional integral differential equation (FIDE) is formed. This research aims to apply the Shehu transform (Տ-transform) to solve this type of linear fractional integral differential equation. The Տ-transform is a powerful tool in math and engineering. This will permit us to convert (FIDE) to algebraic equations, and then by solving this equation, we can acquire the unknown function uti

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16

Abdillah, Muhammad Taufik, Berlian Setiawaty, and Sugi Guritman. "The Solution of Generalization of the First and Second Kind of Abel’s Integral Equation." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (2023): 631. http://dx.doi.org/10.31764/jtam.v7i3.14193.

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Integral equations are equations in which the unknown function is found to be inside the integral sign. N. H. Abel used the integral equation to analyze the relationship between kinetic energy and potential energy in a falling object, expressed by two integral equations. This integral equation is called Abel's integral equation. Furthermore, these equations are developed to produce generalizations and further generalizations for each equation. This study aims to explain generalizations of the first and second kind of Abel’s integral equations, and to find solution for each equation. The method

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17

Damak, Mondher, and Zaid Amer Mohammed. "Variational Iteration Method for Solving Fractional Integro-Differential Equations with Conformable Differointegration." Axioms 11, no. 11 (2022): 586. http://dx.doi.org/10.3390/axioms11110586.

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Multidimensional integro-differential equations are obtained when the unknown function of several independent variable and/or its derivatives appear under an integral sign. When the differentiation or integration operators or both are of fractional order, the integral equation in this case is called a multidimensional fractional integro-differential equation. Such equations are difficult to solve analytically; therefore, as the main objective of this paper, an approximate method—which is the variational iteration method—will be used to solve this type of equation with conformable fractional-or

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18

El-Sayed, Ahmed M. A., and Yasmin M. Y. Omar. "Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation." Demonstratio Mathematica 54, no. 1 (2021): 25–36. http://dx.doi.org/10.1515/dema-2021-0003.

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Abstract In this work, we study the existence of one and exactly one solution x ∈ C [ 0 , 1 ] x\in C\left[0,1] , for a delay quadratic integral equation of Volterra-Stieltjes type. As special cases we study a delay quadratic integral equation of fractional order and a Chandrasekhar cubic integral equation.

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19

Abdulqader, Alan Jalal, Saleh S. Redhwan, Ali Hasan Ali, Omar Bazighifan, and Awad T. Alabdala. "Picard and Adomian decomposition methods for a fractional quadratic integral equation via generalized fractional integral." Iraqi Journal For Computer Science and Mathematics 5, no. 3 (2024): 170–80. http://dx.doi.org/10.52866/ijcsm.2024.05.03.008.

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The primary focus of this paper is to thoroughly examine and analyze a class of a fractional quadraticintegral equation via generalized fractional integral. To achieve this, we introduce an operator that possessesfixed points corresponding to the solutions of the fractional quadratic integral equation, effectively transforming thegiven equation into an equivalent fixed-point problem. By applying the Banach fixed-point theorems, we prove theuniqueness of solutions to fractional quadratic integral equation. Additionally, The Adomian decomposition methodis used, to solve the resulting fractional

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20

Aldosary, Saud Fahad, Mohamed M. A. Metwali, Manochehr Kazemi, and Ateq Alsaadi. "On integrable and approximate solutions for Hadamard fractional quadratic integral equations." AIMS Mathematics 9, no. 3 (2024): 5746–62. http://dx.doi.org/10.3934/math.2024279.

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<abstract><p>This article addressed the integrable and approximate solutions of Hadamard-type fractional Gripenberg's equation in Lebesgue spaces $ L_1[1, e] $. It is well known that the Gripenberg's equation has significant applications in mathematical biology. By utilizing the fixed point (FPT) approach and the measure of noncompactness (MNC), we demonstrated the presence of monotonic integrable solutions as well as the uniqueness of the solution for the studied equation in spaces that are not Banach algebras. Moreover, the method of successive approximations was successfully app

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21

El-Sayed, Ahmed M. A., Hind H. G. Hashem та Shorouk M. Al-Issa. "Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation". Foundations 2, № 1 (2022): 167–83. http://dx.doi.org/10.3390/foundations2010010.

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Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ− fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results.

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22

Arshad, Sadia, Iram Saleem, Ali Akgül, Jianfei Huang, Yifa Tang, and Sayed M. Eldin. "A novel numerical method for solving the Caputo-Fabrizio fractional differential equation." AIMS Mathematics 8, no. 4 (2023): 9535–56. http://dx.doi.org/10.3934/math.2023481.

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<abstract><p>In this paper, a unique and novel numerical approach—the fractional-order Caputo-Fabrizio derivative in the Caputo sense—is developed for the solution of fractional differential equations with a non-singular kernel. After converting the differential equation into its corresponding fractional integral equation, we used Simpson's $ 1/3 $ rule to estimate the fractional integral equation. A thorough study is then conducted to determine the convergence and stability of the suggested method. We undertake numerical experiments to corroborate our theoretical findings.</p&g

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23

Saha Ray, S., and S. Singh. "Numerical solution of nonlinear stochastic Itô – Volterra integral equation driven by fractional Brownian motion." Engineering Computations 37, no. 9 (2020): 3243–68. http://dx.doi.org/10.1108/ec-01-2020-0039.

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Purpose This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis o

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24

Aghili, A. "Fractional Black–Scholes equation." International Journal of Financial Engineering 04, no. 01 (2017): 1750004. http://dx.doi.org/10.1142/s2424786317500049.

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In this paper, it has been shown that the combined use of exponential operators and special functions provides a powerful tool to solve certain class of generalized space fractional Laguerre heat equation. It is shown that exponential operators are powerful and effective method for solving certain singular integral equations and space fractional Black–Scholes equation.

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25

Becker, Leigh, Theodore Burton, and Ioannis Purnaras. "Complementary equations: a fractional differential equation and a Volterra integral equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 12 (2015): 1–24. http://dx.doi.org/10.14232/ejqtde.2015.1.12.

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26

Cheng, Kelong, Chunxiang Guo, and Min Tang. "Some Nonlinear Gronwall-Bellman-Gamidov Integral Inequalities and Their Weakly Singular Analogues with Applications." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/562691.

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Some Gronwall-Bellman-Gamidov type integral inequalities with power nonlinearity and their weakly singular analogues are established, which can give the explicit bound on solution of a class of nonlinear fractional integral equations. An example is presented to show the application for the qualitative study of solutions of a fractional integral equation with the Riemann-Liouville fractional operator.

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27

Iskandarova, Gulistan, and Dogan Kaya. "Symmetry solution on fractional equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 7, no. 3 (2017): 255–59. http://dx.doi.org/10.11121/ijocta.01.2017.00498.

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As we know nearly all physical, chemical, and biological processes in nature can be described or modeled by dint of a differential equation or a system of differential equations, an integral equation or an integro-differential equation. The differential equations can be ordinary or partial, linear or nonlinear. So, we concentrate our attention in problem that can be presented in terms of a differential equation with fractional derivative. Our research in this work is to use symmetry transformation method and its analysis to search exact solutions to nonlinear fractional partial differential eq

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28

Bin Jebreen, Haifa. "The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives." Fractal and Fractional 7, no. 10 (2023): 763. http://dx.doi.org/10.3390/fractalfract7100763.

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We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some n

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Bin Jebreen, Haifa, and Ioannis Dassios. "A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation." Mathematics 10, no. 9 (2022): 1461. http://dx.doi.org/10.3390/math10091461.

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This paper is devoted to the wavelet Galerkin method to solve the Fractional Riccati equation. To this end, biorthogonal Hermite cubic Spline scaling bases and their properties are introduced, and the fractional integral is represented based on these bases as an operational matrix. Firstly, we obtain the Volterra integral equation with a weakly singular kernel corresponding to the desired equation. Then, using the operational matrix of fractional integration and the Galerkin method, the corresponding integral equation is reduced to a system of algebraic equations. Solving this system via Newto

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30

Lukashchuk, S. Yu. "Fractional-order integral generalization of the Rapoport–Leas equation." Multiphase Systems 18, no. 2 (2023): 58–67. http://dx.doi.org/10.21662/mfs2023.2.009.

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A one-dimensional time-fractional integral model for a two-phase flow through a porous medium with power-law memory in the presence of capillary forces is considered. Based on the time-fractional integral generalization of Darcy’s law and the classical mass balance equations for phases, neglecting gravity forces, a time-fractional integral generalization of the Rapoport–Leas equation is derived. It is shown that in the limiting case of constant capillary pressure, the obtained equation coincides with the classical Buckley–Leverett equation. An example of initial boundary value problem for the

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31

Luo, Min-Jie, and Ravinder Krishna Raina. "On the Composition Structures of Certain Fractional Integral Operators." Symmetry 14, no. 9 (2022): 1845. http://dx.doi.org/10.3390/sym14091845.

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This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the various Erdélyi-type hypergeometric integrals. We also derive a derivative formula for the fractional integral operator and some applications of the operator are considered for a certain Volterra-type integral equation, which provide two generalizations to Khudozhnikov’s integral equation (see below). Some specific relationships, examples, and som

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32

Cao Labora, Daniel, and Rosana Rodríguez-López. "Improvements in a method for solving fractional integral equations with some links with fractional differential equations." Fractional Calculus and Applied Analysis 21, no. 1 (2018): 174–89. http://dx.doi.org/10.1515/fca-2018-0011.

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Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a fu

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Zhang, Kangqun. "Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation." Fractional Calculus and Applied Analysis 23, no. 5 (2020): 1381–400. http://dx.doi.org/10.1515/fca-2020-0068.

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Abstract In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the existence and uniqueness of L p -solution of time-fractional Tricomi-Keldysh type equation by use of Mikhlin multiplier theorem.

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Hammad, Hasanen A., Hassen Aydi, and Maryam G. Alshehri. "Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases." AIMS Mathematics 9, no. 6 (2024): 14574–93. http://dx.doi.org/10.3934/math.2024709.

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<abstract><p>This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying period

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Razminia, Kambiz, and Abolhassan Razminia. "Convolution integral for fractional diffusion equation." Chaos, Solitons & Fractals 155 (February 2022): 111728. http://dx.doi.org/10.1016/j.chaos.2021.111728.

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Elwakil, S. A., and M. A. Zahran. "Fractional Integral Representation of Master Equation." Chaos, Solitons & Fractals 10, no. 9 (1999): 1545–48. http://dx.doi.org/10.1016/s0960-0779(98)00176-3.

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CONA, Lale, and Esmahan BAL. "On Existence and Uniqueness of Some Fractional Order Integro-Differential Equation." Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi 16, no. 2 (2023): 297–310. http://dx.doi.org/10.18185/erzifbed.1220243.

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In this study, a sufficient condition for the existence and uniqueness of some fractional order Integral-Differential equations is obtained. Therefore, the fixed point method is used to solve the differential equation problem involving nonlinear degree integrals. In addition, the results found is supported by examples.

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Durdiev, D. K., T. R. Suyarov, and Kh Kh Turdiev. "An inverse coefficient problem for the fractional telegraph equation with the corresponding fractional derivative in time." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 2 (February 26, 2025): 39–52. https://doi.org/10.26907/0021-3446-2025-2-39-52.

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This work investigates an initial-boundary value and an inverse coefficient problem of determining a time dependent coefficient in the fractional wave equation with the conformable fractional derivative and an integral. In the beginning, the initial boundary value problem (direct problem) is considered. By the Fourier method this problem is reduced to equivalent integral equations. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get apriori estimate for the solution via the unknown coefficient which will be used to study the inverse problem.

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Ahmed, Wagdi F. S., Ahmad Y. A. Salamooni, and Dnyaneshwar D. Pawar. "Solution of fractional kinetic equation for Hadamard type fractional integral via Mellin transform." Gulf Journal of Mathematics 12, no. 1 (2022): 15–27. http://dx.doi.org/10.56947/gjom.v12i1.781.

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The aim of this paper is to introduce the generalized form of the fractional kinetic equation including Hadamard type fractional integral. We also present the solution of Kinetic equation including Hadamard type fractional integral with the help of generalized k-Wright function and Mellin transform. We also present the solution of Kinetic equation including Hadamard type fractional integral with the help of generalized Multindex Bessel Function and Mellin transform.

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Bin Jebreen, Haifa, and Ioannis Dassios. "On the Wavelet Collocation Method for Solving Fractional Fredholm Integro-Differential Equations." Mathematics 10, no. 8 (2022): 1272. http://dx.doi.org/10.3390/math10081272.

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An efficient algorithm is proposed to find an approximate solution via the wavelet collocation method for the fractional Fredholm integro-differential equations (FFIDEs). To do this, we reduce the desired equation to an equivalent linear or nonlinear weakly singular Volterra–Fredholm integral equation. In order to solve this integral equation, after a brief introduction of Müntz–Legendre wavelets, and representing the fractional integral operator as a matrix, we apply the wavelet collocation method to obtain a system of nonlinear or linear algebraic equations. An a posteriori error estimate fo

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More, Pravin M., A. N. Patil, and P. S. Amurtrao. "Locally Attractivity Solutions for Nonlinear Functional Quadratic Integral Equations of Fractional Order." Journal of Advances in Mathematics and Computer Science 40, no. 2 (2025): 82–94. https://doi.org/10.9734/jamcs/2025/v40i21971.

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We prove using hybrid fixed point theorem in Banach algebra an existence of solutions to fractional order quadratic functional integral equation in Also locally attractivity solutions for fractional order of nonlinear functional quadratic functional integral equations is proved by using Dhage’s Hybrid fixed point theorem.

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42

Zhang, Xianmin, Zuohua Liu, Shixian Yang, Zuming Peng, Yali He та Liran Wei. "The Right Equivalent Integral Equation of Impulsive Caputo Fractional-Order System of Order ϵ∈(1,2)". Fractal and Fractional 7, № 1 (2022): 37. http://dx.doi.org/10.3390/fractalfract7010037.

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For the impulsive fractional-order system (IFrOS) of order ϵ∈(1,2), there have appeared some conflicting equivalent integral equations in existing studies. However, we find two fractional-order properties of piecewise function and use them to verify that these given equivalent integral equations have some defects to not be the equivalent integral equation of the IFrOS. For the IFrOS, its limit property shows the linear additivity of the impulsive effects. For the IFrOS, we use the limit analysis and the linear additivity of the impulsive effects to find its correct equivalent integral equation

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43

Çetınkaya, Süleyman, and Ali Demir. "Effects of the ARA transform method for time fractional problems." Mathematica Moravica 26, no. 2 (2022): 73–84. http://dx.doi.org/10.5937/matmor2202073c.

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The aim of this study is to establish the solutions of time fractional mathematical problems with the aid of new integral transforms called the ARA transform. The fractional derivative is taken in the sense of Liouville-Caputo derivative. The fractional partial differential equations are reduced into ordinary differential equations. Later solving this fractional equation and applying inverse the ARA transform, the solution is acquired. The implementation of this transform for fractional differential equations is very similar to the implementation of the Laplace transform. However, the ARA tran

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44

Nikam, Vishal, Dhananjay Gopal, and Rabha W. Ibrahim. "Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem." Foundations 1, no. 2 (2021): 286–303. http://dx.doi.org/10.3390/foundations1020021.

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The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of β−G, ψ−G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions f

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45

Hilmi, Hozan, and Karwan H. F. Jwamer. "Existence and Uniqueness Solution of Fractional Order Regge Problem." JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences 30, no. 2 (2022): 80–96. http://dx.doi.org/10.29196/jubpas.v30i2.4186.

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In this paper, we look into a group of fractional boundary value problem equations involving fractional derivative fractional orders and there are two boundary value criteria in this equation. The existence and uniqueness solutions are obtained using the Banach fixed point theorem (Contraction mapping theorem) and the Schauder fixed point theorem. based on the method of fractional integral and integral operator, our primary findings are illustrated using examples.

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46

Chaoui, A., and N. Rezgui. "Solution to fractional pseudoparabolic equation with fractional integral condition." Rendiconti del Circolo Matematico di Palermo Series 2 67, no. 2 (2017): 205–13. http://dx.doi.org/10.1007/s12215-017-0306-x.

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Alshammari, Saleh, Naveed Iqbal, and Mohammad Yar. "Analytical Investigation of Nonlinear Fractional Harry Dym and Rosenau-Hyman Equation via a Novel Transform." Journal of Function Spaces 2022 (September 9, 2022): 1–12. http://dx.doi.org/10.1155/2022/8736030.

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We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to find an error analysis that demonstrates that our final result converges to the exact and approximate result. The convergent series form solution demonstrates

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Li, Lishuang, Xinguang Zhang, Peng Chen, and Yonghong Wu. "The Existence of Positive Solutions for a p-Laplacian Tempered Fractional Diffusion Equation Using the Riemann–Stieltjes Integral Boundary Condition." Mathematics 13, no. 3 (2025): 541. https://doi.org/10.3390/math13030541.

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In this paper, we focus on the existence of positive solutions for a class of p-Laplacian tempered fractional diffusion equations involving a lower tempered integral operator and a Riemann–Stieltjes integral boundary condition. By introducing certain new local growth conditions and establishing an a priori estimate for the Green’s function, several sufficient conditions on the existence of positive solutions for the equation are derived by using a fixed point theorem. Interesting points are that the tempered fractional diffusion equation contains a lower tempered integral operator and that the

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Luchko, Yuri, and Masahiro Yamamoto. "The General Fractional Derivative and Related Fractional Differential Equations." Mathematics 8, no. 12 (2020): 2115. http://dx.doi.org/10.3390/math8122115.

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In this survey paper, we start with a discussion of the general fractional derivative (GFD) introduced by A. Kochubei in his recent publications. In particular, a connection of this derivative to the corresponding fractional integral and the Sonine relation for their kernels are presented. Then we consider some fractional ordinary differential equations (ODEs) with the GFD including the relaxation equation and the growth equation. The main part of the paper is devoted to the fractional partial differential equations (PDEs) with the GFD. We discuss both the Cauchy problems and the initial-bound

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Sene, Ndolane. "Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution." Journal of Fractional Calculus and Nonlinear Systems 2, no. 1 (2021): 60–75. http://dx.doi.org/10.48185/jfcns.v2i1.214.

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In this paper, we propose the approximate solution of the fractional diffusion equation described by a non-singular fractional derivative. We use the Atangana-Baleanu-Caputo fractional derivative in our studies. The integral balance methods as the heat balance integral method introduced by Goodman and the double integral method developed by Hristov have been used for getting the approximate solution. In this paper, the existence and uniqueness of the solution of the fractional diffusion equation have been provided. We analyze the impact of the fractional operator in the diffusion process. We r

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