Displaying similar documents to “On the solution of some simple fractional differential equations.”

On a Differential Equation with Left and Right Fractional Derivatives

Atanackovic, Teodor, Stankovic, Bogoljub (2007)

Fractional Calculus and Applied Analysis

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Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05 We treat the fractional order differential equation that contains the left and right Riemann-Liouville fractional derivatives. Such equations arise as the Euler-Lagrange equation in variational principles with fractional derivatives. We reduce the problem to a Fredholm integral equation and construct a solution in the space of continuous functions. Two competing approaches in formulating differential equations...

Fractional Derivatives in Spaces of Generalized Functions

Stojanović, Mirjana (2011)

Fractional Calculus and Applied Analysis

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MSC 2010: 26A33, 46Fxx, 58C05 Dedicated to 80-th birthday of Prof. Rudolf Gorenflo We generalize the two forms of the fractional derivatives (in Riemann-Liouville and Caputo sense) to spaces of generalized functions using appropriate techniques such as the multiplication of absolutely continuous function by the Heaviside function, and the analytical continuation. As an application, we give the two forms of the fractional derivatives of discontinuous functions in spaces of...

A detailed analysis for the fundamental solution of fractional vibration equation

Li-Li Liu, Jun-Sheng Duan (2015)

Open Mathematics

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In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case...

Bifurcations of the time-fractional generalized coupled Hirota-Satsuma KdV system

Marwan Alquran, Kamel Al-Khaled, Mohammed Ali, Omar Abu Arqub (2017)

Waves, Wavelets and Fractals

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The Hirota-Satsuma model with fractional derivative is considered to provide some characteristics of memory embedded into the system. The modified system is analyzed analytically using a new technique called residual power series method. We observe thatwhen the value of memory index (time-fractional order) is close to zero, the solutions bifurcate and produce a wave-like pattern.

A Fractional LC − RC Circuit

Ayoub, N., Alzoubi, F., Khateeb, H., Al-Qadi, M., Hasan (Qaseer), M., Albiss, B., Rousan, A. (2006)

Fractional Calculus and Applied Analysis

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Mathematics Subject Classification: 26A33, 30B10, 33B15, 44A10, 47N70, 94C05 We suggest a fractional differential equation that combines the simple harmonic oscillations of an LC circuit with the discharging of an RC circuit. A series solution is obtained for the suggested fractional differential equation. When the fractional order α = 0, we get the solution for the RC circuit, and when α = 1, we get the solution for the LC circuit. For arbitrary α we get a general solution...