11 results
Search Results
2. A novel finite difference based numerical approach for Modified AtanganaBaleanu Caputo derivative.
- Author
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Chawla, Reetika, Deswal, Komal, Kumar, Devendra, and Baleanu, Dumitru
- Subjects
FRACTIONAL calculus ,FINITE differences ,FOURIER analysis ,APPROXIMATION theory ,NUMERICAL analysis - Abstract
In this paper, a new approach is presented to investigate the time-fractional advectiondispersion equation that is extensively used to study transport processes. The present modified fractional derivative operator based on Atangana-Baleanu’s definition of a derivative in the Caputo sense involves singular and non-local kernels. A numerical approximation of this new modified fractional operator is provided and applied to an advection-dispersion equation. Through Fourier analysis, it has been proved that the proposed scheme is unconditionally stable. Numerical examples are solved that validate the theoretical results presented in this paper and ensure the proficiency of the numerical scheme. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses.
- Author
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Manjula, M., Kaliraj, K., Botmart, Thongchai, Nisar, Kottakkaran Sooppy, and Ravichandran, C.
- Subjects
EXISTENCE theorems ,UNIQUENESS (Mathematics) ,APPROXIMATION theory ,FRACTIONAL differential equations ,GALERKIN methods - Abstract
This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative.
- Author
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Zhao, Tinggang, Li, Changpin, and Li, Dongxia
- Subjects
- *
FRACTIONAL differential equations , *COLLOCATION methods , *JACOBI method , *ORTHOGONAL functions , *APPROXIMATION theory , *FRACTIONAL calculus , *BURGERS' equation , *SPECTRAL theory - Abstract
Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Generalized proportional fractional integral Hermite–Hadamard's inequalities.
- Author
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Aljaaidi, Tariq A., Pachpatte, Deepak B., Abdeljawad, Thabet, Abdo, Mohammed S., Almalahi, Mohammed A., and Redhwan, Saleh S.
- Subjects
INTEGRAL inequalities ,FRACTIONAL integrals ,FRACTIONAL calculus ,FRACTIONAL differential equations ,APPROXIMATION theory ,CONTINUOUS functions ,PRODUCTION engineering - Abstract
The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
6. Redefined Quintic B-Spline Collocation Method to Solve the Time-Fractional Whitham-Broer-Kaup Equations.
- Author
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Hadhoud, Adel R. and Rageh, Abdulqawi A. M.
- Subjects
COLLOCATION methods ,FRACTIONAL calculus ,APPROXIMATION theory ,VON Neumann algebras ,FRACTIONAL differential equations - Abstract
This article proposes a collocation approach based on a redefined quintic B-spline basis for solving the time-fractional Whitham-Broer-Kaup equations. The presented method involves discretizing the time-fractional derivatives using an L 1 -approximation scheme and then approximating the spatial derivatives using the redefined quintic B-spline basis. The von Neumann technique has been used to demonstrate that the proposed method is unconditionally stable. The error estimates are discussed and show that the proposed method is third-order convergent. The results demonstrate the potential of the proposed method as a reliable tool for solving fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Fractional-Order Logistic Differential Equation with Mittag–Leffler-Type Kernel.
- Author
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Area, Iván and Nieto, Juan J.
- Subjects
DIFFERENTIAL equations ,KERNEL (Mathematics) ,NUMERICAL analysis ,APPROXIMATION theory ,POLYNOMIALS - Abstract
In this paper, we consider the Prabhakar fractional logistic differential equation. By using appropriate limit relations, we recover some other logistic differential equations, giving representations of each solution in terms of a formal power series. Some numerical approximations are implemented by using truncated series. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
8. An Approximate Solution of the Time-Fractional Two-Mode Coupled Burgers Equation.
- Author
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Shokhanda, Rachana, Goswami, Pranay, He, Ji-Huan, and Althobaiti, Ali
- Subjects
LAPLACE transformation ,APPROXIMATION theory ,LINEAR equations ,CAPUTO fractional derivatives ,FRACTIONAL calculus - Abstract
In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into a series of linear equations, which can be effectively and easily solved by the Laplace transform. The solution process is illustrated step by step, and the results show that the present method is extremely powerful for fractional differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
9. Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm.
- Author
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Al-Sawalha, M. Mossa, Alshehry, Azzh Saad, Nonlaopon, Kamsing, Shah, Rasool, and Ababneh, Osama Y.
- Subjects
APPROXIMATION theory ,ANALYTIC functions ,HOMOTOPY theory ,DECOMPOSITION method ,FRACTIONAL calculus ,CAPUTO fractional derivatives - Abstract
With effective techniques like the homotopy perturbation approach and the Adomian decomposition method via the Yang transform, the time-fractional vibration equation's solution is found for large membranes. In Caputo's sense, the fractional derivative is taken. Numerical experiments with various initial conditions are carried out through a few test examples. The findings are described using various wave velocity values. The outcomes demonstrate the competence and reliability of this analytical framework. Figures are used to discuss the solution of the fractional vibration equation using the suggested strategies for different orders of memory-dependent derivative. The suggested approaches reduce computation size and time even when the accurate solution of a nonlinear differential equation is unknown. It is helpful for both small and large parameters. The results show that the suggested techniques are trustworthy, accurate, appealing and effective strategies. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
10. Rational Approximations of Arbitrary Order: A Survey.
- Author
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Colín-Cervantes, José Daniel, Sánchez-López, Carlos, Ochoa-Montiel, Rocío, Torres-Muñoz, Delia, Hernández-Mejía, Carlos Manuel, Sánchez-Gaspariano, Luis Abraham, and González-Hernández, Hugo Gustavo
- Subjects
- *
APPROXIMATION theory , *INTEGRATORS , *CAPUTO fractional derivatives , *FRACTIONAL calculus , *CALCULUS - Abstract
This paper deals with the study and analysis of several rational approximations to approach the behavior of arbitrary-order differentiators and integrators in the frequency domain. From the Riemann–Liouville, Grünwald–Letnikov and Caputo basic definitions of arbitrary-order calculus until the reviewed approximation methods, each of them is coded in a Maple 18 environment and their behaviors are compared. For each approximation method, an application example is explained in detail. The advantages and disadvantages of each approximation method are discussed. Afterwards, two model order reduction methods are applied to each rational approximation and assist a posteriori during the synthesis process using analog electronic design or reconfigurable hardware. Examples for each reduction method are discussed, showing the drawbacks and benefits. To wrap up, this survey is very useful for beginners to get started quickly and learn arbitrary-order calculus and then to select and tune the best approximation method for a specific application in the frequency domain. Once the approximation method is selected and the rational transfer function is generated, the order can be reduced by applying a model order reduction method, with the target of facilitating the electronic synthesis. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
11. Generalized proportional fractional integral Hermite–Hadamard’s inequalities
- Author
-
Saleh S. Redhwan, Mohammed S. Abdo, Thabet Abdeljawad, Deepak B. Pachpatte, Tariq A. Aljaaidi, and Mohammed A. Almalahi
- Subjects
Approximation theory ,Algebra and Number Theory ,Partial differential equation ,Hermite polynomials ,Continuous function ,Applied Mathematics ,Fractional inequalities ,Hermite–Hadamard inequalities ,Fractional calculus ,ψ-proportional fractional operators ,Hadamard transform ,Ordinary differential equation ,QA1-939 ,Applied mathematics ,Uniqueness ,Analysis ,Mathematics - Abstract
The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.
- Published
- 2021
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