scholarly journals Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 958 ◽  
Author(s):  
Sinan Deniz ◽  
Ali Konuralp ◽  
Mnauel De la Sen
Keyword(s):  
Laplace Transform ◽  
Analytical Solutions ◽  
Differential Form ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Optimal Perturbation ◽  
Fractional Differential ◽  
Fractional Type ◽  
Iteration Technique ◽  
Fractional Differential Form

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.

scholarly journals New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

2020 ◽  
Vol 66 (3 May-Jun) ◽  
pp. 297
Author(s):  
Mehmet Senol
Keyword(s):  
Wave Equation ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Algebraic Method ◽  
Symbolic Calculation ◽  
Solution Sets ◽  
Flow Models ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

scholarly journals Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

2019 ◽  
Vol 13 (1) ◽  
pp. 81-93
Author(s):  
Rawid Banchuin ◽  
Roungsan Chaisrichaoren
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Current Mode ◽  
Equation Approach ◽  
Time Dimension ◽  
Current Mode Circuits ◽  
Time Component ◽  
Fractional Differential

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account.  The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

scholarly journals Two Reliable Methods for The Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation

2019 ◽  
Vol 4 (2) ◽  
pp. 523-534 ◽  
Author(s):  
Ali Kurt ◽  
Mehmet Şenol ◽  
Orkun Tasbozan ◽  
Mehar Chand
Keyword(s):  
Water Waves ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Approximate Solutions ◽  
Iteration Algorithm ◽  
Shallow Water Waves ◽  
Fractional Differential ◽  
Reliable Methods ◽  
Coupled Burgers Equation ◽  
Partial Fractional Differential Equations

AbstractIn this article, we attain new analytical solution sets for nonlinear time-fractional coupled Burgers’ equations which arise in polydispersive sedimentation in shallow water waves using exp-function method. Then we apply a semi-analytical method namely perturbation-iteration algorithm (PIA) to obtain some approximate solutions. These results are compared with obtained exact solutions by tables and surface plots. The fractional derivatives are evaluated in the conformable sense. The findings reveal that both methods are very effective and dependable for solving partial fractional differential equations.

scholarly journals On matrix fractional differential equations

2017 ◽  
Vol 9 (1) ◽  
pp. 168781401668335
Author(s):  
Adem Kılıçman ◽  
Wasan Ajeel Ahmood
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Convolution Product ◽  
Operational Matrices ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Fractional Differential

The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.

About Maxwell’s equations on fractal subsets of ℝ3

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Alireza Golmankhaneh ◽  
Ali Golmankhaneh ◽  
Dumitru Baleanu
Keyword(s):  
Differential Form ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Measure Theory ◽  
Computer Programs ◽  
Physical Equation ◽  
Fractional Differential ◽  
Local Derivative ◽  
Fractional Differential Form

AbstractIn this paper we have generalized $$F^{\bar \xi }$$-calculus for fractals embedding in ℝ3. $$F^{\bar \xi }$$-calculus is a fractional local derivative on fractals. It is an algorithm which may be used for computer programs and is more applicable than using measure theory. In this Calculus staircase functions for fractals has important role. $$F^{\bar \xi }$$-fractional differential form is introduced such that it can help us to derive the physical equation. Furthermore, using the $$F^{\bar \xi }$$-fractional differential form of Maxwell’s equations on fractals has been suggested.

scholarly journals Analytical Solution of General Bagley-Torvik Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
William Labecca ◽  
Osvaldo Guimarães ◽  
José Roberto C. Piqueira
Keyword(s):  
Numerical Methods ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Analytical Solutions ◽  
Empirical Data ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Polynomial Form ◽  
Inhomogeneous Term

Bagley-Torvik equation appears in viscoelasticity problems where fractional derivatives seem to play an important role concerning empirical data. There are several works treating this equation by using numerical methods and analytic formulations. However, the analytical solutions presented in the literature consider particular cases of boundary and initial conditions, with inhomogeneous term often expressed in polynomial form. Here, by using Laplace transform methodology, the general inhomogeneous case is solved without restrictions in boundary and initial conditions. The generalized Mittag-Leffler functions with three parameters are used and the solutions presented are expressed in terms of Wiman’s functions and their derivatives.

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 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.

Analytical solutions of some integral fractional differential–difference equations

2019 ◽  
Vol 34 (01) ◽  
pp. 2050009 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng
Keyword(s):  
Invariant Subspace ◽  
Difference Equations ◽  
Analytical Solutions ◽  
Volterra Equation ◽  
Burgers Equation ◽  
Space Time ◽  
Subspace Method ◽  
Lattice Equation ◽  
Fractional Differential

The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as two couple space-time FDDEs. Furthermore, some properties of the analytical solutions are illustrated by graphs.

scholarly journals Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Javed Iqbal ◽  
Khurram Shabbir ◽  
Liliana Guran
Keyword(s):  
Laplace Transform ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Fractional Partial Differential Equations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Variational Iteration ◽  
The Laplace Transform ◽  
Fractional Models ◽  
Iteration Technique

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

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