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1 – 10 of 105Samira Ramdane and Assia Guezane-Lakoud
The paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right…
Abstract
Purpose
The paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.
Design/methodology/approach
Fixed-point theorems.
Findings
No finding.
Originality/value
The obtained results are original.
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In this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and…
Abstract
Purpose
In this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.
Design/methodology/approach
The authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.
Findings
The purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.
Originality/value
Many engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.
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Iqbal M. Batiha, Adel Ouannas, Ramzi Albadarneh, Abeer A. Al-Nana and Shaher Momani
This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional…
Abstract
Purpose
This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.
Design/methodology/approach
In order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.
Findings
In accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.
Originality/value
The existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.
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S. Sahoo, S. Saha Ray and S. Das
In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously…
Abstract
Purpose
In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously variable fractional order) systems, have been presented. The authors will demonstrate via two cases where the frictional damping given by fractional derivative, the order of which varies continuously – while the mass moves in a guide. Here, the continuously changing nature of the fractional-order derivative for dynamic systems has been studied for the first time. The solutions of the fractional continuously variable order mass-spring damper systems have been presented here by using a successive recursive method, and the closed form of the solutions has been obtained. By using graphical plots, the nature of the solutions has been discussed for the different cases of continuously variable fractional order of damping force for oscillator. The purpose of the paper is to formulate the continuously variable order mass-spring damper systems and find their analytical solutions by successive recursion method.
Design/methodology/approach
The authors have used the viscoelastic and viscous – viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of the fractional derivative varies continuously.
Findings
By using the successive recursive method, here, the authors find the solution of the fractional continuously variable order mass-spring damper systems, and then obtain close-form solutions. The authors then present and discuss the solutions obtained in the cases with the continuously variable order of damping for an oscillator through graphical plots.
Originality/value
Formulation of fractional continuously variable order dynamic models has been described. Fractional continuous variable order mass-spring damper systems have been analysed. A new approach to find solutions of the aforementioned dynamic models has been established. Viscoelastic and viscous – viscoelastic dampers are described. The discussed damping nature of the oscillating systems has not been studied yet.
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Mostafa Abbaszadeh, Hossein Pourbashash and Mahmood Khaksar-e Oshagh
This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.
Abstract
Purpose
This study aims to propose a new numerical method for solving non-linear partial differential equations on irregular domains.
Design/methodology/approach
The main aim of the current paper is to propose a local meshless collocation method to solve the two-dimensional Klein-Kramers equation with a fractional derivative in the Riemann-Liouville sense, in the time term. This equation describes the sub-diffusion in the presence of an external force field in phase space.
Findings
First, the authors use two finite difference schemes to discrete temporal variables and then the radial basis function-differential quadrature method has been used to estimate the spatial direction. To discrete the time-variable, the authors use two different strategies with convergence orders
Originality/value
The proposed numerical technique is flexible for different computational domains.
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This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which…
Abstract
Purpose
This paper aims to propose a novel approach based on uniform scale-3 Haar wavelets for unsteady state space fractional advection-dispersion partial differential equation which arises in complex network, fluid dynamics in porous media, biology, chemistry and biochemistry, electrode – electrolyte polarization, finance, system control, etc.
Design/methodology/approach
Scale-3 Haar wavelets are used to approximate the space and time variables. Scale-3 Haar wavelets converts the problems into linear system. After that Gauss elimination is used to find the wavelet coefficients.
Findings
A novel algorithm based on Haar wavelet for two-dimensional fractional partial differential equations is established. Error estimation has been derived by use of property of compactly supported orthonormality. The correctness and effectiveness of the theoretical arguments by numerical tests are confirmed.
Originality/value
Scale-3 Haar wavelets are used first time for these types of problems. Second, error analysis in new work in this direction.
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The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a…
Abstract
Purpose
The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided.
Design/methodology/approach
In this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models.
Findings
From the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration.
Originality/value
With the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.
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Sabri T.M. Thabet, Bashir Ahmad and Ravi P. Agarwal
In this paper, we study a Cauchy-type problem for Hilfer fractional integrodifferential equations with boundary conditions. The existence of solutions for the given problem is…
Abstract
In this paper, we study a Cauchy-type problem for Hilfer fractional integrodifferential equations with boundary conditions. The existence of solutions for the given problem is proved by applying measure of noncompactness technique in an abstract weighted space. Moreover, we use generalized Gronwall inequality with singularity to establish continuous dependence and uniqueness of
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Farshid Mirzaee and Sahar Alipour
The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral…
Abstract
Purpose
The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically.
Design/methodology/approach
For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time.
Findings
By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated.
Originality/value
Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.
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Mushtaq Ali, Mohammed Almoaeet and Basim Karim Albuohimad
This study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them…
Abstract
Purpose
This study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.
Design/methodology/approach
In this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.
Findings
Applying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.
Originality/value
This paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.
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