Article
Mathematics, Applied
S. Behera, S. Saha Ray
Summary: This paper introduces a fractional-order operational matrix method based on Euler wavelets for solving linear Volterra-Fredholm integro-differential equations. The method constructs the operational matrix of fractional integration and reduces the equations to algebraic equation systems. The convergence and numerical convergence rate of the method are analyzed, and error analysis is conducted.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics
Maryam Al-Kandari, Latif A-M. Hanna, Yuri Luchko
Summary: In this paper, the authors discuss the general fractional integrals and derivatives with kernels that have an integrable singularity of power function type. They introduce the sequential fractional derivatives and derive an explicit formula for their projector operator. The main contribution of this paper is the construction of an operational calculus for the general fractional derivatives of arbitrary order, providing a representation and operational relations.
Article
Mathematics, Interdisciplinary Applications
Xiaobin Yu, Yajun Yin, Rekha Srivastava
Summary: In this study, we delve into the general theory of operator kernel functions in operational calculus and establish a mapping relation between the kernel function and the corresponding operator. This research demonstrates the uniqueness of the kernel function and provides a novel perspective on how operational calculus can be understood and applied. The accuracy of the proposed method is substantiated through consistency tests and its application is illustrated in different structures. These results highlight the importance of this study for the understanding and application of operational calculus.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Lijiao Wu, Haixiang Zhang, Xuehua Yang
Summary: This paper presents an efficient numerical method for fourth-order partial integro-differential equations with weakly singular kernel. The method is constructed on graded meshes and achieves second-order convergence for weakly singular solutions. Numerical results demonstrate its effectiveness, and further improvement in convergence order is achieved using the extrapolation method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
I. Zamanpour, R. Ezzat
Summary: The main objective of this study is to develop the operational matrix for fractional integration and use it to solve non-linear fractional weakly singular two-dimensional partial Volterra integral equations. By presenting the findings in the form of figures and tables, a deeper investigation and explanation of the proposed approach was provided, showing the high reliability and accuracy of the method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Zheng Ma, Chengming Huang
Summary: In this paper, we propose a method for solving Volterra integro-differential equations with weakly singular kernels. By increasing the degrees of piecewise fractional polynomials, exponential rates of convergence can be achieved for certain solutions. The method is easy to implement and has the same computational complexity as polynomial collocation methods.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Sayed Arsalan Sajjadi, Hashem Saberi Najafi, Hossein Aminikhah
Summary: This paper proposes an algorithm to address the non-smooth behavior of solutions of nonlinear fractional Volterra integro-differential equations with weakly singular kernels. The convergence of the algorithm is investigated and four numerical examples are solved to test its efficiency and accuracy.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
El Mehdi Lotfi, Houssine Zine, Delfim F. M. Torres, Noura Yousfi
Summary: This paper introduces new and more general definitions for fractional operators with non-singular kernels using the Laplace transform method and the convolution theorem. The new operators are based on a generalized form of the Mittag-Leffler function, characterized by a key parameter p. The power parameter p is important for researchers to choose an appropriate notion of the derivative, provide good mathematical models, and predict future dynamic behaviors. The fundamental properties of the new operators are investigated and rigorously proved. As an application, a Caputo and a Riemann-Liouville fractional differential equation are solved.
Article
Mathematics, Interdisciplinary Applications
Mohamed A. Abdelkawy, Ahmed Z. M. Amin, Antonio M. Lopes, Ishak Hashim, Mohammed M. Babatin
Summary: In this paper, a fractional-order shifted Jacobi-Gauss collocation method is proposed to solve variable-order fractional integro-differential equations with weakly singular kernel. By solving systems of algebraic equations, the approximate solutions of the equations are obtained using Riemann-Liouville fractional integral and derivative as well as fractional-order shifted Jacobi polynomials. The method demonstrates superior accuracy through numerical examples.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics
George A. Anastassiou
Summary: This paper extends the previous univariate high order simultaneous fractional monotone approximation theory to abstract scenarios, covering both left and right constrained approximations, and applying to Prabhakar fractional calculus and non-singular kernel fractional calculus.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Review
Mathematics, Applied
M. Taghipour, H. Aminikhah
Summary: This article focuses on finding the numerical solution of a nonlinear time-fractional partial integro-differential equation. By using operational matrices based on Pell polynomials, the fractional Caputo derivative, nonlinear, and integro-differential terms are approximated. The problem is then transformed into a system of nonlinear equations using collocation points. This system can be solved using the fsolve command in Matlab. The stability and convergence of the method have been studied, and five numerical examples are included to demonstrate its accuracy.
JOURNAL OF FUNCTION SPACES
(2022)
Article
Mathematics, Applied
Hafiz Muhammad Fahad, Arran Fernandez
Summary: Mikusiński's operational calculus is a method for interpreting and solving fractional differential equations, recently extended to cover Caputo derivatives with respect to functions. This approach provides a deeper understanding of the structures involved in fractional calculus and utilizes multivariate Mittag-Leffler functions to solve fractional differential equations.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Khadijeh Sadri, Kamyar Hosseini, Dumitru Baleanu, Soheil Salahshour, Choonkil Park
Summary: This work addresses the numerical solution of fractional delay integro-differential equations with weakly singular kernels using a Vieta-Fibonacci collocation method. The existence and uniqueness of the solution is investigated and proved, and a new formula for extracting the Vieta-Fibonacci polynomials and their derivatives is given. The orthogonality of the derivatives of the polynomials is easily proved, and an error bound for the residual function is estimated. The designed algorithm is examined on four equations, showing its simplicity and accuracy compared to previous methods.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Interdisciplinary Applications
George A. Anastassiou
Summary: In this study, the fractional monotone approximation theory is extended to abstract fractional monotone approximation, which is applied to Prabhakar fractional calculus and non-singular kernel fractional calculi. The constrained approximation is covered on both the left and right sides using linear abstract left or right fractional differential operators. Additionally, the target function can be quantitatively approximated with uniform convergence rates using polynomials and the first modulus of continuity.
FRACTAL AND FRACTIONAL
(2021)
Article
Mathematics
Fatima Cruz, Ricardo Almeida, Natalia Martins
Summary: This work focuses on variational problems involving time delays and higher-order distributed-order fractional derivatives with a new fractional operator. Necessary and sufficient optimality conditions for different types of variational problems are established, and some well-known results can be derived as special cases. The study of generalized fractional derivatives in this work provides a foundation for obtaining important results in the field.
