Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

Băleanu, Dumitru; Yusuf, Abdullahi; Aliyu, Aliyu Isa

doi:10.1186/s13662-018-1468-3

Cited by 32 publications

(14 citation statements)

References 46 publications

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“…The series solution is commonly used to solve various types of fractional differential equations (see [12][13][14][15][16]). Since x = 0, is αregular singular point of Equation (1.1), see [17], we apply the well-known Frobenius method to obtain a solution of the form…”

Section: The Series Solutionmentioning

confidence: 99%

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

Shat

Al-Refai

Alhamayda

et al. 2019

Front. Appl. Math. Stat.

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In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable derivative of order 0 < α < 1. We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solution truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.

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Section: The Series Solutionmentioning

confidence: 99%

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

Shat

Al-Refai

Alhamayda

et al. 2019

Front. Appl. Math. Stat.

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“…Recently, the RKM has been improved and successfully applied in obtaining approximations of solutions for many initial and boundary problems that appear in natural sciences and engineering. The RKM was successfully used for solving the Thomas-Fermi equation [28], the Poisson-Boltzmann equation for semiconductor devices [29], variable-order fractional differential equations [30] and second-order partial differential equations [31] and others [32][33][34]. Moreover, Cui and Lin [24] have efficiently solved obstacle third-order BVP using RKM.…”

Section: Introductionmentioning

confidence: 99%

Two computational approaches for solving a fractional obstacle system in Hilbert space

et al. 2019

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The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0 < α ≤ 1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

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“…Fractional‐order system is more general than classical integer order system and it can provide an excellent instrument for the description of memory and hereditary properties of some materials and processes, which can more truly reveal the internal structure and dynamical behaviors of an actual system. In recent years, fractional dynamical method has been widely used in theoretical and applied aspects of numerous branches, such as applied mathematics, mechanics and physics problems, anomalous diffusion, optimal control, secure communication and encryption, heat conduction, and biological systems . They demonstrate the importance of fractional calculus and motivate the development of new applications.…”

Section: Introductionmentioning

confidence: 99%

Fractional matrix and inverse matrix projective synchronization methods for synchronizing the disturbed fractional‐order hyperchaotic system

Chen

Lei

2018

Math Methods in App Sciences

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In this paper, for synchronizing two actual nonidentical fractional-order hyperchaotic systems disturbed by model uncertainty and external disturbance, the fractional matrix and inverse matrix projective synchronization methods are presented and the methods' correctness and effectiveness are proved. Especially, under certain degenerative conditions, the methods are reduced to study the complete synchronization, antisynchronization, projective (or inverse projective) synchronization, modified (or modified inverse) projective synchronization, and stabilization problem for the disturbed (or undisturbed) fractional-order hyperchaotic systems. In addition, as the fractional matrix and inverse matrix projective synchronization methods' applications, the fractional-order hyperchaotic Chen and Rabinovich systems disturbed by model uncertainty and external disturbance are constructed, and the matrix and inverse matrix projective synchronizations between the two disturbed systems are achieved, respectively. This work constructs a basic theoretical framework of fractional matrix and inverse matrix projective synchronization methods and provides a general method for synchronizing the actual disturbed fractional-order hyperchaotic systems that are related to science and engineering. KEYWORDS disturbed hyperchaotic system, fractional inverse matrix projective synchronization, fractional matrix projective synchronization, fractional-order derivative Math Meth Appl Sci. 2018;41:6907-6920.wileyonlinelibrary.com/journal/mma

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Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

Cited by 32 publications

References 46 publications

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions

Two computational approaches for solving a fractional obstacle system in Hilbert space

Fractional matrix and inverse matrix projective synchronization methods for synchronizing the disturbed fractional‐order hyperchaotic system

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