scholarly journals The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Chun-Guang Zhao ◽  
Ai-Min Yang ◽  
Hossein Jafari ◽  
Ahmad Haghbin
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Local Fractional Derivative ◽  
Fractional Differential

The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.

scholarly journals Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Adel Al-Rabtah ◽  
Shaher Momani ◽  
Mohamed A. Ramadan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Spline Functions ◽  
Exact Analytical Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Good Agreement

Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear fractional differential equations. The proposed method is applicable for0<α≤1andα≥1, whereαdenotes the order of the fractional derivative in the Caputo sense. The results obtained are in good agreement with the exact analytical solutions and the numerical results presented elsewhere. Results also show that the technique introduced here is robust and easy to apply.

scholarly journals Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

2020 ◽  
Vol 26 (1) ◽  
pp. 35-55
Author(s):  
Abdelkader Kehaili ◽  
Ali Hakem ◽  
Abdelkader Benali
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

scholarly journals A generalization of truncated M-fractional derivative and applications to fractional differential equations

2020 ◽  
Vol 5 (1) ◽  
pp. 171-188 ◽  
Author(s):  
Esin İlhan ◽  
İ. Onur Kıymaz
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integer Order ◽  
Derivative Type ◽  
Fractional Differential

AbstractIn this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated ℳ-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integer-order derivatives. Finally, we obtain the analytical solutions of some ℳ-series fractional differential equations.

scholarly journals On the Generalized Laplace Transform

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 669
Author(s):  
Paul Bosch ◽  
Héctor José Carmenate García ◽  
José Manuel Rodríguez ◽  
José María Sigarreta
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Harmonic Oscillator ◽  
Fractional Differential Equations ◽  
Fractional Differential ◽  
Definition Of ◽  
Generalized Harmonic Oscillator

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.

On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 37-46 ◽  
Author(s):  
Mohammad Hossein Derakhshan ◽  
Alireza Ansari
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Ulam Stability ◽  
Nonlinear Fractional Differential Equations ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
The Stability

AbstractIn this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

scholarly journals Analytical solutions for conformable fractional Bratu-type equations

2018 ◽  
Vol 7 (1) ◽  
pp. 15 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Conformable Fractional Derivative ◽  
Fractional Differential

Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.

scholarly journals Pseudo-fractional differential equations and generalized g-Laplace transform

2021 ◽  
Vol 12 (3) ◽  
Author(s):  
J. Vanterler da C. Sousa ◽  
Rubens F. Camargo ◽  
E. Capelas de Oliveira ◽  
Gastáo S. F. Frederico
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Fractional Differential

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

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